3.5 keV X-ray line and R-parity conserving supersymmetry

Abstract

Abstract

We present some R-parity conserving supersymmetric models which can accommodate the 3.5 keV X-ray line reported in recent spectral studies of the Perseus galaxy cluster and the Andromeda galaxy. Within the Minimal Supersymmetric Standard Model (MSSM) framework, the dark matter (DM) gravitino (or the axino) with mass of around 7 keV decays into a massless neutralino (bino) and a photon with lifetime 10^sup 28^ sec. The massless bino contributes to the effective number of neutrino species N ^sub eff^ and future data will test this prediction. In the context of NMSSM, we first consider scenarios where the bino is massless and the singlino mass is around 7 keV. We also consider quasi-degenerate bino-singlino scenarios where the mass scale of DM particles are O(GeV) or larger. In such a scenario we require the mass gap to generate the 3.5 keV line. We comment on the possibility of a 7 keV singlino decaying via R parity violating couplings while all other neutralinos are heavy.

Published for SISSA by Springer Received: July 8, 2014

Revised: October 4, 2014 Accepted: October 6, 2014 Published: November 5, 2014

Bhaskar Dutta,a Ilia Gogoladze,b,1 Rizwan Khalidc and Qaisar Shab

aMitchell Institute of Fundamental Physics and Astronomy,

Department of Physics and Astronomy, Texas A&M University,

College Station, TX 77843-4242, U.S.A.

bBartol Research Institute, Department of Physics and Astronomy, University of Delaware, Newark, DE 19716, U.S.A.

cDepartment of Physics, School of Natural Sciences, National University of Sciences & Technology, H-12, Islamabad, Pakistan

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We present some R-parity conserving supersymmetric models which can accommodate the 3.5 keV X-ray line reported in recent spectral studies of the Perseus galaxy cluster and the Andromeda galaxy. Within the Minimal Supersymmetric Standard Model (MSSM) framework, the dark matter (DM) gravitino (or the axino) with mass of around 7 keV decays into a massless neutralino (bino) and a photon with lifetime 1028 sec. The

massless bino contributes to the e ective number of neutrino species Ne and future data will test this prediction. In the context of NMSSM, we rst consider scenarios where the bino is massless and the singlino mass is around 7 keV. We also consider quasi-degenerate bino-singlino scenarios where the mass scale of DM particles are O(GeV) or larger. In such a scenario we require the mass gap to generate the 3.5 keV line. We comment on the possibility of a 7 keV singlino decaying via R parity violating couplings while all other neutralinos are heavy.

Keywords: Supersymmetry Phenomenology

ArXiv ePrint: 1407.0863

1On leave of absence from: Andronikashvili Institute of Physics, 0177 Tbilisi, Georgia.

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2014)018

Web End =10.1007/JHEP11(2014)018

3.5 keV X-ray line and R-parity conserving supersymmetry

JHEP11(2014)018

Contents

1 Introduction 1

2 MSSM 22.1 Gravitino dark matter and massless bino 32.2 Axino dark matter and massless bino 5

3 NMSSM 6

4 Conclusion 9

A Two massless neutralinos in the NMSSM 10

1 Introduction

Two recent independent studies [1, 2] based on X-ray observation data show a photon emission line at 3.5 keV energy in the spectra from Perseus galaxy cluster and the Andromeda galaxy. This observation can be interpreted as a possible signal of dark matter (DM) decay with the emission of a 3.5keV photon, with the DM mass (mDM) and lifetime (DM) given by,

mDM [similarequal] 7 keV

DM [similarequal] 2 [notdef] 1027 1028 sec. (1.1)

A variety of explanations of this line have already been proposed [332]. However, there exist just a few supersymmetric scenarios which contain such a light neutral particle. For instance, it could be an axino [2729], gravitino [30, 31] or neutralino (bino) [32]. These particles are able to produce the observed X-ray line [1, 2] by decaying through R-parity violating processes [32] to a photon and neutrino, for example.

In this paper we present some simple scenarios which can accommodate the 3.5 keV X-ray line in the context of R-parity conserving supersymmetry (SUSY). They include the minimal supersymmetric standard model (MSSM) and Next-to-MSSM (NMSSM). It is interesting to note that in the MSSM, the lightest neutralino can be massless [3335] while satisfying the current experimental constraints. In order to realize this scenario [33], we assume that the soft supersymmetry breaking (SSB) MSSM gaugino masses are arbitrary, and we impose the requirement that the neutralino mass matrix at the weak scale has zero determinant. This can be achieved by suitable choice of parameters, while having very small ([lessorsimilar] 1 eV) or even zero mass bino, with the charginos (and the next to lightest neutralino ~

~01)

heavier then 420 GeV to satisfy the mass bounds on the chargino from LHC [36, 37]. In our scenarios where the near massless bino is accompanied by a 7 keV gravitino, axino, or

1

JHEP11(2014)018

singlino which behave as warm DM. arising in di erent models around keV scale giving rise to warm DM. The 7 keV DM particle decays to a bino and a photon with an appropriate long lifetime to explain the observed X-ray line. The warm dark matter scenario which is under investigation for a long time [38], proposes solution to the missing satelite problem of the local group of galaxies [39]. The massless bino contributes to the e ective number of neutrino species, Ne , which is expected to be strongly constrained in the near future.

We also consider an almost degenerate bino-singlino scenario in the NMSSM framework, such that the mass scale of cold DM particles are O(GeV) or larger.

We can retain gauge coupling unication in the presence of non-universal gaugino masses at MGUT, which are realized via non-singlet F -terms compatible with the underlying grand unied theory (GUT) [4044]. Nonuniversal gauginos can also be generated from an F -term which is a linear combination of two distinct elds of di erent dimensions [45]. It is also possible to have non-universal gaugino masses [46] in the SO(10) GUT with unied Higgs sector [47, 48], or utilize two distinct sources for supersymmetry breaking [49]. In general, in the gauge mediated supersymmetry breaking (GMSB) scenario, all gaugino masses can be independent of each other [50, 51]. With so many distinct possibilities available for realizing non-universal gaugino masses while keeping universal sfermion mass (m0) at MGUT, we employ non-universal masses for the MSSM gauginos in our study without further justication.

One of the motivations for non-universal gauginos can be related to the interplay between the 125 GeV Higgs boson and the explanation of the apparent muon g-2 anomaly. A universal SSB mass term for sfermions (m0) is needed to suppress avor-changing neutral current processes.1 On the other hand, in order to accommodate the 125 GeV [53, 54] light CP even Higgs boson mass and to resolve the discrepancy between the SM and the measurement of the anomalous magnetic moment of the muon [55, 56] in the framework of universal sfermion SSB masses, we need to have non-universal gaugino masses at MGUT [57].

The outline of our paper is as follows. In section 2, we discuss the 3.5 keV line in the context of MSSM scenarios. In section 3, we discuss possible NMSSM scenarios, followed with our conclusion in section 4. In the appendix we present technical details regarding two massless neutralinos in the NMSSM and provide a few representative solutions of interest.

2 MSSM

In this section, we outline several scenarios that can accommodate a 3.5 keV X-ray line in the MSSM. Let us start by examining how it might be possible to obtain a massless neutralino in the framework of the MSSM. The neutralino mass matrix in the gauge eigenbasis 0 = ( ~B, 0,0d,0u)T has the form [52]

M1 0 MZswc MZcws

0 M2 MZcwc MZcws

MZswc MZcwc 0 MZsws MZcws 0

2

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M~~0 =

1See for instance [52] and references therein.

0

B

B

B

B

B

@

1

C

C

C

C

C

A

. (2.1)

Here M1, M2 are the supersymmetric gaugino mass parameters for the U(1) and SU(2) sector respectively, while is the bilinear Higgs mixing parameter. MZ denotes the Z gauge-boson mass and sw sin w, cw cos w, where w is the weak mixing angle.

s sin , c cos , while tan is the ratio of the vacuum expectation values (VEVs)

of the MSSM Higgs doublets.To realize a massless netralino [33, 35], the following relation must be satised:

M1 = M2M2Z sin(2 )s2w

M2 M2Z sin(2 )c2w

2M2Zs2w

tan . (2.2)

Implementing the chargino mass bound ([notdef][notdef], M2) > 420 GeV in eq. (2.2) leads to M1

(M2, [notdef][notdef]). In the appendix we give one example of an MSSM scenario with very small LSP

neutralino (mostly bino) mass. Such a bino is consistent with current experimental data from LEP, structure formation etc [35]. The LHC provides constraints on the next to lightest neutralino, chargino, and slepton masses when the lightest neutralino is almost massless.

The relation in eq. (2.2) has been obtained at tree level, but radiative corrections do not signicantly modify it. Notwithstanding radiative corrections, since M1, M2 and are free parameters, there is no problem to ensure that the determinant in eq. (2.1) is zero. Thus, it is possible to have an essentially massless neutralino by ne-tuning the parameters in the framework of the MSSM, and an example is presented in the appendix.

The existence of a near massless bino, however, would contribute to Ne Ne Ne ,SM = 1. The reason for this is that the essentially massless bino decouples

from the thermal background around the same time as the neutrinos. The decoupling temperature also depends on the slepton mass which we take around the weak scale. However, if the slepton mass increases, the decoupling temperature also increases, e.g., if the slepton mass is 10 TeV, then the decoupling temperature will be O(GeV). The present observational bound on Ne from Planck + WMAP9 + ACT + SPT + BAO + HST at 2 is Ne = 0.48+0.480.45 [58]. The value of Ne depends on Hubble constant where there is a discrepancy between Planck and HST [59]. A reconciliation can occur using larger Ne [60]. The new BICEP2 data [61] also requires a larger Ne (=0.81 [notdef]0.25) in order

to reconcile with the Planck data [62, 63]. Future data hopefully will settle this issue.

2.1 Gravitino dark matter and massless bino

One way to accommodate a 3.5 keV X-ray line via a massless neutralino comes from the gauge mediated SUSY breaking (GMSB) scenario. As a consequence of the avor blind gauge interactions responsible for generating the SSB terms,2 this senario provides a compelling resolution of the SUSY avor problem. In both the minimal [64] and general [50, 51] GMSB versions, the gravitino, which is the spin 3/2 superpartner of the graviton, acquires mass through spontaneous breaking of local supersymmetry. The gravitino mass can be

1 eV 100 TeV. Additionally, in the general GMSB scenario, the SSB mass terms for

the MSSM gauginos are arbitrary. In particular, it is possible to have a massless neutralino

2See [64] and original references therein.

3

JHEP11(2014)018

eG ! ~~01 + decay.

(essentially a bino) in this framework. With all other sparticles being much heavier, the gravitino dominantly decays to the neutralino (bino) and photon ( ! ~

Figure 1.

~01 + ).

The relevant diagram for this decay is shown in gure 1, and the decay rate is given by [65]

(

eG ! ~~01 ) =

cos 2W m3[tildewide]G

. (2.3)

Using eq. (2.3) and assuming the gravitino mass to be 7 keV, the gravitino lifetime is estimated to be 3 [notdef] 1029 sec, which is approximately a factor of 10 more than what we need

which can be di cult to obtain. However, physics around the Planck scale MP is largely unknown. It has been noted in refs. [6669] that the fundamental mass scale (M ) can be reduced to MP /pN in the presence of a nonzero number of degrees of freedom (N). In fact, it is shown that the scale for quantum gravity in 4D becomes the new scale M where the clas

sical gravity becomes very strong and below this scale no quasi-classical black hole can exist. This becomes the scale of the non-renormalizable operators as well since this mass scale marks the new cuto . In this way the cuto scale can be reduced as required in eq. (2.3).

It is possible to envision a larger e ective coupling~

~01 coupling by assuming new particles providing additional contributions to the e ective~

~01 coupling. For example,

there could be a new operator~

~01 fscalar/M , which can arise from the fundamental interactions, ffermion and fscalar ~

~01ffermion. By integrating the fermion ffermion at the scale M we can get the above operator. The scalar fscalar can have a VEV < fscalar > M

to give us a new tree-level O(1) contribution to the~

~01 coupling. It is possible to have large contributions from many such diagrams to induce a large e ective coupling to yield the desired lifetime for the gravitino as needed in eq. (2.3). However, SUSY needs to be broken in order to preserve equivalence principle.

One important issue for gravitino dark matter is the reproduction of the correct dark matter relic density. The initial thermal abundance is diluted because of a late reheat temperature (TR) arising from heavy eld/moduli decay. The relic density ( h2) of gravitinos which arise from the scattering of gluinos, squarks etc. is given by [7073],

h2 0.27 [parenleftbigg]

To realize h2 0.1 with m~g [greaterorsimilar] 1.4 TeV and M 1017 GeV, we require TR [lessorsimilar]

104 GeV.

4

JHEP11(2014)018

8M2P

100 GeV m

eG

TR 1010 GeV

[parenrightbigg] [parenleftbigg]

[parenrightbigg] [parenleftBig]

m~g

1 TeV

2 2.4 [notdef] 1018 GeV

M

2. (2.4)

JHEP11(2014)018

Figure 2. ! ~

2.2 Axino dark matter and massless bino

A very compelling way of solving the strong CP problem is via the Peccei-Quinn (PQ) mechanism [74], which yields a light pseudo-scalar eld (axion a) associated with the spontaneously broken global U(1) symmetry. An inevitable prediction from a combination of PQ mechanism and low scale supersymmetry is the existence of the supersymmetric partners of the axion, the axino (a) and saxion s [76]. The axion supereld A can be expressed as,

A = 1

p2(s + ia) +

~B B . (2.6)

Here Y = Y 2/4 is the hypercharge gauge coupling constant and CY is a model dependent coupling associated with the U(1)Y gauge anomaly interaction. The axion decay constant is denote by fa. The axino decays to a neutralino (bino) and photon without requiring R-parity violating interaction. The relevant diagram for this decay is shown in gure 2,

3See [75] and references therein.

4See [79] and references therein.

5

~01 + decay.

p2 + FA , (2.5)

where FA denotes the auxiliary eld and is a Grassmann coordinate. In general, the axino mass is very model dependent3 and can lie anywhere from eV to multi-TeV. It was shown that a stable axino with keV mass is a viable warm dark matter candidate [77, 78]. The3.5 keV X-ray line can be explained by a decaying axino dark matter. For this purpose, the authors in [2729] introduce R-parity violating couplings, with strength 101 103 in order to accommodate desired axino life time.

In this paper, we propose an alternative way to explain the X-ray line using 7 keV axino dark matter. As mentioned above, within the MSSM framework, it is possible to have a massless neutralino in the spectrum which is consistent with all experimental constraints. We know that the axino couples to the gauginos and gauge bosons via the anomaly induced term. In particular, we are interested in the interaction of the axino to the bino ( ~B) and the hypercharge vector boson (B). This interaction takes the form,4

i Y CY

16fa 5[ , ]

and the decay rate is given by [78],

( ! ~01 ) =

m3~af2a , (2.7)

where m~a is axino mass, C2a~ = (Cy/ cos W )Z11, and Z11 denotes the bino part of the lightest neutralino.

The axino lifetime can be expressed as:

( ! ~01 ) = 1.3 [notdef] 1023sec [parenleftbigg]

fa 1012 GeV

2emC2a~

1283

2 7.1 keV m~a

3

(2.8)

From eq. (2.8) we see that we need to have fa 1014 GeV is required. On the other hand,

in order not to overproduce axion dark matter, we need to have fa [lessorsimilar] 1012 GeV is preferred. One resolution of this is to invoke a small initial axion mis-alignment angle 0.10.01,5

which yields the required axion dark matter abundance while allowing fa 1014 GeV. An

alternative solution [8184] is to add additional massive elds whose late decay can inject substantial entropy into the universe at times after axion oscillations begin, but before BBN starts.

It is, furthermore, possible to have an axion-like particle (and associated axino) [8587] in the low scale spectrum, which may be obtained from string theory. Axino-like particles can decay into a bino and photon. In this case the bound on fa can be more exible and also the coe cient Cy can be suitably adjusted to be O(102) or so, since it is not tied to the solution of the strong CP problem.

3 NMSSM

As shown in the previous section, in the MSSM it is possible to have a massless bino, while keeping all other neutralinos heavier than 400 GeV. In the NMSSM, the neutralinos have a singlino component from the gauge singlet chiral supereld S (with even Z2 matter parity)

added to the MSSM with new terms in the superpotential:

W HuHd + HuHdS

13 S3, (3.1)

Hu and Hd are the standard MSSM Higgs doublets and and are dimensionless couplings. Once the S eld acquires a VEV [angbracketleft]S[angbracketright], we obtain an e ective -term for MSSM Higgs

elds, e = + [angbracketleft]S[angbracketright]. The neutralino mass matrix in the gauge eigenstate basis 0 =

( ~B, 0,0d,0u, s)T has the following form:

MN =

JHEP11(2014)018

0

B

B

B

B

B

B

B

@

M1 0 mZc sW mZs sW 0

0 M2 mZc cW mZs cW 0

mZc sW mZc cW 0 e vs mZs sW mZs cW e 0 vc

0 0 vs vc 2 [angbracketleft]S[angbracketright]

1

C

C

C

C

C

C

C

A

. (3.2)

5See for instance [80] and references therein.

6

It was shown in [34] that a massless neutralino requires that

= 12 [parenleftbigg]

v

2 0.6m2zM2 0.5M22 sin 2 M1M2 .

This solution is obtained for the case when (e , M1, M2) > MZ and the singlino is the

lightest neutralino. We can, however, easily make the lightest neutralino to be mostly bino and the next to lightest neutralino essentially the singlino. The technical details for obtaining two massless neutralinos in the framework of NMSSM are given in appendix A.

In order to explain the 3.5 keV X-ray line, we propose that one of the neutralinos, which is mostly bino, is almost a massless ([lessorsimilar] 1 eV) particle and does not, therefore, contribute to the warm or cold dark matter relic abundance. The second neutralino, in this scenario, is mostly singlino with a mass of 7 keV and gives rise to the correct dark matter relic abundance [88]. The annihilation of thermal NMSSM Higgs produce singlinos, and it was shown that the correct relic abundance requires the singlino mass to be a few keV. Thus,

~~h2

4(1.2)2 5

!2 M3~~Mplc . (3.3)

Here Ms is the mass of the scalar singlet, g(TR) = 228.75, g(T ) = 2, TR 102 105 GeV,

kT = (43g(T )/45)1/2 and T is the present CMB temperature. Choosing = 3 [notdef] 102,

= 1010, M1 = 0.23 GeV and M2 = = 550 GeV (shown in point 1 of table 1 in the

appendix), we can have the masses for the lightest neutralino (mostly bino) and the next to lightest neutralino (mostly singilino) to be essentially massless and 7 keV respectively. This scenario satises the dark matter relic abundance constraint.

The singlino can radiatively decay to a bino and photon with a long lifetime, which allows us to obtain the 3.5 keV X-ray line. The relevant diagram [89] for this decay is shown in gure 3 and the decay rate is given by

(~

~02 ! ~

~01 )

2 2em 83

JHEP11(2014)018

( /3 + 2)v2 sin 2 MsM~~

2 g(T ) g(TR)

TRT 3

kT v4 sin2 2

m3~~2

M2H

. (3.4)

Here we assume that the charginos (m~~+

i ) and charged Higgs (mH

+ ) have approximately

the same mass.

The ~

~02 lifetime can be written as:

(~

~02 ! ~

~01 ) 2 [notdef] 1027sec [parenleftbigg]

MH 105 GeV

2 1010

2. (3.5)

In the NMSSM, an alternative explanation for the 3.5 keV emission line requires one to have two quasi-degenerate neutralinos (bino and singlino), with mass di erence arranged to be 3.5 keV. We present one such example in the appendix. We require the next to

lightest supersymmetric particle (NLSP), which is a mixture of singlino and bino, to be long-lived on cosmological time scales. The decay of this NLSP to the LSP, which again may be a bino-singlino mixture, can explain the 3.5 keV emission line.

7

Figure 3. Decay of NLSP neutralino to the LSP neutralino with the associated emission of a

photon.

The relevant Feynman diagrams for the NLSP neutralino decay are given in gure 3. The decay width is given by [9092],

2em 2 644

( m~)3

m4H[notdef]

m2~, (3.6)

where em is the electromagnetic coupling constant, m~ is the quasi-degenerate mass of the two lightest neutralinos, m~ is their mass splitting, and mH[notdef] is the mass of the charged

Higgs.Assuming m~02 m~

01 1 GeV and m~ 3.5 keV, and as an example we consider

108 and mH[notdef] = 500 GeV in order to have (~02 ! ~01 ) (1027 1028) sec. The

dark matter in this case is cold compared to the previous scenarios.

The singlino/bino dark matter can be produced non-thermally from the decay of some heavy eld/moduli () with a reheat temperature [greaterorsimilar] 2 MeV in order to avoid problems with big bang nucleosynthesis. As shown in [93, 94], if the abundance of DM production (combination of dilution factor due to decay and branching ratio into DM particles) is small enough to satisfy the DM content, the annihilation cross-section of dark matter becomes irrelevant.

The DM abundance is given as nDM/s = min[(nDM/s)obs(3 [notdef] 1026/ < v >f

)(Tf/TR), Y BrDM], where (nDM/s)obs [similarequal] 5 [notdef] 1010(1 GeV/mDM), TR is the reheat tem

perature, Y = 3TR/4m [similarequal] 1/

JHEP11(2014)018

pcm/MP , and BRDM denotes the branching ratio for decay into singlino/bino. The singlino DM does not reach thermal equilibrium after production from the decay of the heavy eld since the decoupling temperature is much larger than the reheat temperature TR.

It is also interesting to note that the singlino can be the lightest sparticle, and it can then decay via some R-parity violating couplings. We present an example in the appendix. A slight change in the parameter values corresponding to the existence of massless neutralinos will make the neutralino mass around keV. A keV scale singlino LSP can decay at loop level in the presence of R-parity violating couplings. Here we consider only the lepton number violation operators:

L[negationslash]R = iLiHuS + ijkLiLjEck + [prime]ijkQiLjdck + iHuLi. (3.7)

8

Figure 4. Decay of LSP neutralino through R-parity violation term.

The neutralino-neutrino mass matrix in the gauge eigenstate basis 0T ( ~B0, 03,0d,0u,, i) is given by

M~~0 = MN T[negationslash]R

[negationslash]R M 3[notdef]3 [parenrightBigg]

, (3.8)

[prime]v1 p2

gv1p2 0 1 + 1[angbracketleft]s[angbracketright] 1vu

g

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where

g

[negationslash]R =

0

B

B

@

[prime]v2 p2

gv2p2 0 2 + 2[angbracketleft]s[angbracketright] 2vu

g

1

C

C

A

, (3.9)

[prime]v3 p2

gv3p2 0 3 + 3[angbracketleft]s[angbracketright] 3vu

and M 3[notdef]3 is the 3 [notdef] 3 light neutrino majorana mass matrix.

One of the dominant diagrams for the decay ~

~01 ! + is given in gure 4, and the

corresponding decay rate is given by

(~

~01 ! ) em

( 1)2

323

~ . (3.10)

Here we assume, for simplicity, that the charged Higgs and charginos have similar masses MH (m~~

+ i

mH

+ ). The singlino lifetime can be expressed as

(~

~01 ! ) 2 [notdef] 1027sec [parenleftbigg]

MH 105GeV

2 1011 1

2, (3.11)

and if we assume 1 3 [notdef] 106, the desired singlino life time is obtained. The LSP

singlino, as mentioned above, can provide the correct DM abundance.

4 Conclusion

In summary, we have presented several scenarios that can accommodate the 3.5 keV X-ray line in the context of R-parity conserving SUSY. In the MSSM, the LSP neutralino can be massless and the gravitino or axino dark matter of mass around 7 keV can decay into the LSP neutralino and a photon with lifetime 1028 sec. To realize this scenario, we assume

that the soft SUSY breaking MSSM gaugino masses are non-universal and they satisfy

9

the requirement that the determinant of the neutralino mass matrix vanishes at the weak scale. This can always be achieved with a suitable choice of parameters, while keeping the charginos (and second lightest neutralino ~

~02) heavier than 420 GeV to avoid the LHC constraint. A keV mass dark matter is of considerable interest since it can provide potential solutions to the missing satellites problems of the Local Group of Galaxies. The massless bino, however, contributes to Ne and future data should seriously test this scenario. In the context of NMSSM, we consider scenarios where the bino is massless and the dark matter singlino mass is around 7 keV. Within the NMSSM, we also consider quasi-degenerate bino-singlino scenarios where the DM mass scale is O(GeV) or larger. We require, in this scenario, a small mass gap to generate the 3.5 keV X-ray line. In passing, we also consider scenarios where the singlino is the lightest SUSY particle, and it decays via R parity violating couplings which give rise to the 3.5 keV X-ray line.

Acknowledgments

We would like to thank R. Allahverdi, Y. Gao for very useful discussions. This work is supported in part by the DOE Grants Nos. DE-FG02-13ER42020 (B.D.) and DE-FG02-12ER41808 (I.G. and Q.S.). I.G. acknowledges support from the Rustaveli National Science Foundation No. 31/98.

A Two massless neutralinos in the NMSSM

The neutralino mass matrix is given in eq. (3.2) and we seek a solution with two massless neutralinos. Assuming that is an eigenvalue of MN , we can write the characteristic

equation in the form

[notdef]MN I5[notdef] = 5 + A 4 + B 3 + C 2 + D + E = 0, (A.1)

where I5 is the 5 [notdef] 5 identity matrix, and A, B, C, D, E, of course, depend on the entries in MN . It is known that A, B, C, D and E are invariants (under similarity transformations) of

the matrix and, in particular, E is the determinant of MN . We can express the coe cients

in eq. (A.1) in terms of the mass eigenstates:

E = m21m22m23m24m25; D =

n

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Xi[negationslash]=j[negationslash]=km2i m2j m2k;

Xi

6=j[negationslash]=k[negationslash]=l

m2i m2j m2k m2l; C =

n

Xi[negationslash]=j[negationslash]=km2i m2j A =

5

B =

n

Xi=1m2i. (A.2)

A necessary and su cient condition for any one eigenvalue to be zero is for the determinant of the matrix to be zero (i.e. E = 0). The quintic characteristic equation then reduces to a quadratic one. Proceeding in this fashion, if we now also set D = 0, we will ensure that two eigenvalues of the mass matrix are zero. It is then possible to adjust the parameters to get the desired small mass eigenvalues.

10

Point 1 Point 2 Point 3

M2 (GeV) 550 500 550 (GeV) 550 500 550x (GeV) 0.0001 1 7 [notdef] 106 tan 30 30 50

M1 0.234 1.267 550 3.5 [notdef] 102 0.5 0.5

1010 109 105 m~~01 (GeV) 6.69 [notdef] 1013 1 7 [notdef] 106

~

~01 composition [similarequal] 100%

~B 99% ~B [similarequal] 100%

m~~02 (GeV) 7 [notdef] 106 1 1.08

~

~02 composition [similarequal] 100%

99% mixture m~~03 (GeV) 498 445 550 m~~04 (GeV) 554 505 554 m~~05 (GeV) 605 560 617

Table 1. Three representative solutions.

While the general expression for the the determinant and the coe cient of in the characteristic equation (variously known as the fourth invariant) is rather complicated, the conditions to obtain two massless neutralinos simplies in the limit of large tan . Setting s ! 1 and c ! 0 in the neutralino mass matrix, we obtain the following conditions for

two massless neutralinos,

D = M1M2( 2v2 + 2) 2 x2(M1 + M2)+

2 xm2Z(M1c2W + M2s2W ) + m2Zv2 2 = 0

E = 2M1M2 x2 m2Z 2(M1c2W + M2s2W ) = 0 (A.3) There can, however, be issues while using this approximation because of the large di erences in orders of magnitudes of the various terms. In practice it is much simpler to numerically ne-tune the parameters in the exact expressions to obtain two zero eigenvalues. We are essentially interested in a quasi-degenerate ([lessorsimilar] 1 GeV) bino-singlino mixture. With

small, there is very little mixing between the singlino and the higgsinos, particularly for [greaterorsimilar] 100 GeV (which is needed as previously explained). Furthermore, if we choose M1, 2 x

1 GeV and M2 [greaterorsimilar] 400 GeV, we should naively expect to get the required neutralino masses.

In table 1 we display three representative solutions that correspond to the three scenarios for obtaining the 3.5 keV X-ray line within the NMSSM framework. Point 1 corresponds to a massless bino with a 7 keV singlino. Point 2 shows the quasi-degenerate scenario involving the bino and singlino, with a mass of 1GeV and a mass splitting of 3.5keV. Point 3 describes the scenario in which the singlino is 7 keV and all other neutralinos are heavy.

As far as the MSSM case is concerned, things are even simpler. For example, one could take, tan = 30, M2 = = 550 GeV, M1 = 0.23 GeV where, M1 is chosen to obtain a massless bino. The masses of the three heavier neutralinos are 499 GeV, 555 GeV and 606 GeV.

11

JHEP11(2014)018

Open Access. This article is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/

Web End =CC-BY 4.0 ), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

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