Dark matter and a new gauge boson through kinetic mixing
aa r X i v : . [ h e p - ph ] N ov Preprint typeset in JHEP style - HYPER VERSION
KIAS-P010015
Dark matter and a new gauge boson throughkinetic mixing
Eung Jin Chun and Jong-Chul Park
Korea Institute for Advanced StudyHeogiro 87, Dongdaemun-guSeoul 130-722, KoreaEmails: [email protected], [email protected]
Stefano Scopel
Department of Physics, Sogang University1 Sinsu-dong, Mapo-guSeoul 121-742, KoreaEmail: [email protected]
Abstract:
We consider a hidden sector model of dark matter which is charged undera hidden U (1) X gauge symmetry. Kinetic mixing of U (1) X with the Standard Modelhypercharge U (1) Y is allowed to provide communication between the hidden sectorand the Standard Model sector. We present various limits on the kinetic mixingparameter and the hidden gauge coupling constant coming from various low energyobservables, electroweak precision tests, and the right thermal relic density of thedark matter. Saturating these constraints, we show that the spin-independent elasticcross section of the dark matter off nucleons is mostly below the current experimentallimits, but within the future sensitivity. Finally, we analyze the prospect of observingthe hidden gauge boson through its dimuon decay channel at hadron colliders. Keywords:
Hidden gauge symmetry, Kinetic mixing, Thermal dark matter. ontents
1. Introduction 12. Hidden U (1) X model and gauge interactions 23. Low-energy and electroweak constraints 5 g µ − ρ parameter 63.4 EWPT 9
4. Thermal relic abundance of dark matter 105. Direct detection of dark matter 126. Tevatron and LHC probes of U ( ) X
1. Introduction
One of the popular scenarios for a TeV-scale physics beyond the Standard Modelis postulating an additional gauge interaction other than the Standard Model one SU (3) c × SU (2) L × U (1) Y . A classic example is an extra U (1) interaction that arisesfrom a grand unification theory [1]. Such a possibility has been well studied as itmight be discovered in the early stage of the LHC experiment. Another interestingpossibility for an extended gauge sector is to assume a U (1) X interaction in thehidden sector, in the sense that Standard Model particles are neutral under U (1) X .However, the hidden sector and the Standard Model sector can couple to each otherthrough the kinetic mixing of U (1) X and U (1) Y [2]. Since the kinetic mixing termis gauge-invariant, it can be present at the tree-level. In the hidden U (1) X model,one can introduce a massive Dirac fermion charged under U (1) X which is a darkmatter (DM) candidate. This kind of scenario has been used to implement MeVDM [3], a sub-GeV X boson for the Sommerfeld enhancement employing kinetic and– 1 –iggs mixing [4], light DM with Sommerfeld enhancement in the NMSSM with gaugemediation [5], a TeV scale hidden sector through Higgs mixing [6], and a 10 GeVDM through kinetic mixing [7]. Let us also remark that Stuekelberg Z ′ models havesimilar features [8, 9, 10].In this paper, we examine various phenomenological implications of the kineticmixing of U (1) X and of a DM candidate which has a preferable mass scale of 0 . − ǫ , the dark matter mass m ψ , the X boson mass m X , andthe X gauge coupling g X . These parameters are constrained by various low-energyand electroweak observables, Tevatron II results, and the thermal relic density ofdark matter. After considering these constraints, we will look for perspectives forthe direct detection of dark matter and the LHC discovery of the X boson.This paper is organized as follows. In Section 2, we set up the hidden sectordark matter model and present interaction vertices relevant for further discussions.In Section 3, we work out various constraints on the kinetic mixing parameter ǫ from low-energy observables [11, 12] and electroweak precision tests [13, 14] usedin the phenomenological discussion of the ensuing Sections. In Section 4, the DMannihilation rate is calculated and normalized to the observed DM relic density. Inparticular, by using this procedure we fix a combination of ǫ and g X ( ≃ ǫg X inthe limit ǫ ≪
1) as a function of the two masses m ψ and m X . In Section 5, thespin-independent DM-nucleon cross section is calculated and compared to the recentCDMS II [15] and XENON100 [16] results. In Section 6, we analyze the TevatronII limit on the X boson production and dimuon decay and the LHC perspective fordetection of the same quantity, which depends on the branching ratio of the X bosondecay to the dark matter particle. Finally, we conclude in Section 7.
2. Hidden U (1) X model and gauge interactions We consider a hidden sector containing a gauge symmetry U (1) X and a Dirac fermiondark matter candidate at the TeV scale, which couples to the Standard Model sectorthrough kinetic mixing. The full Lagrangian including kinetic mixing is: L = L SM −
12 sin ǫ ˆ B µν ˆ X µν −
14 ˆ X µν ˆ X µν − g X ˆ X µ ¯ ψγ µ ψ + 12 m X ˆ X + m ψ ¯ ψψ, (2.1)where the U (1) X is assumed to be broken spontaneously leading to the gauge bosonmass m ˆ X . In the Standard Model sector, the ˆ Z gauge boson has the mass m ˆ Z andthe gauge couplings are denoted by ˆ g = ˆ e/s ˆ W and ˆ g ′ = ˆ e/c ˆ W . Diagonalizing away thekinetic mixing term and mass mixing terms is made by the following transformation– 2 –17]: ˆ B = c ˆ W A − ( t ǫ s ξ + s ˆ W c ξ ) Z + ( s ˆ W s ξ − t ǫ c ξ ) X , ˆ W = s ˆ W A + c ˆ W c ξ Z − c ˆ W s ξ X , ˆ X = s ξ c ǫ Z + c ξ c ǫ X , (2.2)where the angle ξ is determined by:tan 2 ξ = − m Z s ˆ W sin 2 ǫm X − m Z ( c ǫ − s ǫ s W ) . (2.3)Then, the X and Z gauge bosons get the redefined masses: m Z = m Z (1 + s ˆ W t ξ t ǫ ) , (2.4) m X = m X c ǫ (1 + s ˆ W t ξ t ǫ ) . (2.5)Moreover, from Eqs. (2.3) - (2.5) one can find t ξ as a function of r X ≡ m X /m Z : t ξ = − s ˆ W t ǫ , (2.6) t ξ = 1 − r X ± q (1 − r X ) − s W t ǫ r X s ˆ W t ǫ r X . (2.7)The solution given by Eq.(2.6) is unphysical since it corresponds to m Z = 0 and m X = ∞ . In addition, the other two solutions (2.7) are physical only in the range r X ≤ s W t ǫ − q s W t ǫ (1 + s W t ǫ ) or r X ≥ s W t ǫ + 2 q s W t ǫ (1 + s W t ǫ ), where(1 − r X ) − s W t ǫ r X >
0. In the following analysis we exclude the unphysical region,which is limited to a small interval around m X = m Z . As we will see shortly, theweak mixing angle s ˆ W is very close to the physical angle s W due to the strong ρ parameter limit.Let us now list all the interaction vertices relevant for our analysis. The W , Z ,and X gauge bosons have the following fermion couplings: L W = − e √ s ˆ W W + µ { ¯ νγ µ P L e + ¯ uγ µ P L d } + c.c. , (2.8) L Z = − ec ˆ W s ˆ W c ξ Z µ ¯ f γ µ (cid:8) P L (cid:2) T (1 + s ˆ W t ǫ t ξ ) − Q ( s W + s ˆ W t ǫ t ξ ) (cid:3) − P R (cid:2) Q ( s W + s ˆ W t ǫ t ξ ) (cid:3)(cid:9) f − g X s ξ c ǫ Z µ ¯ ψγ µ ψ , (2.9) L X = − ec ˆ W s ˆ W c ξ X µ ¯ f γ µ (cid:8) P L (cid:2) T ( s ˆ W t ǫ − t ξ ) + Q ( s W t ξ − s ˆ W t ǫ ) (cid:3) + P R (cid:2) Q ( s W t ξ − s ˆ W t ǫ ) (cid:3)(cid:9) f − g X c ξ c ǫ X µ ¯ ψγ µ ψ . (2.10)– 3 –he couplings of the Z and X gauge bosons to the W gauge bosons are given by: L V W W = et ˆ W c ξ [[ ZW + W − ]] − et ˆ W s ξ [[ XW + W − ]] , (2.11)where [[ V W + W − ]] ≡ i [( ∂ µ W + ν − ∂ ν W + µ ) W µ − V ν − ( ∂ µ W − ν − ∂ ν W − µ ) W µ + V ν +(1 / ∂ µ V ν − ∂ ν V µ )( W µ + W ν − − W µ − W ν + )]. The couplings of the Higgs scalar h to the Z and X gauge bosons are: L hV V = m Z v c ξ h " (1 + s ˆ W t ξ t ǫ ) Z µ Z µ + t ξ (cid:18) − s ˆ W t ǫ t ξ (cid:19) X µ X µ +2 t ξ (cid:18) − s ˆ W t ǫ t ξ + s W t ǫ (cid:19) X µ Z µ (cid:21) . (2.12)Summarizing the interaction vertices in Eqs. (2.8) - (2.12), let us define the variouscouplings, g ’s, as follows: L = W + µ g Wf [¯ νγ µ P L e + ¯ uγ µ P L d ] + c.c. + Z µ (cid:2) g ZfL ¯ f γ µ P L f + g ZfR ¯ f γ µ P R f + g Zψ ¯ ψγ µ ψ (cid:3) + g ZW [[ ZW + W − ]]+ X µ (cid:2) g XfL ¯ f γ µ P L f + g XfR ¯ f γ µ P R f + g Xψ ¯ ψγ µ ψ (cid:3) + + g XW [[ XW + W − ]]+ h (cid:2) g hZZ Z µ Z µ + g hXX X µ X µ + g hXZ X µ Z µ (cid:3) . (2.13)Unlike the Z and X gauge bosons, the mass of the W gauge boson is not modifiedby the above transformation (2.2): m W = m W = m Z c W . (2.14)Then, the ρ parameter is given by: ρ ≡ m W m Z c W = c W (1 + s ˆ W t ξ t ǫ ) c W . (2.15)Consequently, the current bound on the ρ parameter, ρ − +8 − × − [18],provides a constraint on the parameter ǫ as a function of the gauge boson masses.Let us note that the photon coupling does not change (ˆ e = e ) and the identity m Z / ( g + g ′ ) = m Z / (ˆ g + ˆ g ′ ) [17] leads to the relation between the original andredefined weak mixing angles: c W s W = c W s W s ˆ W t ξ t ǫ . (2.16)Therefore, the ρ parameter can be recast as ρ = s W s W . (2.17)– 4 –efining ˆ δ ≡ ρ − s W /s W − δ , we find: ω ≡ s W t ξ t ǫ ≃ − (1 − t W )ˆ δ . (2.18)In the redefined physical basis, the above couplings can be rewritten in the firstorder of ω (or ˆ δ ) as follows: L W = − e √ s W (cid:18) − ω − t W ) (cid:19) W + µ { ¯ νγ µ P L e + ¯ uγ µ P L d } + c.c. , (2.19) L Z = − ec W s W c ξ Z µ ¯ f γ µ (cid:26) P L T h ω i − Q (cid:20) s W + ω (cid:18) − t W − t W ) (cid:19)(cid:21)(cid:27) f − g X s ξ c ǫ Z µ ¯ ψγ µ ψ , (2.20) L X = − ec W s W c ξ X µ ¯ f γ µ (cid:26) P L T (cid:20) s W t ǫ − t ξ + 12 ω (cid:18) t ξ + s W t W t ǫ − t W (cid:19)(cid:21) + Q (cid:20) s W t ξ − s W t ǫ + 12 t W ω (cid:18) t ξ − s W t ǫ − t W (cid:19)(cid:21)(cid:27) f − g X c ξ c ǫ X µ ¯ ψγ µ ψ , (2.21) L V W W = et W (cid:18) − ω c W − s W ) (cid:19) (cid:8) c ξ [[ ZW + W − ]] − s ξ [[ XW + W − ]] (cid:9) , (2.22) L hV V = m Z v c ξ h (cid:26) [1 + ω ] Z µ Z µ + (cid:20) t ξ + s W t ǫ − ω (cid:18) t ξ − s W t W t ǫ − t W (cid:19)(cid:21) X µ X µ +2 (cid:20) s W t ǫ − t ξ + ω (cid:18) t ξ + s W t W t ǫ − t W (cid:19)(cid:21) X µ Z µ (cid:27) . (2.23)From these expressions, we can read off the couplings defined in Eq. (2.13). Theexplicit expressions of the redefined interaction couplings are given in Appendix A.
3. Low-energy and electroweak constraints
The dark matter model with a hidden U (1) X sector introduced in the previous Sectionhas four free parameters: ǫ , g X , m X , and m ψ . In this Section, we show how sizablethe U (1) X contributions are for the muon g −
2, atomic parity-violation, ρ parameterand Electro–Weak Precision Tests (EWPT). From these analyses, we will obtain anupper limit on the kinetic mixing parameter ǫ as a function of the hidden gaugeboson mass m X . g µ − X induces a contribution to the anomalousmagnetic moment of the muon, a µ = ( g µ − /
2. The modified couplings of the Z – 5 –oson induce an additional contribution. Adopting the formula in Ref. [12], we find: δa µ ≈ ( g XµV ) − g XµA ) π m µ m X + ∆ ( g ZµV ) − g ZµA ) π m µ m Z , (3.1)where g X,ZµV,A ≡ g X,ZµR ± g X,ZµL are given in Appendix A and ∆ represents the deviationfrom the value calculated in the SM due to the modification of Z couplings. Thedifference between the improved Standard Model (SM) prediction of a µ and the latestexperimental value for a µ is [19] δa µ = (31 . ± . · − . (3.2)In Fig. 1, we present the contribution to a µ from the X gauge boson and the modified Z couplings as a function of m X for various values of sin ǫ in the ranges m X < m Z (upper panel) and m X > m Z (lower panel). As shown in Fig. 1, the contributionfrom our model is well below the current limit with the exception of very small m X and sizable sin ǫ . The strength of the vector part of the Z weak neutral current (i.e. the weak force)between interacting quarks and leptons can be characterized by their weak charge.This weak charge governs parity-violation effect in atomic physics. The deviationof the present experimental results on the weak charge for cesium from the theoret-ical SM predictions corresponds to an uncertainty of less than 1%. Consequently,the parity-violation effect in atomic physics can provide strong constraints for low m X [11, 12], if the new effect from the couplings of the Z and X gauge bosons toelectrons and quarks violates parity. Adopting the result in Ref. [12], the limit onthe product of the axial coupling to the electron and its (average) vector coupling isgiven by − . × − GeV − . APV ≡ g XeA g XqV m X + ∆ g ZeA g ZqV m Z . . × − GeV − , (3.3)where g Z,XfV = ( g Z,XfL + g Z,XfR ) / g Z,XfA = ( g Z,XfL − g Z,XfR ) /
2, and ∆ again represents thedeviation from the Standard Model value. Fig. 2 shows the atomic parity-violation(APV) effect from the X gauge boson and the modified Z couplings as a function of m X for sin ǫ = 0 . , . , . , and 0.005 in the range m X < m Z (upper panel), andsin ǫ = 0 . , . , . , and 0.05 in the range m X > m Z (lower panel). ρ parameter As mentioned in Section 2, the ρ parameter is defined as: ρ = m W m Z c W = s W s W , (3.4)– 6 – .1 0.5 1.0 5.0 10.0 50.0 100.010 - - - - m X H GeV L ∆ a Μ
200 400 600 800 100010 - - - - - - - m X H GeV L ∆ a Μ Figure 1:
Contribution to the anomalous magnetic moment of the muon, δa µ from thehidden U (1) X model. In the upper panel, the red dashed, green dotted, blue dot-dashed,and purple long-dashed lines show the four cases sin ǫ = 0 . , . , . , and 0.001 for m X < m Z ; in the lower panel, the same line–styles and colors show sin ǫ = 0 . , . , . , and0.05 for m X > m Z . The horizontal solid line is the current limit on the difference betweenthe SM prediction and the latest experimental value [19]. and the deviation of ρ from 1, ˆ δ ≡ ρ −
1, is determined by sin ǫ according to Eq. (2.18).Therefore, the global fit for the ρ parameter, ρ − +8 − × − [18], results in a limiton the parameters ǫ and m X . In Fig. 3, we present the deviation of the ρ parameteras a function of m X for sin ǫ = 0 . , . , . , and 0.003 in the range m X < m Z – 7 – .1 0.5 1.0 5.0 10.0 50.0 100.010 - - - - m X H GeV L A P V H G e V - L
200 400 600 800 100010 - - - - m X H GeV L - A P V H G e V - L Figure 2:
Atomic parity-violation (APV) effect from the hidden U (1) X model. In theupper panel, the red dashed, green dotted, blue dot-dashed, and purple long-dashed linesshow the APV effect corresponding to sin ǫ = 0 . , . , . , and 0.005 for m X < m Z ;in the lower panel, the same line–styles and colors show sin ǫ = 0 . , . , . , and 0.05 for m X > m Z . The horizontal solid line represents the upper bound given by the differencebetween the experimental values on the weak charge of cesium and the SM predictions [11]. (upper panel) and for sin ǫ = 0 . , . , . , and 0.05 in the range m X > m Z (lowerpanel). This constraint is stronger than limits from the g µ − ǫ from the ρ parameteris shown as a function of m X in Fig. 4. To obtain this bound we use the 2 σ limit onthe ρ parameter. – 8 –
20 40 60 80 10010 - - m X H GeV L - ∆ `
200 400 600 800 100010 - - m X H GeV L ∆ ` Figure 3:
Difference from unity of the ρ parameter, ˆ δ = ρ −
1, due to the hidden U (1) X model. In the upper panel, the deviation of ρ is shown for sin ǫ = 0 . , . , . , and 0.003as red dashed, green dotted, blue dot-dashed, and purple long-dashed lines, respectively,and for the case m X < m Z . In the lower panel, the same line–styles and colors show thecases sin ǫ = 0 . , . , . , and 0.05 for m X > m Z . The solid horizontal line shows the 2 σ limit from the global fit [18]. The constraints on the hidden U (1) X model from electroweak precision tests (EWPT)have been analyzed in Refs. [13, 14]. Here, we adopt the result of Ref. [13] which– 9 –
200 400 600 800 1000 1200 14000.00.10.20.30.40.5 m X H GeV L s i n Ε Figure 4:
Upper bounds on the kinetic mixing parameter sin ǫ . The solid line shows thelimit from the ρ parameter [18], while the dashed line shows the limit from EWPT [13]. puts a conservative limit: (cid:18) tan ǫ . (cid:19) (cid:18)
250 GeV m X (cid:19) < ∼ . (3.5)This limit is more stringent than that from the ρ parameter except for m X around m Z . The ρ parameter and EWPT bounds in the sin ǫ – m X plane are shown in Fig. 4.
4. Thermal relic abundance of dark matter
The relic abundance of the DM candidate ψ depends on the ψψ annihilation crosssection to Standard Model particles, which proceeds through s -channel exchange of Z and X bosons in the zero-velocity limit. In particular, the annihilation modesinclude ψψ → f f , W + W − , Xh, and Zh , which give the following annihilation cross– 10 –ection times velocity: h σ A v i ≃ X f N f π m ψ s − m f m ψ " ( G fL + G fR ) − m f m ψ ! + 32 G fL G fR m f m ψ + 1 π m ψ G W − m W m ψ ! / (cid:20) m ψ m W + 5 m ψ m W + 34 (cid:21) + 18 π G Xh vuut − m Z m ψ + ∆ m hZ m ψ ! m ψ m Z − ∆ m hZ m ψ + ∆ m hZ m ψ ! + 18 π G Zh vuut − m X m ψ + ∆ m hX m ψ ! m ψ m X − ∆ m hX m ψ + ∆ m hX m ψ ! , (4.1)where: G fL = g Zψ g ZfL m ψ − m Z + g Xψ g XfL m ψ − m X ,G fR = g Zψ g ZfR m ψ − m Z + g Xψ g XfR m ψ − m X ,G W = g Zψ g ZW m ψ − m Z + g Xψ g XW m ψ − m X ,G V h = g Vψ g hXZ m ψ − m V . (4.2)Here, we define m V = ( m h + m V ) / m hV = m h − m V . For the annihilationto hZ ( hX ) through s -channel exchange of the X ( Z ) boson, (∆ m hV / m ψ ) ≪ m h =115 GeV.The present relic density of a hidden Dirac fermion ψ can be calculated from thefollowing analytic formula:Ω ψ h ≈ × . × GeV − M pl x F √ g ∗ h σ A v i , (4.3)where g ∗ is the number of relativistic degrees of freedom at the freeze-out temper-ature T F and x F ≡ m ψ /T F [20]. The extra factor of two on the right-hand side ofEq. (4.3) results from the fact that the annihilation can only occur between particleand antiparticle, since the dark matter candidate is a Dirac fermion. In the following,we will use the ψ relic abundance to constrain our parameter space. In particular,the annihilation cross section h σ A v i given in Eq. (4.1) is dominated either by thecoupling G Xh in the Xh final state channel ( m X < ∼
120 GeV) or by the couplings– 11 – fL and G fR in the f ¯ f final state. One can see from Eq.(A.1) that in both cases h σ A v i depends on the parameters g X and ǫ through the same multiplicative factor, g X s ξ /c ǫ . For each dark matter mass m ψ and hidden gauge boson mass m X , wewill, therefore, determine the combination g X s ξ /c ǫ by imposing the recent bound onthe dark matter relic density, (Ω DM h ) obs ≃ . ǫ toits experimental upper bound, the same combination will be used in Section 6 todetermine the remaining coupling g X .
5. Direct detection of dark matter
Satisfying all the previous constraints, we now discuss prospect of observing DM indirect detection experiments. The dark matter particle ψ can elastically scatter offa nucleus through t -channel X and Z gauge boson exchange. The spin-independent(SI) DM-nucleon scattering cross section can be calculated from the following effec-tive operator: L eff = b f ¯ ψγ µ ψ ¯ f γ µ f , (5.1)where b f = g Zψ ( g ZfL + g ZfR )2 m Z + g Xψ ( g XfL + g XfR )2 m X . (5.2)From the interaction couplings presented in Section 2, one calculates b u and b d to get b p = + ˆ gg X c ˆ W c ξ c ǫ t ξ m Z (cid:20) (1 − s W ) (cid:18) − r X (cid:19) − s ˆ W t ǫ t ξ (cid:18) t ξ + 1 r X (cid:19)(cid:21) ≃ eg X c W s W c ξ c ǫ t ξ m Z (cid:26) (1 − s W ) (cid:18) − r X (cid:19) − r X s W t ǫ t ξ − ω (cid:20) (cid:18) − r X (cid:19) (cid:18)
12 + 2 s W t W − t W (cid:19) − r X s W t ǫ t ξ t W − t W (cid:21)(cid:27) ,b n = − ˆ gg X c ˆ W c ξ c ǫ t ξ m Z (cid:20)(cid:18) − r X (cid:19) + s ˆ W t ǫ t ξ (cid:18) t ξ + 1 r X (cid:19)(cid:21) ≃ − eg X c W s W c ξ c ǫ t ξ m Z (cid:26)(cid:20) − r X (cid:18) − s W t ǫ t ξ (cid:19)(cid:21) + ω (cid:20) r X (cid:18) s W t W t ǫ (1 − t W ) t ξ (cid:19)(cid:21)(cid:27) . (5.3)For a given nucleus AZ N , one has b N = Zb p + ( A − Z ) b n . Finally, the ψ -nucleon elasticscattering cross section is given by [22]: σ n,p = 164 π µ n,p A b N , (5.4)where µ n,p is the DM–nucleon reduced mass. In Fig. 5, we present the SI DM-nucleonscattering cross section as a function of m X for m ψ = 50 , , , and 1000 GeV.From this figure, one can see that small DM masses around m ψ ∼
100 GeV are at thelevel of the current sensitivity of direct detection experiments. On the other hand,– 12 – - - - m X H GeV L Σ S I H c m L Figure 5:
Spin-independent DM-nucleon scattering cross section σ SI calculated by normal-izing the DM–nucleon couplings to the values that provide the observed DM relic density(see text). The red dashed, green dotted, blue dot-dashed, and purple long-dashed linescorrespond to m ψ = 50 , , , and 1000 GeV, respectively. The experimental limitsfrom CDMS II [15] corresponding to m ψ = 100 , , and 1000 GeV are shown by the dot-ted, dot-dashed, and long-dashed lines, respectively . The experimental limit for m ψ = 50,which is taken from XENON100 [16], is shown by the dashed line. for a larger mass up to around m ψ ∼
300 GeV and smaller m X , DM signals can bedetectable in the near future.To calculate the SI DM-nucleon scattering cross sections shown in Fig. 5, we havenormalized the b f coefficients given in Eq. (5.2) by using the relic abundance con-straint. Notice that, as in the case of the annihilation cross section, also b f dependson g X and ǫ through the multiplicative factor g X s ξ /c ǫ . An important consequence ofthis is that, at fixed values of m X and m ψ , the dependence on g X and ǫ cancels out inthe ratio σ n,p / h σ A v i . This implies that in our model the direct detection cross section σ n,p is potentially able to put robust constraints on m X and m ψ , once the ψ particleis assumed to provide the observed DM relic density in the Universe. On the otherhand, this degeneracy in g X and ǫ is not present in the calculation of acceleratorsignals, as will be discussed in the next Section.
6. Tevatron and LHC probes of U ( ) X For the analysis of collider searches we concentrate on the dimuon signal from pro-duction and decay of the hidden gauge boson X at the Tevatron and LHC. For this,– 13 –e need to calculate the corresponding branching ratios, which depend on the kine-matically available decay channels. The relevant decay rates of the X gauge bosonare given by:Γ( X → f ¯ f ) = N f π r m X − m f (cid:26) [( g XfL ) + ( g XfR ) ] (cid:18) − m f m X (cid:19) + 6( g XfL ) ( g XfR ) m f m X (cid:27) , Γ( X → ψ ¯ ψ ) = ( g Xψ ) π r m X − m ψ (cid:18) m ψ m X (cid:19) , Γ( X → hZ ) = ( g hXZ ) πm X q [ m X − ( m Z + m h ) ][ m X − ( m Z − m h ) ] × (cid:20) (cid:18) m X m Z + m Z m X − m h m X m Z (cid:19)(cid:21) , Γ( X → W + W − ) = ( g XW ) π m X (cid:18) m X m W (cid:19) (cid:18) − m W m X (cid:19) / (cid:20) m W m X + 12 m W m X (cid:21) . (6.1)We wish now to combine the constraints summarized in the previous Sections toput bounds on the parameter space of our model, spanned by the four parameters m X , m ψ , g X , and sin ǫ . In particular, the bounds which turn out to be the mostconstraining, and that will be the most relevant for the present discussion are: i)the upper bound on sin ǫ from EWPT discussed in Section 3.4; ii) the condition thatthe ψ relic density Ω ψ h is equal to the observed dark matter relic density value(Ω DM h ) obs ≃ . σ ( p ¯ p → X ) BR ( X → µ ¯ µ ) from CDFwhich will be discussed below in this Section. As a first approach to this problem,we fix sin ǫ to its EWPT upper bound and g X to the value required to provide(Ω DM h ) obs ≃ . σ ( p ¯ p → X ) BR ( X → µ ¯ µ ) as a function of m X at fixed values of m ψ .Here and in the following, we calculate the production cross section σ ( p ¯ p → X ) bymaking use of the PYTHIA code [23]. In the same Figure, the shaded area showsthe upper bound from CDF on the same quantity [24]. From this Figure, one cansee that, depending on the value of m ψ , the expected number of dimuon events atthe Tevatron σ ( p ¯ p → X ) BR ( X → µ ¯ µ ) can exceed the CDF upper bound.This is studied in detail in Fig. 7 in the m X – m ψ plane. In this figure, the regionto the left of the black solid line boundary is excluded by the CDF upper bound.In particular, one can see from this plot that the range m X < ∼
600 GeV is excludedunless m ψ < ∼
200 GeV. In fact, in this latter case the process p ¯ p → X → µ ¯ µ at theTevatron can be suppressed weakening the CDF bound because the invisible decay– 14 – igure 6: Expected number of events for the production of the X boson and its decayto µ ¯ µ at Tevatron, calculated for different values of m ψ . The shaded area is excluded bydimuon searches at CDF [24]. Notice that the curve for m ψ =1000 GeV is cut at m X ∼
360 GeV due to the perturbativity bound shown as a shaded area in Fig. 7. channel X → ψ ¯ ψ can become sizable. An exception to this is when m ψ ≃ m Z / ψ particles in the calculation ofthe relic density requires very low values of the coupling g X in order to provide thecorrect amount dark matter. As a consequence of this, one has BR ( X → ψ ¯ ψ ) ≃ m X jumps to the same value m X < ∼
600 GeV that one findsfor higher values of m ψ . However, in Fig. 7, the values of m X and m ψ to the leftof the solid line boundary are only excluded if sin ǫ is fixed to its EWPT upperbound. Consequently, in this region of the m X – m ψ parameter space, the quantity σ ( p ¯ p → X ) BR ( X → µ ¯ µ ) can be brought below the CDF constraint by using a valuefor sin ǫ smaller than the EWPT bound. In other words, Fig. 7 shows that for m X < ∼
600 GeV the CDF constraint can be stronger than the EWPT bound.In order to see the combined effect of the two constraints on the parameter spaceof our model, we provide Fig. 8 where we show the upper bound on sin ǫ as a functionof m X for different fixed values of m ψ when the CDF constraint is combined to the– 15 – igure 7: The region to the left of the solid line is excluded by CDF dimuon X searchesin the m X – m ψ plane when the sin ǫ parameter is set to its upper bound from EWPT. Thedashed lines show the contour plots in the same plane of the minimal values of sin ǫ compati-ble to the cosmological constraint and with the perturbativity requirement ( g Xψ ) / (4 π ) < ǫ is already exceeding the upper boundshown in Fig. 8 (see text). EWPT one. In this Figure, the upper thick solid line shows the constraint from the ρ parameter [18], as discussed in Section 3.3, while the lower thick solid line shows theconstraint from EWPT [13], as discussed in Section 3.4. The latter bound is mostlymore constraining than the former, and is saturated for m X > ∼
600 GeV. On the otherhand, the set of curves at m X < ∼
600 GeV show how the CDF constraint becomesimportant at lower X masses for different values of m ψ . As already discussed inconnection to Figs. 6 and 7, one can see from Fig. 8 that the CDF bound on theparameter space of our model can be stronger than the EWPT one for m X < ∼ m ψ ≃ m Z / σ ( p ¯ p → X ) BR ( X → µ ¯ µ ) hasbeen calculated by fixing g X as before using the relic abundance of the ψ particle.In particular, following this procedure one has to check that the upper bound onsin ǫ is consistent to the perturbativity bound ( g Zψ ) / (4 π ),( g Xψ ) / (4 π ) <
1. Indeed, for– 16 –ome intervals of m X and m ψ , the combination of the relic abundance, the CDF, andEWPT constraints require ( g Xψ ) / (4 π ) >
1. The corresponding excluded region in the m X – m ψ plane is shown as a shaded area in Fig. 7. In particular, in this region ofthe parameter space, the f ¯ f –final state contribution to the annihilation cross section h σ A v i (which dominates at larger masses) is suppressed, while the upper bound onsin ǫ is particularly constraining, driving g Xψ to large values in order to keep Ω ψ h equal to the observed cosmological DM abundance.Note that the shaded area excluded by the perturbativity limit does not extendbelow m X ∼
90 GeV. This is due to the fact that in this region of the parameter space,where the f ¯ f –final state contribution in h σ A v i is suppressed, the Xh -final state takesover as the dominant channel, thanks to the sizable value of the t ξ parameter in the g hXZ coupling when m X ∼ m Z (see Eqs. (2.3) and (A.1)). The ensuing enhancementof h σ A v i implies that for m X < ∼
90 GeV the observed relic abundance can be obtainedfor ( g Zψ ) / (4 π ),( g Xψ ) / (4 π ) <
1. Notice that, for consistency, the curves in Figs. 6 and9 for m ψ =1000 GeV have been cut at m X ∼
360 GeV. Moreover, since at fixed m X and m ψ , the cosmological constraint fixes the combination g X s ξ /c ǫ which vanisheswhen ǫ →
0, keeping the value of Ω ψ h fixed when ǫ → g X → ∞ . So,imposing ( g Zψ ) / (4 π ),( g Xψ ) / (4 π ) < ǫ as afunction of m X and m ψ . The result of this procedure is also shown in Fig.7, wherecontour plots at fixed values of (sin ǫ ) min are plotted as dashed lines in the m X – m ψ plane.The curves given in Fig. 8 can be used to estimate the perspectives of detectionof our model at the LHC. This is shown in Fig. 9, where the quantity σ ( pp → X ) BR ( X → µ ¯ µ ) at the center-of-mass energy of 14 TeV is shown by a set of linesas a function of m X at fixed values of m ψ . In each of these curves, the value of sin ǫ is fixed to the upper bound shown in Fig. 8 for the corresponding m ψ . The peculiarhollow shape of some of the curves for m X < ∼
600 GeV corresponds to the region ofthe parameter space where the CDF constraint on sin ǫ overcomes that from EWPT.In the same Figure, an estimation is also given of the 5 σ discovery reach at the LHCfor an exposure of 10 f b − [25]. We finally recall once again that, at variance withthe analysis of signals at accelerators, the calculation of the ψ -nucleon elastic crosssection relevant for direct DM searches does not rely on any assumption on ǫ becauseat fixed values of m X and m ψ the cross section σ n,p is unambiguously determined bythe relic abundance.
7. Conclusions
We postulated the existence of a hidden U (1) X gauge symmetry and a Dirac darkmatter fermion charged under U (1) X . Assuming that the hidden sector commu-nicates with the Standard Model sector only through the kinetic mixing between U (1) X and U (1) Y , the phenomenology of such scenario depends on four parameters:– 17 – igure 8: Upper bounds on the sin ǫ parameter as a function of m X for different values of m ψ . The upper thick solid line shows the constraint from the ρ parameter [18], as discussedin Section 3.3, while the lower thick solid line shows the constraint from EWPT [13], asdiscussed in Section 3.4. For m X < ∼
600 GeV, the sin ǫ parameter is also constrained bydimuon searches at CDF [24], depending on the value of m ψ , as explained in detail inSection 6. the kinetic mixing angle, the mass of the X boson, the dark matter mass, the cou-pling between the DM particle and the X gauge boson. We have considered variousobservables constraining the kinetic mixing parameter and the U (1) X coupling con-stant as a function of the X gauge boson mass. Since the dark matter annihilationto the Standard Model fields proceeds through kinetic mixing, we have used the relicdensity of the DM particle (determined by the standard thermal freeze-out process)to constrain a combination of the kinetic mixing and the X gauge boson coupling, re-quiring Ω ψ h to be equal to the observed DM density in the Universe. Saturating allthese constraints, we analyzed the spin-independent elastic cross-section of the darkmatter off nucleons showing that a large parameter space is within the sensitivity offuture direct detection experiments. We have also analyzed collider searches of the X gauge boson, concentrating on the dimuon signal at the Tevatron and the LHC.– 18 – igure 9: Maximal expected number of events for the production of the X boson andits decay to µ ¯ µ at LHC for different values of m ψ compared to an estimation of the LHCsensitivity for an exposure of 10 fb − [25]. For each value of m ψ , the parameter ǫ is fixedto the corresponding upper bound given in Fig. 8. Notice that the curve for m ψ =1000GeV is cut at m X ∼
360 GeV due to the perturbativity bound shown as a shaded area inFig. 7.
In particular, we found that the current Tevatron result puts the stronger bound onthe kinetic mixing term ǫ for m X < ∼
600 GeV, while at larger masses Electro–WeakPrecision Tests are more constraining. We have also found some intervals in the m X and m ψ masses ( m X < ∼
350 GeV and m ψ > ∼
320 GeV) that result to be excluded by aperturbativity requirement, since the combination of the relic abundance, the CDFand EWPT constraints require ( g Xψ ) / (4 π ) >
1. The same perturbativity constraint,combined to the requirement that the relic density of our DM candidate matchesthe observed value, allowed us to put also a lower bound on the s ǫ parameter as afunction of m χ and m ψ .Finally, the LHC prospects for observing the X gauge boson have been analyzedtaking the center-of-mass energy of 14 TeV and the integrated luminosity of 10 fb − .We found that an X gauge boson mass up to 4.5 TeV is within the LHC reach when– 19 –he dark matter mass m ψ is heavy enough to suppress the invisible decay X → ψ ¯ ψ ,or in the resonant limit m ψ ∼ m Z /
2, when the X → ψ ¯ ψ decay is negligible becausethe X –dark matter coupling g X is suppressed due to the cosmological bound onΩ ψ h . Moreover, an X boson mass up to 2.5 TeV can be probed if the dark mattermass is heavier than about 200 GeV. These results are obtained assuming that thekinetic mixing parameter saturates all the constraints including the Tevatron limit,and would be weakened by choosing smaller values. Acknowledgment:
SS is supported by NRF with CQUEST grant 2005-0049049and by by the Sogang Research Grant 2010.
A. Interaction couplings
Comparing Eq. (2.13) with Eqs. (2.19) - (2.23), one can easily find the redefinedcouplings expressed by the physical observables (unhatted parameters): g Wf = − e √ s W (cid:18) − ω − t W ) (cid:19) ,g ZfL = − ec W s W c ξ (cid:26) T h ω i − Q (cid:20) s W + ω (cid:18) − t W − t W ) (cid:19)(cid:21)(cid:27) ,g ZfR = ec W s W c ξ Q (cid:20) s W + ω (cid:18) − t W − t W ) (cid:19)(cid:21) ,g Zψ = − g X s ξ c ǫ ,g XfL = − ec W s W c ξ (cid:26) T (cid:20) s W t ǫ − t ξ + 12 ω (cid:18) t ξ + s W t W t ǫ − t W (cid:19)(cid:21) + Q (cid:20) s W t ξ − s W t ǫ + 12 t W ω (cid:18) t ξ − s W t ǫ − t W (cid:19)(cid:21)(cid:27) ,g XfR = − ec W s W c ξ Q (cid:20) s W t ξ − s W t ǫ + 12 t W ω (cid:18) t ξ − s W t ǫ − t W (cid:19)(cid:21) ,g Xψ = − g X c ξ c ǫ ,g ZW = et W c ξ (cid:18) − ω c W − s W ) (cid:19) ,g XW = − et W s ξ (cid:18) − ω c W − s W ) (cid:19) ,g hZZ = m Z v c ξ (1 + ω ) ,g hXX = m Z v c ξ (cid:20) t ξ + s W t ǫ − ω (cid:18) t ξ − s W t W t ǫ − t W (cid:19)(cid:21) ,g hXZ = m Z v c ξ (cid:20) s W t ǫ − t ξ + ω (cid:18) t ξ + s W t W t ǫ − t W (cid:19)(cid:21) . (A.1)– 20 – eferences [1] For a review, see, P. Langacker, Rev. Mod. Phys. (2008) 1199 [arXiv:0801.1345[hep-ph]].[2] B. Holdom, Phys. Lett. B , 196 (1986).[3] J. H. Huh, J. E. Kim, J. C. Park and S. C. Park, Phys. Rev. D , 123503 (2008)[arXiv:0711.3528 [astro-ph]].[4] E. J. Chun and J. C. Park, JCAP (2009) 026 [arXiv:0812.0308 [hep-ph]].[5] Z. Kang, T. Li, T. Liu, C. Tong and J. M. Yang, arXiv:1008.5243 [hep-ph].[6] S. Gopalakrishna, S. J. Lee and J. D. Wells, Phys. Lett. B (2009) 88[arXiv:0904.2007 [hep-ph]].[7] Y. Mambrini, arXiv:1006.3318 [hep-ph].[8] K. Cheung and T. C. Yuan, JHEP , 120 (2007) [arXiv:hep-ph/0701107].[9] D. Feldman, Z. Liu and P. Nath, Phys. Rev. D , 115001 (2007)[arXiv:hep-ph/0702123].[10] F. Fucito, A. Lionetto, A. Mammarella and A. Racioppi, Eur. Phys. J. C , 455(2010) [arXiv:0811.1953 [hep-ph]].[11] C. Bouchiat and P. Fayet, Phys. Lett. B , 87 (2005) [arXiv:hep-ph/0410260].[12] P. Fayet, Phys. Rev. D , 115017 (2007) [arXiv:hep-ph/0702176].[13] J. Kumar and J. D. Wells, Phys. Rev. D (2006) 115017 [arXiv:hep-ph/0606183].[14] W. F. Chang, J. N. Ng and J. M. S. Wu, Phys. Rev. D (2006) 095005[Erratum-ibid. D (2009) 039902] [arXiv:hep-ph/0608068].[15] Z. Ahmed et al. [The CDMS-II Collaboration], arXiv:0912.3592 [astro-ph.CO].[16] E. Aprile et al. [XENON100 Collaboration], arXiv:1005.0380 [astro-ph.CO].[17] K. S. Babu, C. F. Kolda and J. March-Russell, Phys. Rev. D , 6788 (1998)[arXiv:hep-ph/9710441].[18] C. Amsler et al. [Particle Data Group], Phys. Lett. B , 1 (2008).[19] T. Teubner, K. Hagiwara, R. Liao, A. D. Martin and D. Nomura, arXiv:1001.5401[hep-ph].[20] G. Bertone, D. Hooper and J. Silk, Phys. Rept. , 279 (2005)[arXiv:hep-ph/0404175].[21] E. Komatsu et al. , arXiv:1001.4538 [astro-ph.CO]. – 21 –
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