Direct Detection of Electroweak-Interacting Dark Matter
Junji Hisano, Koji Ishiwata, Natsumi Nagata, Tomohiro Takesako
aa r X i v : . [ h e p - ph ] J u l IPMU 11-0046ICRR-Report-583-2010-16CALT 68-2824
Direct Detection of Electroweak-Interacting Dark Matter
Junji Hisano a,b , Koji Ishiwata c , Natsumi Nagata a,d , and Tomohiro Takesako ea Department of Physics, Nagoya University, Nagoya 464-8602, Japan b Institute for the Physics and Mathematics of the Universe, University of Tokyo,Kashiwa 277-8568, Japan c California Institute of Technology, Pasadena, CA 91125, USA d Department of Physics, University of Tokyo, Tokyo 113-0033, Japan e Institute for Cosmic Ray Research, University of Tokyo, Kashiwa 277-8582, Japan
Abstract
Assuming that the lightest neutral component in an SU (2) L gauge multiplet isthe main ingredient of dark matter in the universe, we calculate the elastic scatteringcross section of the dark matter with nucleon, which is an important quantity for thedirect detection experiments. When the dark matter is a real scalar or a Majoranafermion which has only electroweak gauge interactions, the scattering with quarksand gluon are induced through one- and two-loop quantum processes, respectively,and both of them give rise to comparable contributions to the elastic scattering crosssection. We evaluate all of the contributions at the leading order and find that thereis an accidental cancellation among them. As a result, the spin-independent crosssection is found to be O (10 − (46 − ) cm , which is far below the current experimentalbounds. Introduction
The existence of dark matter (DM) is one of the mysteries of the universe. Its energydensity in the universe, which is about six times larger than that of baryon [1], cannotbe explained in the standard model (SM) in particle physics. Weakly interacting massiveparticles (WIMPs) beyond the standard model are regarded as promising candidates forthe DM. If they have TeV-scale masses, their relic abundance in the thermal history ofuniverse may naturally account for the observed value. For the past years, a lot of effortshave been dedicated to the direct detection of WIMP DM, and their sensitivities havebeen improving. The XENON100 experiment [2], for example, has already started andannounced its first result, which gives a stringent constraint on the spin-independent (SI)WIMP-nucleon elastic scattering cross section σ SI N ( σ SI N < . × − cm for WIMPs witha mass 50 GeV [3]). Furthermore, ton-scale detectors for the direct detection experimentsare now planned and expected to have significantly improved sensitivities.Introduction of new particles with masses TeV scale is one of the simplest extensionsof the SM in order to explain the DM in the universe if they are an electrically neutralcomponent in an SU (2) L gauge multiplet ( n -tuplet ) with the hypercharge Y . We callthem electroweak-interacting massive particles (EW-IMPs) in this article. EW-IMPs areassumed to interact with quarks and leptons only via weak gauge interactions. NeutralWino and Higgsino in the minimal supersymmetric standard model (MSSM) are examplesof such particles when the mixing with other states is negligible and squarks and slep-tons are heavy enough. There is an earlier work [4] which studied the direct detection ofHiggsino-like DM. They gave the cross section of Higgsino-like DM with nucleon by point-ing out large loop contribution to Yukawa couplings. The cross sections of Wino/HiggsinoDM with nucleon were calculated in Ref. [5]. In their study, they took into account elec-troweak loop corrections and showed that the SI interaction does not vanish in the largeDM mass limit, which is a distinctive feature of EW-IMP DM. There are several other ar-ticles which give theoretical prediction for the cross section of EW-IMP DM. The authorsin Ref. [6] intensively studied EW-IMP DM (they refer to it as Minimal dark matter),and a similar calculation was also performed in Ref. [7]. There is, however, inconsistencyamong Refs. [5, 6, 7]. In addition, interaction of gluon with EW-IMP, which is also oneof the leading contributions in the EW-IMP DM-nucleon scattering, was neglected in thereferences. This point was first pointed out in Ref. [8].In this paper, we give accurate prediction for EW-IMP-nucleon scattering in the directdetection experiments. Assuming that EW-IMPs are the main component of the DM inthe universe, we calculate the cross section at the leading order of SM gauge couplings,including the gluon contribution. We provide the complete formulae for the EW-IMPDM-nucleon scattering cross section in a general form, and show that the SI cross sectionis below the current experimental bounds in all the cases we have studied. We also givenumerical results for spin-dependent (SD) cross section.This paper is organized as follows: in Section 2 we explain the EW-IMP DM scenario.In Section 3 general formulae of the elastic scattering cross section of dark matter withnucleon are summarized. In Section 4 we derive the cross section of the EW-IMP with1ucleon and show the numerical results. Section 5 is devoted to conclusion. As we described in the Introduction, EW-IMPs are an electrically neutral component in n -tuplet of SU (2) L with the hypercharge Y . In this section, we explain characteristics ofEW-IMP DM assuming the EW-IMPs are fermionic. Fermionic EW-IMPs are popular, e.g. , Wino or Higgsino in the MSSM. We will discuss scalar EW-IMPs in the end of thissection.When the fermionic n -tuplet has only SU (2) L × U (1) Y gauge interactions in the SMand all the components have a common mass at tree level, the charged components becomeheavier than the neutral one because of quantum loop corrections [6]. The typical valueof the mass difference ∆ M is O (100) MeV. Thus, the neutral component of the multipletcould be a DM candidate in the universe if it is stable. It is found that fermionic EW-IMPs with n ≥ R parity in the MSSM, since the gauge and Lorentz invariance prevent theDM candidate from decaying via the renormalizable interactions, and they turn out tobe stable even if non-renormalizable effective operators are considered under the cut-offscale as large as the GUT-scale or Planck-scale [6, 9] .The thermal relic abundance could also explain the observed DM energy density whenthe EW-IMP mass is over 1 TeV for n ≥
2. In Ref. [10], the thermal relic abundanceof Wino in the MSSM (which is EW-IMP with n = 3 and Y = 0) is evaluated. Thethermal relic abundances for EW-IMPs with n = 2 and 5 ( Y = 0) are also evaluated [9].According to those studies, the EW-IMP mass 1 , . , and 10 TeV is suggested in orderto explain the observed DM abundance when n = 2 , , and 5, respectively, and Y = 0.Now we explicitly show the Lagrangian of EW-IMP DM scenario for our calculation.In the case of the multiplet with the hypercharge Y = 0, the following Lagrangian isintroduced to the SM : ∆ L = 12 ˜ χ ( i / D − M ) ˜ χ , (1)where ˜ χ is an SU (2) L n -tuplet fermion, and D µ = ∂ µ − ig Y B µ − ig W aµ T a ( a = 1 , , g and g is U (1) Y and SU (2) L gauge couplingconstants, respectively, and T a is for the generators of SU (2) L gauge group. The tree-level mass of the multiplet is denoted as M . The neutral component of the n -tuplet is aMajorana fermion, while the other charged components are Dirac fermions. The neutralcomponent has no interaction with gauge bosons by itself so that the elastic scattering ofEW-IMP with nucleon is induced via loop diagrams. In this article we implicitly suppose a symmetry to prevent EW-IMPs from decaying for 2 ≤ n < In the evaluation of thermal relic abundance of EW-IMPs with n >
2, the Sommerfeld effect in theannihilation cross section of EW-IMP [11, 12] should be included [10].
2n the other hand, for Y = 0 n -tuplet, the following term is added to the SM La-grangian, ∆ L = ˜ ψ ( i / D − M ) ˜ ψ . (2)Here, ˜ ψ is an SU (2) L n -tuplet Dirac fermion so that the fermion have an SU (2) L × U (1) Y invariant mass term. Note that Dirac fermions as the DM in the universe havebeen severely constrained by the direct detection experiments because of its large elasticscattering cross section via coherent vector interaction with target nuclei.The situation changes if introducing the extra effective operators which give rise toa mass splitting δm between the left- and right-handed components of the neutral Diracfermion after the electroweak symmetry breaking. In this case, the neutral component isno longer mass eigenstate; the mass splitting decomposes the neutral Dirac fermion, ˜ ψ ,into two Majorana fermions, ˜ χ and ˜ η :˜ ψ = 1 √ χ + i ˜ η ) . (3)Hereafter we treat ˜ χ as the lighter one, without loss of generality. Now the lightercomponent could be the DM candidate. It does not have any vector interaction by itselfso that the DM-nucleon elastic scattering by Z -boson exchange is forbidden. In addition, ifthe mass splitting is large enough ( δm ≫ O (10) keV), the DM-nucleon inelastic scatteringis suppressed kinematically. Thus, the EW-IMPs avoid the constraints from the directdetection experiments. On the other hand, large δm may also induce sizable contributionof the Higgs-boson exchange to the DM-nucleon scattering at tree level. To avoid all theseconstraints, in this article, we simply assume O (10) keV ≪ δm ≪ ∆ M so that the elasticscattering of EW-IMP with nucleon is dominated by loop diagrams.In the following discussion, we consider the scenarios with Majorana fermion EW-IMPs, with either Y = 0 or Y = 0. We simply express it as ˜ χ and the charged Diraccomponents which couple with ˜ χ as ˜ ψ ± .It is straightforward to extend our calculation to scalar EW-IMPs. Contrary to thefermionic EW-IMPs, the scalar EW-IMPs have renormalizable self-interactions with Higgsboson, which contribute to the SI cross section of scalar EW-IMP with nucleon. Whenthe interaction is suppressed enough to be ignored, elastic scattering of the EW-IMP withnucleon is induced by loop diagrams. In that case, the SI cross section of scalar EW-IMPwith nucleon should agree with that of the fermionic EW-IMP when the EW-IMP mass ismuch larger than weak gauge boson masses. This is because the Lagrangians for both offermionic and scalar EW-IMPs are the same in the non-relativistic limit or at the leadingof the velocity expansion. Needless to say, the SD cross section of scalar EW-IMP withnucleon vanishes since the scalar EW-IMP has spin zero. In this section, we briefly review evaluation of the elastic scattering cross section of DMwith nucleon, which is an important quantity for the direct detection experiments, as-3uming the DM is a Majorana fermion. The formulation given here is originally derivedin Ref. [13]. See also Refs. [14, 15] for further details.First, we write down the effective interactions of DM with light quarks ( q = u, d, s )and gluon : L eff = X q = u,d,s L eff q + L eff g , (4) L eff q = d q ˜ χ γ µ γ ˜ χ ¯ qγ µ γ q + f q m q ˜ χ ˜ χ ¯ qq + g (1) q M ˜ χ i∂ µ γ ν ˜ χ O qµν + g (2) q M ˜ χ ( i∂ µ )( i∂ ν ) ˜ χ O qµν , (5) L eff g = f G ˜ χ ˜ χ G aµν G aµν . (6)Here, M and m q are the masses of the DM and quarks, respectively. The field strengthtensor of the gluon field is denoted by G aµν . The last two terms in Eq. (5) include thequark twist-2 operator, O qµν , which is defined as, O qµν ≡
12 ¯ qi (cid:18) D µ γ ν + D ν γ µ − g µν / D (cid:19) q . (7)The coefficients of the operators in Eqs. (5) and (6) are to be determined in the followingsection.The cross section of DM with nucleon ( N = p, n ) is calculated from the effectiveLagrangian in terms of SI and SD effective couplings, f N and a N , σ N = 4 π m R (cid:2) | f N | + 3 | a N | (cid:3) , (8)where m R ≡ M m N / ( M + m N ) ( m N is nucleon mass). The SI and SD effective couplingsare given as scattering matrix element of the effective Lagrangian between initial and finalstates. The results are f N /m N = X q = u,d,s f q f T q + X q = u,d,s,c,b
34 ( q (2) + ¯ q (2)) (cid:0) g (1) q + g (2) q (cid:1) − π α s f T G f G , (9) a N = X q = u,d,s d q ∆ q N , (10)where h N | m q ¯ qq | N i /m N = f T q , f
T G = 1 − X u,d,s f T q , (11) h N ( p ) |O qµν | N ( p ) i = 1 m N ( p µ p ν − m N g µν ) ( q (2) + ¯ q (2)) , (12) h N | ¯ qγ µ γ q | N i = 2 s µ ∆ q N . (13) We only keep the operators which give rise to the leading contributions for the DM-nucleon scatteringwith the non-relativistic velocity. Moreover, in order to remove the redundant terms, we use the integra-tion by parts and the classical equation of motion for the operators when we construct the low-energyeffective Lagrangian [16]. f T u f T d f T s f T u f T d f T s µ = m Z (for proton) u (2) 0.22 ¯ u (2) 0.034 d (2) 0.11 ¯ d (2) 0.036 s (2) 0.026 ¯ s (2) 0.026 c (2) 0.019 ¯ c (2) 0.019 b (2) 0.012 ¯ b (2) 0.012 Spin fraction(for proton)∆ u p d p -0.49∆ s p -0.15Table 1: Parameters for quark and gluon matrix elements used in this paper.Here α s ≡ g s / π ( g s is the coupling constant of SU (3) C ), q (2) and ¯ q (2) are the secondmoments of the parton distribution functions (PDFs), and s µ is the spin of the nucleon.Note the factor 1 /α s in front of f G in Eq. (9). It makes the gluon contribution comparableto the light-quark contribution, despite the interactions of DM with gluon are induced byhigher-loop processes than those with light quarks [8]. In the present case, as discussedin the previous section, the DM-quark tree-level scattering is highly suppressed, and thusthe one-loop processes become dominant for the DM-light quark effective interactions.Therefore, in order to give accurate prediction of the scattering cross section, we shouldevaluate not only the one-loop diagrams with light quarks but also the two-loop diagramswith gluon.In Table 1 we list the numerical values for the parameters that we used in this article.The mass fractions of light quarks, f T q ( q = u, d, s ), are calculated using the results inRefs. [17, 18, 19]. The procedure for evaluating them is described in these references andRef. [14]. The second moments of PDFs of quarks and anti-quarks are calculated at thescale µ = m Z ( m Z is Z -boson mass) using the CTEQ parton distribution [20] . Finally,the spin fractions, ∆ q N , in Eq. (13) are obtained from Ref. [22]. The second moments andthe spin fractions for neutron are to be obtained by exchanging the values of up quarkfor those of down quark. Now we evaluate the coefficients of the effective interactions displayed in Eqs. (5) and (6).From Eqs. (2) and (3), it is found that the EW-IMP, ˜ χ , interacts with the weak gauge As will be described later, the terms with quark twist-2 operators in Eq. (5) are induced by theone-loop diagrams in which the loop momentum around the weak-boson mass scale yields dominantcontribution. This fact verifies the use of the second moments of PDFs at the m Z scale. See alsodiscussion relevant to it in Ref. [21]. χ ˜ χ h q q ˜ χ ˜ χ q ˜ ψ ± (˜ η ) W ± W ± q ′ ( q ) qW ± ( Z ) ˜ ψ ± (˜ η )( a ) ( b )( Z ) ( Z ) Figure 1: One-loop diagrams which induce effective interactions of EW-IMP DM withlight quarks. There are also W -( Z -) boson crossing diagrams, which are not shown here. ˜ χ ˜ χ ˜ ψ ± W ± Z , γq q ˜ χ ˜ χ W ± ˜ ψ ± ˜ ψ ± Z , γq q Figure 2: One-loop diagrams which correspond to the one-loop quantum correction to theEW-IMP- Z ( γ ) interaction vertex. These contributions turn out to vanish.bosons ( W ± µ , Z µ ) as∆ L int . = h g p n − (2 Y + 1) ˜ χ γ µ ˜ ψ − W + µ + g p n − (2 Y − ˜ χ γ µ ˜ ψ + W − µ + h . c . i + ig ( − Y )cos θ W ˜ χ γ µ ˜ η Z µ , (14)where θ W is the weak mixing angle. The Majorana field ˜ η is introduced for the cases of Y = 0. (See Eq. (3).) In either case ( Y = 0 or Y = 0), the EW-IMP does not have anyinteraction by itself. Thus, it is loop diagrams that yield the leading contribution to theEW-IMP-nucleon elastic scattering cross section.First, we consider the one-loop processes. The relevant diagrams are shown in Figs. 1and 2. The diagrams in Fig. 1 give rise to the coefficients in Eq. (5) as6 q = α m h (cid:20) n − (4 Y + 1)8 m W g H ( w ) + Y m Z cos θ W g H ( z ) (cid:21) + (cid:0) ( a Vq ) − ( a Aq ) (cid:1) Y cos θ W α m Z g S ( z ) , (15) d q = n − (4 Y + 1)8 α m W g AV ( w ) + 2 (cid:0) ( a Vq ) + ( a Aq ) (cid:1) Y cos θ W α m Z g AV ( z ) , (16) g (1) q = n − (4 Y + 1)8 α m W g T1 ( w ) + 2 (cid:0) ( a Vq ) + ( a Aq ) (cid:1) Y cos θ W α m Z g T1 ( z ) , (17) g (2) q = n − (4 Y + 1)8 α m W g T2 ( w ) + 2 (cid:0) ( a Vq ) + ( a Aq ) (cid:1) Y cos θ W α m Z g T2 ( z ) . (18)Here, m h and m W are the masses of Higgs and W bosons, respectively, and α = g / π .We also define the vector and axial-vector couplings of quarks with Z boson as a Vq = 12 T q − Q q sin θ W , a Aq = − T q , (19)where T q and Q q denote the weak isospin and the charge of quark q , respectively. Fur-thermore, we parametrize w ≡ m W /M and z ≡ m Z /M in the above expressions. Thefirst term in Eq. (15) is induced by the Higgs-boson exchange process, shown in diagram(a) of Fig. 1. The other terms in Eqs. (15-18) are all obtained from diagram (b). Themass functions, g H ( x ) , g S ( x ) , g AV ( x ) , g T ( x ), and g T ( x ), in Eqs. (15-18) are given in Ap-pendix A. We ignored the mass differences between ˜ χ and ˜ ψ ± and also between ˜ χ and˜ η here. The loop integrals for the diagrams are finite, and they are dominated by theloop momentum around the weak-boson mass scale. We note here that some of these massfunctions do not vanish in the limit of w, z → f G . These diagrams are presented in Fig. 3. For each diagram, thegluon contributions are classified into two types in terms of the momentum scale whichdominates in the loop integration. So-called “short-distance” contribution means that themomentum is typically masses of heavy particles, such as DM particle or the weak gaugebosons in the diagrams, and “long-distance” one means that the momentum is typicallymass of quark in the loop diagrams. Among the latter one, the contributions in which thelight quarks run are already incorporated in the mass fractions f T q defined in Eq. (11).Therefore, we do not need to add them in the calculation of the gluon contribution;otherwise we would doubly count them [14]. Consequently, the gluon contribution fromeach diagram is given by f ( i ) G = X q = u,d,s,c,b,t f ( i ) G | SD q + X Q = c,b,t c Q f ( i ) G | LD Q . (20)7 χ ˜ χ g Q/qQ ′ /q ′ ˜ ψ ± (˜ η ) W ± W ± ˜ χ ˜ χ ˜ ψ ± W ± Q ′ ( Q ) Qg gg ˜ χ ˜ ψ ± (˜ η ) ˜ χ h W ± ( Z ) Q gg ( a ) ( b ) ( c )( Z ) ( Z ) ( Z ) W ± ( Z )( Q/q ) Figure 3: Relevant two-loop diagrams which contribute to effective scalar coupling ofEW-IMP DM with gluon.There are also W -( Z -) boson crossing diagrams, which are notshown here.Here f ( i ) G | SD q and f ( i ) G | LD q denote the short-distance and long-distance contributions ofquark q in the loop in diagram ( i ) ( i = a, b, c ) of Fig. 3, respectively. We also takeinto account large QCD corrections in the long-distance contributions [23] by using c Q =1+11 α s ( m Q ) / π ( Q = c, b, t ). We take c c = 1 . , c b = 1 .
19, and c t = 1 for α s ( m Z ) = 0 . f G G aµν G aµν is evaluated from scalar-type effec-tive operator f q m q ¯ qq [13, 14]. (See also later discussion where the explicit calculationsare given.) Consequently, the gauge invariance of the short-distance contribution is guar-anteed since summation of the both contribution is obviously gauge invariant. Then, theeffective coupling of EW-IMP with gluon is obtained as f G = f ( a ) G + f ( b ) G + f ( c ) G . (21)Let us see each diagram closely. It is obvious that diagram (a) gives the long-distancecontribution. Thus, we sum up for heavy quarks in the loop, and get f ( a ) G = − α s π × α m h X Q = c,b,t c Q (cid:20) n − (4 Y + 1)8 m W g H ( w ) + Y m Z cos θ W g H ( z ) (cid:21) . (22)Here the first and second terms in the bracket come from W - and Z -boson exchanges,respectively. As we described above, this long-distance contribution is given by effectivescalar-type coupling (of the Higgs contribution) as − α s π f Q .For the calculation of diagrams (b) and (c), on the other hand, we follow the stepssupplied in Ref. [14]. In the work, the systematic calculation for the W -boson exchangediagrams at two-loop level in the Wino DM scenario is given. The procedure is applicableto compute the two-loop diagrams in our case. For the W -boson exchange diagrams, the8nalytic result is simply given by multiplying a factor ( n − (4 Y + 1)) / Z -boson exchangecontribution, the calculation is straightforward extension. In the following we outline thecalculation and give the result of the Z -boson exchange contribution.First, we calculate the vacuum polarization tensor of Z boson in the gluon backgroundfield, Π Zµν ( p ). It is given as i Π Zµν ( p )= − X [ q ] Z d l (2 π ) g cos θ W Tr L+C { γ µ ( a Vq + a Aq γ ) S q ( l ) γ ν ( a Vq + a Aq γ ) ˜ S q ( l − p ) } , (23)where Tr L+C denotes the trace over the Lorentz and color indices, and [ q ] means thesum is taken over all quarks for the short-distance contribution and heavy quarks for thelong-distance contribution. The quark propagators S q ( p ) and ˜ S q ( p ) are under the gluonbackground field with the Fock-Schwinger gauge. (See Appendix A in Ref. [14].) Wedecompose the polarization function asΠ Zµν ( p ) = (cid:18) − g µν + p µ p ν p (cid:19) Π ZT ( p ) + p µ p ν p Π ZL ( p ) . (24)By the explicit calculation, we found that the longitudinal part Π L ( p ) does not contributeto f G . Thus, we only calculate transverse component, Π ZT ( p ). The contributions fromdiagrams (b) and (c) to Π ZT ( p ) are given as (cid:2) Π ZT ( p ) (cid:3) q ( b ) | GG = α α s θ W ( G aµν ) (cid:2) ( a Vq ) + ( a Aq ) (cid:3) h p ( B (2 , + B (2 , ) + 6 B (2 , i , (cid:2) Π ZT ( p ) (cid:3) q ( c ) | GG = 2 α α s cos θ W ( G aµν ) m q (cid:2) { ( a Vq ) + ( a Aq ) } ( − p B (4 , + 2 B (4 , ) − a Aq ) ( p B (4 , + 4 B (4 , ) (cid:3) , (25)where (cid:2) Π ZT ( p ) (cid:3) q ( i ) denotes the contribution from diagram ( i ) ( i = b, c ) with quark q running in the loop. Here, “ | GG ” represents that we keep only the terms which areproportional to G aµν G aµν in the polarization function, since the terms proportional to G aµν G aµν contribute to f G . The loop functions in Eq. (25) are defined as follows: p µ B ( n,m )1 ≡ Z d kiπ k µ [ k − m q ] n [( k + p ) − m q ] m , (26) p µ p ν B ( n,m )21 + g µν B ( n,m )22 ≡ Z d kiπ k µ k ν [ k − m q ] n [( k + p ) − m q ] m . (27) This is consistent with the fact that EW-IMPs have no tree-level coupling with Higgs boson andGoldstone bosons of the electroweak symmetry breaking, which turn into the longitudinal modes of Z and W bosons.
9s is the same in the calculation of two-loop W -boson exchange diagrams discussed inRef. [14], the diagrams (b) and (c) yield the short-distance contribution and long-distancecontribution, respectively. We have checked this identification by explicit calculation.This is also confirmed from the fact that the long distance-contribution corresponds to theone which is calculated from quark triangle diagram using effective scalar-type coupling f q in the M and m W/Z → ∞ limit [14], as we mentioned before. Therefore, Π ZT ( p ) | GG isobtained as Π ZT ( p ) | GG = X q = u,d,s,c,b,t (cid:2) Π ZT ( p ) (cid:3) q ( b ) | GG + X Q = c,b,t c Q (cid:2) Π ZT ( p ) (cid:3) Q ( c ) | GG . (28)The results of (cid:2) Π ZT ( p ) (cid:3) q ( b ) | GG and (cid:2) Π ZT ( p ) (cid:3) Q ( c ) | GG are given in Appendix B.With these Z -boson polarization functions, we get the contribution from Z -bosonexchange diagrams at two-loop level. Combining it with the contribution from W -bosonexchange diagrams, we eventually obtain the effective coupling f G from diagrams (b) and(c) as f ( b ) G + f ( c ) G = α s α π (cid:20) n − (4 Y + 1)8 m W g W ( w, y ) + Y m Z cos θ W g Z ( z, y ) (cid:21) , (29)where y ≡ m t /M ( m t is the top quark mass), and the mass functions, g W ( z, y ) and g Z ( z, y ), are given in Appendix B. We ignore the mass of quarks except that of top quarkin this computation. With the effective couplings derived above, we evaluate the cross section of EW-IMPwith nucleon. First, we discuss the SI cross section. In order to look over the behaviorof the cross section, we plot the SI cross section as a function of the EW-IMP mass inFig. 4. In this figure, we set n = 5 and Y = 2 as an example of the case where both W - and Z -boson exchange diagrams contribute, and take Higgs-boson mass as m h =130,115, 300 and 500 GeV from bottom to top. It is found that the SI cross section has littledependence on the EW-IMP mass when the mass is larger than O (1) TeV. We havechecked such a behavior for other n and Y cases. The cross section ranges from 10 − cm to 10 − cm in this figure when Higgs-boson mass increased from m h = 115 GeV to500 GeV. (Although it is not monotonic.)In Fig. 5, we plot the EW-IMP-proton SI cross section as a function of Higgs-bosonmass for n = 5 (left panel) and both n = 2 and n = 3 (right panel). The EW-IMP massis taken to be equal to 1, 2.7, and 10 TeV for n = 2, 3, and 5, respectively [9, 10]. It isseen that the cross section is enhanced as n is larger and Y is smaller. This behavior will Recall that the EW-IMP mass M = 1 , . , and 10 TeV is suggested in order to explain the observedDM abundance when n = 2 , , and 5, respectively, and Y = 0. The thermal relic abundance of theEW-IMP has not been evaluated when Y = 0. We use those suggested values in Y = 0 case since the SIcross section is insensitive to the EW-IMP mass as mentioned in the text. -48 -47 -46 -45
1 10 S I c r o ss s e c t i on [ c m ] DM mass [TeV]115GeV n=5, Y=210 -48 -47 -46 -45
1 10 S I c r o ss s e c t i on [ c m ] DM mass [TeV]115GeV130GeV10 -48 -47 -46 -45
1 10 S I c r o ss s e c t i on [ c m ] DM mass [TeV]115GeV130GeV300GeV10 -48 -47 -46 -45
1 10 S I c r o ss s e c t i on [ c m ] DM mass [TeV]115GeV130GeV300GeVm h = 500GeV Figure 4: DM-proton SI cross section for n = 5 and Y = 2. We take Higgs-boson massas m h =130, 115, 300 and 500 GeV from bottom to top in this figure.be explained later. We also found that the SI cross section of EW-IMP with nucleon isfar below the current experimental bound. This is the consequence of the calculation inwhich all the relevant terms at leading order are taken into account. The suppression ofthe cross section originates in an accidental cancellation within the SI effective coupling, f N . Such an cancellation was already pointed out for the Wino dark matter, i.e. n = 3with Y = 0 [8]. In our calculation, we found that the similar cancellation also occurs ingeneral n -tuplet cases with both W -boson and Z -boson contributions.Let us examine what happens in the SI effective coupling. In Fig. 6, we show thecontribution to f p /m p of each effective coupling in Eq. (9). The left (right) panel inthe figure shows each contribution due to the W -boson ( Z -boson) loops as a function ofHiggs-boson mass. The plot is given in units of ( n − (4 Y + 1)) / Y in left andright panels, respectively. In the figure, we show the contribution of the Higgs-bosonexchange including both one- and two-loop contributions (solid), the gluon contributionexcept for the Higgs-boson contribution (dashed), the quark twist-2 operator contribution(dash-dotted), and the contribution to the quark scalar-type operators with the coefficient f q from the Z -boson box diagrams (double-dashed). The W -boson box diagrams do notgenerate the scalar-type operators [8]. It is found that the contributions from the twist-2operators generated by both W -boson and Z -boson loops are positive, while all the othercontributions are negative. In addition, they have roughly the same order in absolutevalue. This is why the cancellation happens in the SI effective coupling induced byeach W -boson and Z -boson contribution. We also find that the Z -boson contribution in11 -48 -47 -46 -45 -44
115 200 300 400 500 S I c r o ss s e c t i on [ c m ] Higgs mass [GeV]n=5, Y=010 -48 -47 -46 -45 -44
115 200 300 400 500 S I c r o ss s e c t i on [ c m ] Higgs mass [GeV]n=5, Y=110 -48 -47 -46 -45 -44
115 200 300 400 500 S I c r o ss s e c t i on [ c m ] Higgs mass [GeV]n=5, Y=2 10 -48 -47 -46 -45 -44
115 200 300 400 500 S I c r o ss s e c t i on [ c m ] Higgs mass [GeV]n=2, Y=1/210 -48 -47 -46 -45 -44
115 200 300 400 500 S I c r o ss s e c t i on [ c m ] Higgs mass [GeV]n=3, Y=010 -48 -47 -46 -45 -44
115 200 300 400 500 S I c r o ss s e c t i on [ c m ] Higgs mass [GeV]n=3, Y=1
Figure 5: SI cross sections of DM-proton elastic scattering as a function of Higgs-bosonmass for n = 5 (left panel) and n = 2 , M = 1, 2.7, 10 TeV for n =2, 3, and 5, respectively. In left panel, solid, dashed, and dash-dotted lines represent n = 5 with Y = 0, 1, and 2 cases, respectively. In right panel ( n, Y ) = (3 , , , /
2) correspond to solid, dashed, and dash-dotted lines, respectively. -1-0.5 0 0.5 1 115 200 300 400 500 f p / m p i n un i t s o f [ ( n - ( Y + )) / ] × - Higgs mass [GeV]twist2(W)-1-0.5 0 0.5 1 115 200 300 400 500 f p / m p i n un i t s o f [ ( n - ( Y + )) / ] × - Higgs mass [GeV]Higgs(W)-1-0.5 0 0.5 1 115 200 300 400 500 f p / m p i n un i t s o f [ ( n - ( Y + )) / ] × - Higgs mass [GeV]gluon(W) -1-0.5 0 0.5 1 115 200 300 400 500 f p / m p i n un i t s o f Y × - Higgs mass [GeV]twist2(Z)-1-0.5 0 0.5 1 115 200 300 400 500 f p / m p i n un i t s o f Y × - Higgs mass [GeV]Higgs(Z)-1-0.5 0 0.5 1 115 200 300 400 500 f p / m p i n un i t s o f Y × - Higgs mass [GeV]gluon(Z)-1-0.5 0 0.5 1 115 200 300 400 500 f p / m p i n un i t s o f Y × - Higgs mass [GeV]scalar(Z)
Figure 6: The contributions of each effective coupling in Eq. (9) to f p /m p . Left panelshows each contribution from W -boson loops as a function of Higgs-boson mass, whileright one illustrates contribution by Z -boson loops. These plots are shown in units of( n − (4 Y + 1)) / Y , respectively. We take M = 10 TeV. In the both panels,solid line represents the contribution of the Higgs-boson exchange (including both one-and two-loop contribution), dashed line indicates the rest of gluon contribution in thetwo-loop contribution, and dash-dotted line denotes that from quark twist-2 operators.Double-dashed line in right graph shows contribution of scalar-type operator coming frombox diagrams. 12 -48 -47 -46 -45 -44 -43
1 10 S D c r o ss s e c t i on [ c m ] DM mass [TeV]n=5, Y=010 -48 -47 -46 -45 -44 -43
1 10 S D c r o ss s e c t i on [ c m ] DM mass [TeV]n=5, Y=110 -48 -47 -46 -45 -44 -43
1 10 S D c r o ss s e c t i on [ c m ] DM mass [TeV]n=5, Y=2 10 -48 -47 -46 -45 -44 -43
1 10 S D c r o ss s e c t i on [ c m ] DM mass [TeV]n=2, Y=1/210 -48 -47 -46 -45 -44 -43
1 10 S D c r o ss s e c t i on [ c m ] DM mass [TeV]n=3, Y=010 -48 -47 -46 -45 -44 -43
1 10 S D c r o ss s e c t i on [ c m ] DM mass [TeV]n=3, Y=1
Figure 7: SD cross section of DM-proton elastic scattering for n = 5 (left) and n = 2 , Y = 0, 1, 2,while in right graph they represents ( n, Y ) = (3 , Y is generally rather small compared with the W -boson contribution in unitsof ( n − (4 Y + 1)) /
8. Consequently, the cross section is suppressed when Y gets larger(because the coupling between W boson and EW-IMP is suppressed). When Higgs-bosonmass is relatively small as m h ∼ −
200 GeV, the contribution of Higgs-boson exchangebecomes large. Then, there is much more significant cancellation within each W - and Z -boson contribution. As a result, these cancellations give rise to a very suppressed SI crosssection.Next, we show the SD cross section in Fig. 7 for completeness. Since the SD crosssection is independent of Higgs-boson mass, we plot it as a function of the EW-IMP mass.Contrary to the SI cross section, the SD cross section is suppressed when the EW-IMPmass is larger [5]. In the left panel, the SD cross sections of n = 5 cases with Y = 0 , , and 2 are shown in the solid line, the dashed line, and the dash-dotted line, respectively.In the right panel, we give the cross sections by taking ( n, Y ) = (3 , As expected,the cross section is enhanced by larger n in a similar way to the SI cross section. Thefigure also shows that the cross section decreases when one sets Y larger, because thecontribution from W -boson exchange reduces. We also find that the contribution from Z -boson exchange has the opposite sign and rather small significance compared to thatfrom W -boson loop diagrams.Throughout the calculation, we have ignored the mass splitting δm between the neutralcomponents for Y = 0 case. The magnitude of δm is model-dependent. If δm is relativelylarge against our assumption, the Higgs-boson exchange contribution at tree level becomessignificant, and one might expect a larger SI cross section. Even in such a case, the There was an numerical error in the result of SD cross section in Ref. [8]. We have corrected it here. F (0) S ( x ) and F (0) T ( x ) in Ref. [5]. (This was pre-viously described in Ref. [8]. ) On the other hand, gluon contribution was taken intoaccount without explicit calculation; however, the sign of the contribution is the samewith our result. As a consequence, similarly to our result, the cancellations between eachcontribution happen and the cross sections computed in their works are comparable toour results.On the other hand, the SI cross sections in Refs. [6, 7] are larger than our results bymore than an order of magnitude. In Ref. [6] only the scalar-type operator is taken intoaccount in the limit M ≫ m W ≫ m q , which does not agree with our result in the samelimit. Although they mention the twist-2 type contribution, they took into account thiscontribution in the same sign as the scalar-type one. Moreover, the gluon contribution isomitted. Thus, there is no cancellation which we found in the effective coupling, to giverise to the large cross section.In Ref. [7], the explicit results of the loop calculation are given. However, all of theloop functions except f ZI ( x ) in Eq. (26) in Ref. [7] are different from ours. To check theirresult further, we compare them with ours in the limit m W/Z /M →
0. In Eq. (15), whichis scalar-type contribution, the first term (of the first parenthesis) agree with their resultexcept for the sign, and the other terms are − / i.e. , Eq. (17)) is the same as their result in thelimit, while another contribution ( i.e. , Eq. (18)) is neglected in their paper. The gluoncontribution is considered, but not evaluated properly. As a result, the cancellation wefound in our calculation does not occur, which makes the resultant cross section large. We have studied the direct detection of EW-IMP DM, which is the lightest neutral com-ponent of an SU (2) L multiplet interacting with quarks and leptons only through the SU (2) L × U (1) Y gauge interactions. Although EW-IMP DM does not scatter off nucleonat tree level, it does at loop level, and the SI scattering cross section is not suppressedeven if the mass of EW-IMP is much larger than gauge boson mass. We evaluate thetwo-loop processes for the EW-IMP-gluon interaction in addition to the one-loop pro-cesses of EW-IMP scattering with light quarks, since both of them yield considerablecontribution to the SI scattering cross section. As a result, the SI cross section is foundto be O (10 − (46 − ) cm , depending on Higgs-boson mass, the number of components n inthe multiplet, and the hypercharge Y of the EW-IMP DM. Such small value of the crosssection is due to an accidental cancellation in the SI effective coupling. This cancellationis a general feature for EW-IMP DM, and thus makes it difficult to catch EW-IMP DMin the direct detection experiments in the near future.14 cknowledgments This work is supported by Grant-in-Aid for Scientific research from the Ministry of Edu-cation, Science, Sports, and Culture (MEXT), Japan, No. 20244037, No. 20540252, andNo. 22244021 (J.H.), and also by World Premier International Research Center Initiative(WPI Initiative), MEXT, Japan. This work was also supported in part by the U.S. De-partment of Energy under contract No. DE-FG02-92ER40701, and by the Gordon andBetty Moore Foundation (K.I.).
Appendix
In this Appendix we provide the mass functions and the Z -boson vacuum polarizationfunctions, both presented in Section 4. A Mass Functions from One-loop Diagrams
The mass functions used in Eqs. (15-18) in the calculation of one-loop diagrams are g H ( x ) = − b x (2 + 2 x − x ) tan − (cid:18) b x √ x (cid:19) + 2 √ x (2 − x log( x )) ,g S ( x ) = 14 b x (4 − x + x ) tan − (cid:18) b x √ x (cid:19) + 14 √ x (2 − x log( x )) ,g AV ( x ) = 124 b x √ x (8 − x − x ) tan − (cid:18) b x √ x (cid:19) − x (2 − (3 + x ) log( x )) ,g T1 ( x ) = 13 b x (2 + x ) tan − (cid:18) b x √ x (cid:19) + 112 √ x (1 − x − x (2 − x ) log( x )) ,g T2 ( x ) = 14 b x x (2 − x + x ) tan − (cid:18) b x √ x (cid:19) − √ x (1 − x − x (2 − x ) log( x )) , (30)with b x ≡ p − x/ B Mass functions from Two-loop Diagrams
Here, we present the polarization functions and the mass functions for Z boson. Perform-ing the integration in Eq.(25), we derive the polarization functions as (cid:2) Π ZT ( p ) (cid:3) q ( b ) | GG = α α s θ W ( G aµν ) (cid:2) ( a Vq ) + ( a Aq ) (cid:3) × p − m q )( p − m q ) + 4 m q ( p − m q ) q m q p − − (cid:18)q m q p − (cid:19) p ( p − m q ) , Π ZT ( p ) (cid:3) q ( c ) | GG = − α α s θ W ( G aµν ) " ( a Vq ) (cid:0) p − m q p + 24 m q (cid:1) − ( a Aq ) ( p − m q )(2 p − m q ) p ( p − m q ) +2 m q q m q p − (cid:8)(cid:0) p − m q p + 48 m q (cid:1) ( a Vq ) + ( a Aq ) ( p − m q )( p + 6 m q ) (cid:9) cot − (cid:18)q m q p − (cid:19) p ( p − m q ) , (31) and in the limit of zero quark mass, these functions lead to (cid:2) Π ZT ( p ) (cid:3) q ( b ) | GG → α α s θ W ( G aµν ) (cid:2) ( a Vq ) + ( a Aq ) (cid:3) p , (cid:2) Π ZT ( p ) (cid:3) q ( c ) | GG → − α α s θ W ( G aµν ) (cid:2) ( a Vq ) − ( a Aq ) (cid:3) p . (32)Using the polarization functions, we compute g Z ( z, y ) in Eq.(29) as g Z ( z, y ) = " X q = u,d,s,c,b (cid:8) ( a Vq ) + ( a Aq ) (cid:9) − X Q = c,b c Q (cid:8) ( a Vq ) − ( a Aq ) (cid:9) × g B1 ( z ) + g t ( z, y ) . (33)Here, the first term comes from the contribution of all quarks except top quark, and c Q in it represents the QCD corrections in the long-distance contributions [23]. The secondterm g t ( z, y ) is the contribution of top quark. The function g B1 ( x ) is given as g B1 ( x ) = − √ x ( x log( x ) −
2) + ( x − x + 4) tan − ( b x √ x )24 b x , (34)which is equal to the one in Ref. [8]. In the calculation of g t ( z, y ), we decompose it intotwo parts: g t ( z, y ) = g no-log t ( z, y ) + g log t ( z, y ) . (35) g no-log t ( z, y ) is analytically obtained as g no-log t ( z, y ) = ( a Vt ) G t ( z, y ) + ( a At ) G t ( z, y ) , (36)where G t ( z, y ) = − √ z (12 y − zy + z )3(4 y − z ) + z / (48 y − zy + 12 z y − z )6(4 y − z ) log z + 2 z / y (4 y − z )3(4 y − z ) log(4 y ) − z / √ y (16 y − z ) y + 14(2 + z ) y + 5 z )3(4 y − z ) √ − y tan − (cid:18) √ − y √ y (cid:19) − tan − (cid:18) √ − z √ z (cid:19) × z − z + 4) y − z (5 z − z + 44) y + 12 z ( z − y − z ( z − z + 4)3(4 y − z ) √ − z , t ( z, y ) = √ z (2 y − z )(4 y − z ) − z / (8 y − zy + z )2(4 y − z ) log z − z / y (4 y − z ) log(4 y )+ 4 z / √ y (2 y − y − y − z ) √ − y tan − (cid:18) √ − y √ y (cid:19) − z ( z − z + 1) y − ( z − z + 4)(8 y + z )(4 y − z ) √ − z tan − (cid:18) √ − z √ z (cid:19) . (37)On the other hand, we calculate g log t ( z, y ) numerically. For convenience, we rewrite it as g log t ( z, y ) = 4 z / y ( A y [ I + I ] + A [ I + I ]) , (38)with A = − a Vt ) + 4( a At ) ,A = − ( a Vt ) + ( a At ) , (39)and then carry out the following integrals numerically, I = Z ∞ d t ( √ t + 4 − √ t ) (cid:0) log (cid:2) √ t + 4 y + √ t (cid:3) − log (cid:2) √ t + 4 y − √ t (cid:3)(cid:1) [ t + z ] [ t + 4 y ] / t ,I = Z ∞ d t × ( t + 2 − √ t √ t + 4) (cid:0) log (cid:2) √ t + 4 y + √ t (cid:3) − log (cid:2) √ t + 4 y − √ t (cid:3)(cid:1) [ t + z ] [ t + 4 y ] / t / ,I = Z ∞ d t ( √ t + 4 − √ t ) (cid:0) log (cid:2) √ t + 4 y + √ t (cid:3) − log (cid:2) √ t + 4 y − √ t (cid:3)(cid:1) [ t + z ] [ t + 4 y ] / ,I = Z ∞ d t × √ t ( t + 2 − √ t √ t + 4) (cid:0) log (cid:2) √ t + 4 y + √ t (cid:3) − log (cid:2) √ t + 4 y − √ t (cid:3)(cid:1) [ t + z ] [ t + 4 y ] / . Lastly, we present the mass function g W ( w, y ) in Eq. (29), which is readily obtainedby following the similar procedure described in Ref. [8]: g W ( w, y ) = 2 g B1 ( w ) + g B3 ( w, y ) (40)with g B3 ( x, y ) = g (1)B3 ( x, y ) + c b g (2)B3 ( x, y ) . (41)Here the first term of g W ( w, y ) is coming from the first- and second-generation quarkloop diagrams, while the second term is from the third-generation quark loop diagrams.The mass function g B1 ( x ) is displayed in Eq. (34). Although we use the same symbol for g B3 ( x, y ) as in Ref. [8], it is reevaluated with the QCD correction for the long-distancecontributions, which we illustrate with the factor c b in Eq. (41) explicitly. We have checkednumerically that including the QCD correction changes the SI cross section by up to a17ew %. The functions g (1)B3 ( x, y ) and g (2)B3 ( x, y ) are analytically given as g (1)B3 ( x, y ) = − x / y − x ) + − x / y y − x ) log y − x / ( x − y )24( y − x ) log x − x / √ y ( y + 2) √ − y y − x ) tan − (cid:18) √ − y √ y (cid:19) + x ( x − y + 1) x + 4( y + 1) x + 4 y )12( y − x ) √ − x tan − (cid:18) √ − x √ x (cid:19) ,g (2)B3 ( x, y ) = − x / y y − x ) + − x / y y − x ) log y + x / y y − x ) log x + x / √ y ( − y + xy − xy − x )12( y − x ) √ − y tan − (cid:18) √ − y √ y (cid:19) + − xy ( x y − xy − x − y )12( y − x ) √ − x tan − (cid:18) √ − x √ x (cid:19) , (42)respectively. References [1] E. Komatsu et al. [WMAP Collaboration], Astrophys. J. Suppl. , 18 (2011).[2] E. Aprile et al. [XENON100 Collaboration], Phys. Rev. Lett. , 131302 (2010).[3] E. Aprile et al. [XENON100 Collaboration], arXiv:1103.0303 [hep-ex].[4] M. Drees, M. M. Nojiri, D. P. Roy and Y. Yamada, Phys. Rev. D , 276 (1997)[Erratum-ibid. D , 039901 (2001)][5] J. Hisano, S. Matsumoto, M. M. Nojiri and O. Saito, Phys. Rev. D , 015007 (2005).[6] M. Cirelli, N. Fornengo, A. Strumia, Nucl. Phys. B , 178 (2006) .[7] R. Essig, Phys. Rev. D , 015004 (2008).[8] J. Hisano, K. Ishiwata and N. Nagata, Phys. Lett. B , 311 (2010).[9] M. Cirelli, A. Strumia, and M. Tamburini, Nucl. Phys. B , 152 (2007) .[10] J. Hisano, S. Matsumoto, M. Nagai, O. Saito and M. Senami, Phys. Lett. B , 34(2007).[11] J. Hisano, S. Matsumoto, M. Nojiri, Phys. Rev. Lett. , 031303 (2004).[12] J. Hisano, S. Matsumoto, M. M. Nojiri and O. Saito, Phys. Rev. D , 063528 (2005).1813] M. Drees and M. Nojiri, Phys. Rev. D , 3483 (1993)[14] J. Hisano, K. Ishiwata and N. Nagata, Phys. Rev. D , 115007 (2010).[15] G. Jungman, M. Kamionkowski and K. Griest, Phys. Rept. , 195 (1996).[16] H. D. Politzer, Nucl. Phys. B , 349 (1980).[17] A. Corsetti and P. Nath, Phys. Rev. D , 125010 (2001).[18] H. Ohki et al. , Phys. Rev. D , 054502 (2008).[19] H. Y. Cheng, Phys. Lett. B , 347 (1989).[20] J. Pumplin, D. R. Stump, J. Huston, H. L. Lai, P. M. Nadolsky and W. K. Tung,JHEP , 012 (2002).[21] J. Hisano, K. Ishiwata, N. Nagata and M. Yamanaka, arXiv:1012.5455 [hep-ph].[22] D. Adams et al. [Spin Muon Collaboration], Phys. Lett. B , 248 (1995).[23] A. Djouadi and M. Drees, Phys. Lett. B484