Light neutralino dark matter in $U(1)_{X}$SSM

There are several existences of dark matter in the universe, and dark matter contribution is more important than the visible matter. The earliest and the most compelling evidences for dark matter are the luminous objects that move faster than one expectsrotation1 . The other evidences for the dark matter can be found in Refs.account1 ; account2 . Besides the gravitational interaction, dark matter can take part in weak interactionOE3 . To keep the relic density of dark matter, dark matter should be stable and live long. People have paid much attention to dark matter for many years, but they have not known its mass and interaction property. The non-baryonic matter density is $\Omega h^{2}=0.1186\pm 0.0020$ pdg , and the standard model(SM) can not explain this problem. It implies that there must be new physics beyond the SM. From the present researches, axions, sterile neutrinos, weakly interacting massive particles (WIMPs)OE3 ; WIMP etc. are dark matter candidates.

Considering the shortcoming of SM, physicists extend it and obtain a lot of extended models. In these new models, the minimal supersymmetric standard model (MSSM)MSSM is the favorite one, where the lightest neutralino can be dark matter candidateNDMMSSM . Furthermore, MSSM is also extended by people, and its U(1) extensions are interestingUMSSM1 ; UMSSM2 . In this work, we extend the MSSM with the $U(1)_{X}$ gauge groupU1X . On the base of MSSM, we add three right-handed neutrinos and three singlet Higgs superfields $\hat{\eta},~{}\hat{\bar{\eta}},~{}\hat{S}$. The right-handed neutrinos can not only give tiny masses to light neutrinos but also produce the lightest scalar neutrino possessing dark matter character. Our $U(1)_{X}$ extension of MSSM is called as $U(1)_{X}$SSMSarah ; ZSMJHEP , which relieves the so called little hierarchy problem that is in the MSSM. The baryon number violating operators are avoided because the $U(1)_{X}$ gauge symmetry breaks spontaneously. So, the proton is stable.

In $U(1)_{X}$SSM, there are the terms $\mu\hat{H}_{u}\hat{H}_{d}$ and $\lambda_{H}\hat{S}\hat{H}_{u}\hat{H}_{d}$. $\hat{S}$ is the singlet Higgs superfield and possesses a non-zero VEV ($v_{S}/\sqrt{2}$). Therefore, $U(1)_{X}$SSM has an effective $\mu_{eff}=\mu+\lambda_{H}v_{S}/\sqrt{2}$, which will probably relieve the $\mu$ problem. As discussed in Ref.NMSSM , one can particularly put the $\mu$ term to zero by a redefinition $\frac{v_{S}}{\sqrt{2}}\rightarrow\frac{v_{S}}{\sqrt{2}}-\frac{\mu}{\lambda_{H}}$. The singlet $S$ can improve the lightest CP-even Higgs mass at tree level. Then large loop corrections to the 125 GeV Higgs mass are not necessary. $S$ can also make the second light neutral CP-even Higgs heavy at TeV order. This can easily satisfy the constraints for heavy Higgs from experiments(such as LHC). If we delete S, the second light neutral CP-even Higgs is light, whose mass varies from 150 GeV to 400 GeV. Considering the limit for the heavy neutral CP-even Higgs, we should make the second light neutral CP-even Higgs heavier. The singlet $S$ can produce this effect.

In our previous workZSMJHEP , the lightest CP-even scalar neutrino is supposed as dark matter candidate in the framework of $U(1)_{X}$SSM. Its relic density and the cross section scattering from nucleus have been researched in detail. Some works of scalar neutrino dark matter can be found in Refs.Sneudark1 ; Sneudark2 ; CJJSN ; SND . Here, we study the lightest neutralino as dark matter candidateNDMEMSSM . In the base of the neutral bino ($\tilde{B}$), wino ($\tilde{W}^{0}$) and higgsinos ($\tilde{H}^{0}_{d},\tilde{H}^{0}_{u}$), the neutralino mass eigenstates in the MSSM has the parameters $\tan\beta,~{}M_{1},~{}M_{2}$ and $\mu$. The lower limit on the lightest neutralino $\chi^{0}_{1}$ mass is about 46 GeV, which can be derived from the Large Electron Positron (LEP) chargino mass limitDELP . While, this limit increases to well above 100 GeV, in the constrained MSSM (cMSSM)cMSSM . In pMSSM, the authors research the lightest neutralino below 50 GeV satisfying the constraints from LHC and XENON100pMSSM . Many people have studied the phenomenology of lightest neutralino in MSSMNDMMSSM and there are a lot of works of neutralino dark matter in several models. They enrich the dark matter research and give light to the direct research of dark matter.

After this introduction, some content of $U(1)_{X}$SSM is introduced in section II. In section III, we suppose the lightest neutralino as a dark matter candidate and we study its relic density. The direct detection of the lightest neutralino scattering off nuclei is reseached in section IV, which includes both the spin-independent cross section and spin-dependent cross section. The numerical results of the relic density and cross sections of dark matter scattering are all calculated in section V. We give our discussion and conclusion in section VI. Some formulae are collected in the appendix.

Extending the local gauge group from $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$ to $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes U(1)_{X}$ and adding three Higgs singlets $\hat{\eta},~{}\hat{\bar{\eta}},~{}\hat{S}$, right-handed neutrinos $\hat{\nu}_{i}$ to MSSM, one can obtain $U(1)_{X}$SSMZSMJHEP . The introduction of right-handed neutrinos can explain the neutrino experiments. The mass squared matrix of CP-even Higgs is $5\times 5$, because the CP-even parts of $\eta,~{}\bar{\eta},~{}S$ mix with the neutral CP-even parts of $H_{u},~{}H_{d}$. We take into account the one loop corrections for the lightest CP-even Higgs with 125 GeV. The condition is similar for the CP-odd Higgs, whose mass squared matrix is also $5\times 5$. The sneutrinos are departed into CP-even sneutrinos and CP-odd sneutrinos, whose mass squared matrixes are both $6\times 6$. Here, we show the $U(1)_{X}$ charges of the MSSM superfields: $\hat{Q}_{i}(0)$, $\hat{u}^{c}_{i}(-\frac{1}{2})$, $\hat{d}^{c}_{i}(\frac{1}{2})$, $\hat{L}_{i}(0)$, $\hat{e}^{c}_{i}(\frac{1}{2})$, $\hat{H}_{u}(\frac{1}{2})$, $\hat{H}_{d}(-\frac{1}{2})$. In table I, the superfields beyond MSSM are collected in detail.

The concrete forms of the Higgs superfields are

$v_{u}$ and $v_{d}$ are the VEVs of the Higgs doublets $H_{u}$ and $H_{d}$. While, $v_{\eta}$,  $v_{\bar{\eta}}$ and $v_{S}$ are the VEVs of the Higgs singlets $\eta$, $\bar{\eta}$ and $S$. The angles $\beta$ and $\beta_{\eta}$ are defined as $\tan\beta=v_{u}/v_{d}$ and $\tan\beta_{\eta}=v_{\bar{\eta}}/v_{\eta}$.

The sneutrino fields $\tilde{\nu}_{L}$ and $\tilde{\nu}_{R}$ read as

We show the superpotential and the soft breaking terms in $U(1)_{X}$SSM

Here, $\mathcal{L}_{soft}^{MSSM}$ represents the soft breaking terms of MSSM. Obviously, the $U(1)_{X}$SSM is more complicated than the MSSM. In our previous work, $Y^{Y}$ represents the $U(1)_{Y}$ charge and $Y^{X}$ denotes the $U(1)_{X}$ charge. We have proven that $U(1)_{X}$SSM is anomaly free, and the details can be found in Ref.ZSMJHEP . In $U(1)_{X}$SSM, there are two Abelian groups $U(1)_{Y}$ and $U(1)_{X}$, which cause the gauge kinetic mixing. This effect is the characteristic beyond MSSM and it can also be induced through RGEs.

We write the covariant derivatives of $U(1)_{X}$SSM in the general form

with $A_{\mu}^{\prime Y}$ ($A^{\prime X}_{\mu}$) denotes the gauge field of $U(1)_{Y}$ ($U(1)_{X}$). Considering the fact that the two Abelian gauge groups are unbroken, we change the basis through a correct matrix $R$ and redefine the $U(1)$ gauge fields

Different from MSSM, the $U(1)_{X}$SSM gauge bosons $A^{X}_{\mu},~{}A^{Y}_{\mu}$ and $V^{3}_{\mu}$ mix together at the tree level. In the basis $(A^{Y}_{\mu},V^{3}_{\mu},A^{X}_{\mu})$, the corresponding mass matrix reads as

with $v^{2}=v_{u}^{2}+v_{d}^{2}$ and $\xi^{2}=v_{\eta}^{2}+v_{\bar{\eta}}^{2}$. One can diagonalize the above mass matrix by an unitary matrix including two mixing angles $\theta_{W}$ and $\theta_{W}^{\prime}$. $\theta_{W}$ is Weinberg angle and $\theta_{W}^{\prime}$ is defined as

The lightest neutralino is supposed as dark matter candidate, and we obtain the mass matrix of neutralino in the basis $(\lambda_{\tilde{B}},\tilde{W}^{0},\tilde{H}_{d}^{0},\tilde{H}_{u}^{0},\lambda_{\tilde{X}},\tilde{\eta},\tilde{\bar{\eta}},\tilde{s})$. This is caused by the super partners of the added three Higgs singlets and new gauge boson, which mix with the MSSM neutralino superfields.

The concrete forms of $\mathcal{A}$, $\mathcal{B}$ and $\mathcal{C}$ are

This matrix is diagonalized by $N$

It is too difficult to obtain exactly the analytic forms of the eigenvalues, eigenvectors and N for $M_{\tilde{\chi}^{0}}$. With some supposition, we can deduce the lightest neutralino mass and eigenvector approximately. Comparing with $\mathcal{A}$ and $\mathcal{C}$, the matrix $\mathcal{B}$ is very small. In this condition, we can use $Z_{N}^{T}$ to simplify $M_{\tilde{\chi}^{0}}$ with matrix $\zeta$, whose elements are all small parameters of the order $\mathcal{B}/\mathcal{A}$.

$\zeta^{T}$ can be calculated from the equation $\mathcal{A}\zeta^{T}+\mathcal{B}-\zeta^{T}\mathcal{C}=0$. If we take the simplest approximation, it is

In this work, we suppose the lightest neutralino $m_{\chi^{0}_{1}}$ is different from the MSSM condition, that is to say $m_{\chi^{0}_{1}}$ dominantly comes from the matrix $\mathcal{C}$ and MSSM neutralinos in the matrix $\mathcal{A}$ are heavy. Therefore, we calculate the mass eigenstates of $\mathcal{C}$, which is tedious to solve the common quartic equation with one unknown quantity. Considering the constraint that $\tan\beta_{\eta}$ is near 1, we use $s_{m}=v_{\bar{\eta}}-v_{\eta}$ with the relation $v_{\eta}\gg s_{m}$. So, $\mathcal{C}$ turns to

with $\lambda_{C}^{\prime}=\lambda_{C}/\sqrt{2}$.

The eigenvalues of $\mathcal{C}$ are deduced to the leading order according to the small parameter $s_{m}$.

We take $M_{BL}$ and $m_{\tilde{s}\tilde{s}}$ are both positive parameters. To satisfy the constraint from $m_{Z^{\prime}}$, $v_{\eta}$ is large and bigger than $v_{S}$. So it is easy to see that $m_{\chi^{0}_{a}}$ and $m_{\chi^{0}_{c}}$ are large values. Using $m_{\tilde{s}\tilde{s}}\gg M_{BL}$ and $\mu_{\eta}$, $m_{\chi^{0}_{d}}$ is smaller than $m_{\chi^{0}_{b}}$, so $m_{\chi^{0}_{d}}$ is the lightest neutralino mass $m_{\chi^{0}_{1}}$. For the lightest neutralino mass, we consider the correction at the order $s_{m}$.

At the leading order, the eigenvector of $m_{\chi^{0}_{d}}$ is

Here, $a_{0}$ is a small parameter. The $s_{m}$ correction to eigenvector $V^{(0)}_{\chi^{0}_{d}}$ is $V^{(1)}_{\chi^{0}_{d}}$

$c_{1}$ is much smaller than $b_{1}$, then $V^{(1)}_{\chi^{0}_{d}}$ can be simplified as

In the whole, the eigenvector of the lightest neutralino is dominatly composed by the linear combination of $\tilde{\eta}$ and $\tilde{\bar{\eta}}$.

In the MSSM, the lightest CP-even Higgs mass at tree level is no more than 90 GeV, and the loop corrections to the lightest CP-even Higgs mass can be large. Including the leading-log radiative corrections from stop and top particles higgsOL , we write the mass of the lightest CP-even Higgs boson in the following form

Here, $m_{h_{1}}^{0}$ represents the lightest Higgs boson mass at tree level and $\Delta m_{h}^{2}$ is shown analytically

with $\alpha_{3}$ denoting the strong coupling constant. The parameter $\tilde{A}_{t}$ is $\tilde{A}_{t}=A_{t}-\mu\cot\beta$ with $A_{t}$ representing the trilinear Higgs stop coupling. $M_{\tilde{T}}=\sqrt{m_{\tilde{t}_{1}}m_{\tilde{t}_{2}}}$ and $m_{\tilde{t}_{1,2}}$ are the stop masses. To save space in the text, other used couplings are collected in the appendix.

Supposing the lightest neutralino($\chi^{0}_{1}$) as dark matter candidate, we calculate the relic density. The constraint of dark matter relic density is severe, and the concrete value is $\Omega_{D}h^{2}=0.1186\pm 0.0020$ pdg . The $\chi^{0}_{1}$ number density $n_{\chi^{0}_{1}}$ should satisfy the Boltzmann equation boltzmann ; XFBO1

For $\chi^{0}_{1}$, we take into account self-annihilation and co-annihilation with another particle $\phi$. At the temperature $T_{F}$, the annihilation rate of $\chi^{0}_{1}$ is approximately equal to the Hubble expansion rate, and the lightest neutralino freezes out. We suppose $\chi^{0}_{1}$ is the lightest SUSY particle and $m_{\phi}$ is larger than $m_{\chi^{0}_{1}}$. The relevant formulae are ExpYD

After we study the self-annihilation cross section $\sigma(\chi^{0}_{1}\chi^{0}_{1}\rightarrow$ anything) and co-annihilation cross section $\sigma(\chi^{0}_{1}\phi\rightarrow$ anything), $\langle\sigma v\rangle_{SA}$ and $\langle\sigma v\rangle_{CA}$ are gotten. The annihilation results can be written as $\sigma v_{rel}=a+bv_{rel}^{2}$ in the mass center frame. Here, $v_{rel}$ is the relative velocity of the two particles in the initial states. Using the following formula, we can approximately calculate the freeze-out temperature ($T_{F}$)NDMMSSM ; XFCW ; XFBO1

with $M_{Pl}$ denoting the Planck mass.

The relativistic degrees of freedom with mass less than $T_{F}$ is represented by $g_{*}$. The cold non-baryonic dark matter density is simplified in the following formrotation1 ; XFBO1 ; zhaosm .

It is well known that, the self-annihilation processes are dominant in general condition. We show the researched concrete self-annihilation processes: $\chi^{0}_{1}+\chi^{0}_{1}\rightarrow A+B$, A and B represent final states $(Z,~{}h),~{}(W,~{}W),~{}(Z,~{}Z),~{}(h,~{}h),~{}(\bar{u}_{i},u_{i}),~{}(\bar{d}_{i},~{}d_{i}),~{}(\bar{l}_{i},~{}l_{i}),~{}(\bar{\nu}_{i},~{}\nu_{i})$. Here $i=1,~{}2,~{}3$ and $h$ represents the lightest CP-even Higgs. The neutrinos in final state are just three light neutrinos not including heavy neutrinos.

For co-annihilation processes, if the mass of another particle is almost equal to the mass of $\chi^{0}_{1}$, they give considerable contributions to the annihilation cross section.

a. The lightest neutralino $\chi^{0}_{1}$ annihilates with heavier neutralinos $\chi^{0}_{k}(k=2\dots 8)$, whose final states are same as those produced by self-annihilation processes.

b. $\chi^{0}_{1}+\chi^{-}\rightarrow\{(\nu,l^{-}),(\bar{u},d),(W^{-},Z),(W^{-},\gamma),(W^{-},h^{0})\}$. The corresponding processes are obtained by the charge conjugate transformation.

c. $\chi^{0}_{1}+\tilde{L}^{-}\rightarrow\{(\gamma,l^{-}),(Z,l^{-}),(h^{0},l^{-})\}$. Similar as the condition b, condition c also has charge conjugate processes.

d. $\chi^{0}_{1}+\tilde{\nu}^{R}(\tilde{\nu}^{I})\rightarrow\Big{\{}(\nu,Z),(l^{-},W^{+}),(l^{+},W^{-})$}.

The co-annihilation between $\chi_{1}^{0}$ and scalar quarks($\tilde{U},~{}\tilde{D}$) are neglected, because scalar quark masses are very heavy and much larger than $m_{\chi^{0}_{1}}$.

The experiment constraints for the direct detection of dark matter become strict more and more. The lightest neutralino scatters off nucleus, and the process is $\chi^{0}_{1}+q\rightarrow\chi^{0}_{1}+q$. The exchanged particles can be CP-even Higgs $H^{0}_{j}$, CP-odd Higgs $A^{0}_{j}$, gauge bosons $Z,~{}Z^{\prime}$. For the CP-odd Higgs $A^{0}_{j}$ contribution, there are two suppression factors: 1. The Yukawa coupling $Y_{q}$ of light quark; 2. The operators $\bar{\chi}^{0}_{1}\chi^{0}_{1}\bar{q}\gamma_{5}q$ and $\bar{\chi}^{0}_{1}\gamma_{5}\chi^{0}_{1}\bar{q}\gamma_{5}q$ are suppressed by the factors $q^{2}$ and $q^{4}$ respectivelyLJandHe . Therefore, we neglect the CP-odd Higgs contribution. Because neutralino is Majorana particle, the operator $\bar{\chi}^{0}_{1}\gamma_{\mu}\chi^{0}_{1}\bar{q}\gamma^{\mu}q$ disappears. The dominant operators at quark level are $\bar{\chi}^{0}_{1}\chi^{0}_{1}\bar{q}q$ and $\bar{\chi}^{0}_{1}\gamma_{\mu}\gamma_{5}\chi^{0}_{1}\bar{q}\gamma^{\mu}\gamma_{5}q$ obtained from CP-even Higgs and vector bosons $Z,~{}Z^{\prime}$ contributionsLJandHe .

The quark level operators should be converted to the effective nucleus operators. To convert the operator $\bar{\chi}^{0}_{1}\chi^{0}_{1}\bar{q}q$, we use the following formulaeLJandHe

Integrating out heavy quark loops, the coupling to gluons is induced, which is included in $f_{N}$. We show the concrete values of the parameters $f_{Tq}^{(N)}$ DarkSUSY1 ,

$\bar{\chi}^{0}_{1}\gamma_{\mu}\gamma_{5}\chi^{0}_{1}\bar{q}\gamma^{\mu}\gamma_{5}q$ is a spin-dependent operator, which is converted to the effective nucleus operator with the following formulae.

with $s^{(N)}_{\mu}$ denoting the spin of nucleus. In the numerical calculation, we use the parameters of DarkSUSY

For the spin-independent operator $\bar{\chi}^{0}_{1}\chi^{0}_{1}\bar{q}q$, the scattering cross section reads asLJandHe

with $Z_{p}$ denoting the number of proton, and $A$ representing the number of atom.

The scattering cross section for the spin-dependent operator $\bar{\chi}^{0}_{1}\gamma_{\mu}\gamma_{5}\chi^{0}_{1}\bar{q}\gamma^{\mu}\gamma_{5}q$ is shown as LJandHe