Sensitivity of indirect detection of Neutralino dark matter by Sommerfeld enhancement mechanism
Mikuru Nagayama, Joe Sato, Yasutaka Takanishi, Kazuhiro Tsunemi
SSensitivity of indirect detection of Neutralino dark matter bySommerfeld enhancement mechanism
Mikuru Nagayama, ∗ Joe Sato, † Yasutaka Takanishi, ‡ and Kazuhiro Tsunemi Department of Physics, Saitama University,Shimo-Okubo 255, 338-8570 Saitama Sakura-ku, Japan (Dated: February 9, 2021)
Abstract
We have investigated neutralino dark matter in the framework of minimal supersymmetric Stan-dard Model focusing on the coannihilatioin region. In this region, where the particle whose massis tightly degenerated with the neutralino dark matter exists, we can solve the Lithium problem inthe case of lepton flavor being violated. It turns out that Sommerfeld enhancement is importantin the coannihilation region so that the dark matter signal becomes large enough to be observedby the current sensitivity of indirect experiments. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ h e p - ph ] F e b . INTRODUCTION The discovery of neutrino oscillation in 1998 opens a new era for physics Beyond theStandard Model (SM) [1]. After this discovery, the flavor structure of the lepton sector mustbe considered and its research becomes a very important topic. On the other hand, anotherroad of Beyond the Standard Model is a quest for Dark Matter (DM) problem. The existenceof DM is according to various astrophysical observations, including gravitational effects onthe visible matter in the infrared and gravitational lensing of background radiation [2].Furthermore, the predictions of the SM of cosmology are confirmed by various cosmologicalobservations namely the total mass-energy of Universe contains about 23% DM and 68% ofa form of dark energy so that our present Universe contains only about 4% of the “ordinary”matter and energy. However, we do not really know what DM is.The total abundance of DM, which has important implications for the evolution of theUniverse, has been precisely measured by the WMAP collaboration [3] during the last fewdecades. This requires that a different kind of matter beyond the SM of particle physics mustbe considered. One of the most popular and most intensive studied candidates is the so-called weakly interacting massive particle (WIMP) that may constitute most of the matterin the Universe [4]. Cosmology provides, therefore, a good motivation for Supersymmetry(SUSY) that has a natural candidate for DM i.e. the lightest supersymmetric particle (LSP)can be a stable particle if R-parity conservation is held [5] (for reviews see e.g. [6]). As manyother physicists have already suggested and studied the scenarios of lightest neutralino beingthe candidate for LSP in various different frameworks of SUSY models.We assume that the dark matter particle is neutralino, and we focus on the so-called coan-nihilation region. In such a region, the mass between LSP neutralino and next-LSP(NLSP)slepton degenerates, we are able to reduce the abundance of DM by an order or more or-ders of magnitude with a fixed value of DM mass. In addition, Lithium problem [7–9] canbe solved if we admit lepton flavor violation [10–15]. For this reason Sommerfeld Enhance-ment [16–19] effect plays a crucial role in indirect DM observation experiments such as HESSexperiment and Fermi-LAT experiment [20, 21]. In fact, this effect has been used in manystudies that referred to indirect detection [22–25]. The phenomenon is that the annihilationcross section is enhanced by forming a bound state when DMs move non-relativistically andannihilate into SM particles [26]. Thus our results bring attention to current and future2xperiments of astrophysics.This article is organized as follows: in the next section, we define the Lagrangian relatedto neutralino and the lightest slepton, and we clarify our notation. In section 3, the methodof two-body effective action is reviewed and discussed. Calculations of cross sections andfluxes from dark matter annihilation are also given. In section 4, our calculation results willbe presented and discussed. Finally, we conclude in section 5. We summarize the technicaldetails in Appendices A, B, and C.
II. LAGRANGIAN
In this section, we consider the neutralino-slepton coannihilation region in the frameworkof Minimal Supersymmetric Standard Model (MSSM). Here we assume that DM is LSPBino-like neutralino and NLSP are the lightest slepton which is required very tight massdegeneracy with LSP neutralino. It is important to note that we use terminology in thisarticle that the lightest slepton is “stau” because (cid:101) (cid:96) almost consists of right-handed stau inthe flavor base.It is convenient to use a mass base for calculation, so we begin by formulating Lagrangianwith this base. In MSSM, neutralino is a linear combination of Higgsino neutral componentof ( (cid:101) H u , (cid:101) H d ), Wino neutral component ( (cid:102) W ), and Bino ( (cid:101) B ). In this base neutralino massmatrix is given by L neutralino mass = −
12 ( (cid:101) ψ ) T M N (cid:101) ψ + c.c. (1)Here (cid:101) ψ = ( (cid:101) B, (cid:102) W , (cid:101) H u , (cid:101) H d ) T , and M N = M − c β s w m z s β s w m z M c β c w m z − s β c w m z − c β s w m z c β c w m z − µs β s w m z − s β c w m z − µ , (2)where M is Bino mass, M is Wino mass, µ is Higgsino mass, m z is Z boson mass, respec-tively. θ w is Weinberg angle, and we use shorthand notation: s w = sin θ w and c w = cos θ w .We also denote tan β = v u /v d , where v u and v d are vacuum expectation values of H u and H d , respectively. Also s β = sin β and c β = cos β .3he (cid:101) ψ base can be transformed into mass base (cid:101) χ using unitary matrix (cid:0) N (cid:101) G (cid:1) a b ( a, b =1 , · · · , (cid:101) χ a = (cid:0) N (cid:101) G (cid:1) a b (cid:101) ψ b . (3)In this article we do consider only the case of the lightest neutralino being the candidate ofDM, thus we fix the value of a = 1.The existence of mass degenerated particle with DM in the coannihilation process isnecessary, thus we assume such particle being the lightest slepton. This slepton mass matrixfollows L slepton mass = − (cid:101) ψ † l M (cid:101) l (cid:101) ψ l (4)in flavor base (cid:101) ψ l = ( (cid:101) e L (cid:101) µ L (cid:101) τ L (cid:101) e R (cid:101) µ R (cid:101) τ R ) T , and M (cid:101) l is given by( M (cid:101) l ) I J = ( m L ) I J + y † I K y K J v d + m z ( s w − / c β δ I J (for
I, J = 1 , , , K = 4 , , − µv u y † I J + v d a † I J (for I = 1 , , , J = 4 , , − µ ∗ v u y I J + v d a I J (for I = 4 , , , J = 1 , , m R ) I J + y I K y † K J v d + m z s w c β δ I J (for
I, J = 4 , , , K = 1 , , , (5)where m L , m R is soft-breaking mass parameter. y I J is Yukawa coupling and a I J is oc-curred from A-term, where I ( I = 4 , ,
6) represents right-hand subscript and J ( J = 1 , , (cid:101) l using unitarymatrix N (cid:101) lAB ( A, B = 1 , · · · , (cid:101) l A = N (cid:101) lAB (cid:101) ψ lB , (6)Especially, we assume that the lightest slepton (cid:101) l almost consists of stau (cid:101) τ , so we note (cid:101) l as (cid:101) τ . In the same way, we can write down lagrangian of Chargino in the mass base. Charginois linear combination of Higgsino charged components ( (cid:101) H − d , (cid:101) H + u ) and Wino charged compo-nents (cid:16)(cid:102) W + , (cid:102) W − (cid:17) . The mass matrix is written L chargino mass = −
12 ( (cid:101) ψ ± ) T M C (cid:101) ψ ± + h.c. (7)4n (cid:101) ψ ± = ( (cid:102) W + , (cid:101) H + u , (cid:102) W − , (cid:101) H − d ) T base. Here M C is M C = X T X , where X = M √ s β m w √ c β m w µ . By using two unitary matrices U and V , the mass base becomes (cid:101) C +1 (cid:101) C +2 = V (cid:102) W + (cid:101) H + u , (cid:101) C − (cid:101) C − = U (cid:102) W − (cid:101) H − d . (8)The main contributions of Lagrangian which appear in the below diagram are described inthe following in terms of the above fields (for reader interest in all interaction, see AppendixA.). L = L KT + L int ∼
12 ¯ (cid:101) χ ( i / ∂ − m ) (cid:101) χ + ¯ e i (cid:16) i / ∂δ ij − ( m e ) ij (cid:17) e j − (cid:101) τ ∗ ( ∂ + m (cid:101) τ ) (cid:101) τ + 12 Z µ ( ∂ + m Z ) Z µ + 12 A µ ∂ A µ + L gauge + · · · , (9)where i, j = 1 , , m e ) i j = diag( m e , m µ , m τ ). We pick up L gauge as an example of L int which is shown in Eq. (49). L gauge is written in the following form. L gauge = ieA µ (cid:101) τ ∗ ←→ ∂ µ (cid:101) τ − ig z Z µ (cid:18) s w − N (cid:101) l i N † (cid:101) l i (cid:19) (cid:101) τ ∗ ←→ ∂ µ (cid:101) τ − i √ g (cid:16) W + µ (cid:101) ν ∗ i ←→ ∂ µ N † (cid:101) l i (cid:101) τ + W − µ (cid:101) τ ∗ N (cid:101) l i ←→ ∂ µ (cid:101) ν i (cid:17) + e A | (cid:101) τ | + g z (cid:18) s w − N (cid:101) l i N † (cid:101) l i (cid:19) Z | (cid:101) τ | − eg z (cid:18) s w − N (cid:101) l i N † (cid:101) l i (cid:19) A µ Z µ | (cid:101) τ | + g N (cid:101) l i N † (cid:101) l i W + µ W − µ | (cid:101) τ | , (10)where the sum of i is taken from 1 to 3, and we have used the following definition (cid:101) τ ∗ ←→ ∂ µ (cid:101) τ = (cid:101) τ ∗ ∂ µ (cid:101) τ − ( ∂ µ (cid:101) τ ∗ ) (cid:101) τ . (11)Here we note g and g (cid:48) as SU (2) and U (1) coupling respectively, and g = c w g z . (cid:1) φ = Z, γ, f · · · (cid:101) χ (cid:101) χ (cid:101) χ, (cid:101) τ FIG. 1: DM annihilation ladder diagram.
Z, γ and f are taken as an example of φ . II. FORMALISMA. Two-body effective action
In this section, we derive non-relativistic two-body effective action that has been inves-tigated in Ref. [27]. We will apply their method to calculate the cross sections as of ourinterest. The steps are as follows: (i) We integrate out the fields except for (cid:101) χ, (cid:101) τ . (ii) We inte-grate out large momentum mode of (cid:101) χ and (cid:101) τ . (iii) The non-relativistic action obtained in (ii)is expanded by DM velocity. (iv) We introduce auxiliary fields that represent a two-bodystate and integrate out all fields except these auxiliary fields.We begin by integrating out the fields except for (cid:101) χ, (cid:101) τ and obtain 1-loop effective action.For example, we consider integrating out A µ . First, we choose the terms related to A µ fromEq. (9), and get S A = − i ln (cid:90) DA exp i (cid:20)(cid:90) d x (cid:18) A µ ∂ g µν A ν + ieA µ (cid:101) τ ∗ ←→ ∂ µ (cid:101) τ + e A | (cid:101) τ | − eg z (cid:18) s w − N (cid:101) l i N † (cid:101) l i (cid:19) A µ Z µ | (cid:101) τ | (cid:19)(cid:21) , (12)where the sum of i is taken from 1 to 3. Next, we replace A µ by A µ → A µ + i (cid:90) d y D Aµν ( x − y ) J ν ( y )and use the Green function’s relation L µνA D Aνρ ( x − y ) = iδ ( x − y ) δ µρ . (13)Then Eq. (12) becomes S A = − i ln Det − ( − L µνA ) + i (cid:90) d xd y J µ ( x ) D Aµν ( x − y ) J ν ( y ) , (14)where L µνA =( ∂ + 2 e | (cid:101) τ | ) g µν , J µ ( x ) = ie (cid:101) τ ∗ ←→ ∂ µ (cid:101) τ − eg z (cid:18) s w − N (cid:101) l i N † (cid:101) li (cid:19) Z µ | (cid:101) τ | . ˜ τ ˜ τ ∗ ˜ τ ∗ γγ ˜ τ FIG. 2: 1-loop interaction diagram mediated by photons.
Here, we replace the first term of Eq. (14) as follows. − i ln Det − ( − L µνA ) = i (cid:2) ln (cid:0) − ∂ − e | (cid:101) τ | ) g µν (cid:1)(cid:3) ≡ i A + δA )]= i (cid:20) ln A + A − δA − A − δAA − δA (cid:21) ∼ ie tr (cid:90) d x d x | (cid:101) τ | ( x ) | (cid:101) τ | ( x ) D Aµν ( x − x ) D Aνρ ( x − x ) . (15)Note that Tr() donates operator trace and tr() donates Dirac trace. Also D Aµν ( x − y ) repre-sents photon propagator, A − = iD Aµν ( x − y ) = i (cid:90) d q (2 π ) − ig µν q + i(cid:15) e − iq ( x − y ) . (16)The second term of Eq. (14), D Aµν ∼ D Aµν at the lowest order of expansion. Finally, we geteffective action on A µ as follows S A = ie tr (cid:90) d x d x | (cid:101) τ | ( x ) | (cid:101) τ | ( x ) D Aµν ( x − x ) D Aνρ ( x − x )+ i (cid:90) d xd y J µ ( x ) D Aµν ( x − y ) J ν ( y ) . (17)By this calculation, we can represent the 1-loop interaction shown in Fig. 2.After all fields except (cid:101) χ, (cid:101) τ are integrated out, the effective action becomes S eff = (cid:90) d x (cid:20)
12 ¯ (cid:101) χ ( i / ∂ − m ) (cid:101) χ − (cid:101) τ ∗ ( ∂ + m (cid:101) τ ) (cid:101) τ (cid:21) + S (cid:48) A + S Z + S (cid:48) W + S (cid:101) ν + S e + S (cid:101) C + S h . (18)One can see other action of S A in Appendix B.Next, we integrate out the large momentum modes of (cid:101) χ, (cid:101) τ . Here we divide the fields two7arts, namely relativistic part and non-relativistic part. For the case of (cid:101) χ , it will be (cid:101) χ ( x ) = (cid:101) χ ( x ) R + (cid:101) χ ( x ) NR , (cid:101) χ ( x ) R = (cid:90) R d q (2 π ) φ ( q ) e − iqx , (cid:101) χ ( x ) NR = (cid:90) NR d q (2 π ) φ ( q ) e − iqx , (19)where φ is the Fourier coefficient of the DM field. After this division, we integrate out (cid:101) χ R .The same operation is done for (cid:101) τ , in the result we obtain S NR = (cid:90) d x (cid:20)
12 ¯ (cid:101) χ NR ( i / ∂ − m ) (cid:101) χ NR − (cid:101) τ ∗ NR ( ∂ + m (cid:101) τ ) (cid:101) τ NR (cid:21) + S P ot ( (cid:101) χ NR , (cid:101) τ NR ) + S Im ( (cid:101) χ NR , (cid:101) τ NR ) , (20)Here we note S P ot is the real part except the kinematic part, and S Im is the imaginary part.In the following, we omit the subscript NR for (cid:101) χ and (cid:101) τ .Then, we expand this action by DM velocity. For this expansion, we use two-componentspinors of neutralino and stau. These spinor fields are defined in the following form.¯ χ = e − imt ζ + ie imt −→∇· σ m ζ c e imt ζ c − ie − imt −→∇· σ m ζ , (cid:101) τ = 1 √ m ηe − imt + 1 √ m ξe imt , (21)where ζ c = − iσ ζ † T . In this form, S NR becomes S NR = S KT + S P ot + S Im , (22)where S KT = 12 (cid:90) d x ¯ (cid:101) χ ( x )( i / ∂ − m ) (cid:101) χ ( x ) − (cid:90) d x (cid:101) τ ∗ ( ∂ + m (cid:101) τ ) (cid:101) τ = (cid:90) d x (cid:20) ζ † (cid:18) i∂ + ∇ m (cid:19) ζ + η ∗ (cid:18) i∂ + ∇ m − δm (cid:19) η − ξ ∗ (cid:18) i∂ − ∇ m + δm (cid:19) ξ (cid:21) . (23)We define to quantify δm as δm = m (cid:101) τ − m m , (24)8nd the potential part, S P ot = (cid:90) d xd yδ ( x − y ) (cid:34) αr + g z (cid:18) s w − N (cid:101) l i N † (cid:101) l i (cid:19) e − m z r πr + C h e − ( m h ) r πm (cid:35) η ∗ ( x ) η ( x ) ξ ∗ ( y ) ξ ( y )+ (cid:90) d xd y e − m ei r δ ( x − y )16 πmr (cid:16) C i ( m e ) ij C † j − C i ( m e ) ij C † j (cid:17) × (cid:0) η ∗ ( x ) ζ c † ( x ) ξ ( y ) ζ ( y ) − ξ ∗ ( x ) ζ † ( x ) η ( y ) ζ c ( y ) (cid:1) ≡ (cid:90) d xd yδ ( x − y ) (cid:104) S (1) P ot η ∗ ( x ) η ( x ) ξ ∗ ( y ) ξ ( y )+ S (2) P ot (cid:0) η ∗ ( x ) ζ c † ( x ) ξ ( y ) ζ ( y ) − ξ ∗ ( x ) ζ † ( x ) η ( y ) ζ c ( y ) (cid:1)(cid:105) , (25)where i, j = 1 , , i is taken from 1 to 3. Note that C i , C i and C h are definedas in Eq. (66) , Eq. (67), and Eq. (75), respectively, in Appendix B. Also, the imaginarypart becomes S Im = S γ + S e + S Z + S Zh + S h + S W + S ν . (26)For example, S γ is calculated the diagram drawn above (see Fig. 2) by optical theorem, thenwe obtain S γ = i e πm (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) ≡ i Γ γγ (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) . (27)One can see the other actions except for S γ in Appendix C.Next, we introduce auxiliary fields σ (cid:101) χ , σ (cid:101) τ to make two-body states (cid:101) χ (cid:101) χ and (cid:101) τ (cid:101) τ , whichsatisfy the relationship:1 = (cid:90) Dσ (cid:101) τ Ds † (cid:101) τ exp (cid:20) i (cid:90) d ( xy ) σ (cid:101) τ ( t, x , y ) (cid:16) s † (cid:101) τ ( t, x , y ) − iη ∗ ( t, x ) ξ ( t, y ) (cid:17)(cid:21) , (cid:90) Dσ † (cid:101) τ Ds (cid:101) τ exp (cid:20) i (cid:90) d ( xy ) σ † (cid:101) τ ( t, x , y ) ( s (cid:101) τ ( t, x , y ) − iξ ∗ ( t, y ) η ( t, x )) (cid:21) , (cid:90) Dσ (cid:101) χ Ds † (cid:101) χ exp (cid:20) i (cid:90) d ( xy ) σ (cid:101) χ ( t, x , y ) (cid:18) s † (cid:101) χ ( t, x , y ) − ζ † ( t, x ) ζ c ( t, y ) (cid:19)(cid:21) , (cid:90) Dσ † (cid:101) χ Ds (cid:101) χ exp (cid:20) i (cid:90) d ( xy ) σ † (cid:101) χ ( t, x , y ) (cid:18) s (cid:101) χ ( t, x , y ) − ζ c † ( t, y ) ζ ( t, x ) (cid:19)(cid:21) , where d ( xy ) = dtd xd y . With this relationship, we integrate out the fields except for σ (cid:101) χ σ (cid:101) τ . Then two-body state effective action is obtained as S II = (cid:90) d xd r Φ † ( x, r ) ∇ r m + i∂ x + ∇ x m − δm
00 0 + 12 − V Φ( x, r ) , (28)where Φ( x, r ) is represented asΦ( x, r ) = V − σ (cid:101) τ ( x, r ) σ (cid:101) χ ( x, r ) , V = − S (1) P ot − iδ ( x − y )Γ − i S (2) P ot i S (2) P ot . (29)Note that x denotes the center of mass coordinate in two-body system and r is the relativecoordinate, and here we note Γ asΓ = Γ e i e i + Γ γγ + Γ Z Z + Γ Z γ + Γ Z h + Γ h h + Γ ν i ν i . (30)The symbols Γ f ( f = e i e i , γγ, Z Z , Z γ, Z h , h h , ν i ν i ) are listed in Appendix C. B. cross section
In this subsection, we calculate DM annihilation cross section with the method written in[27, 28]. For S-wave, annihilation cross section is obtained by determining radial componentof Green function which satisfies Schwinger-Dyson equation derived from Eq. (28): (cid:20) ∇ r m + i∂ x + ∇ x m − V ( r ) + i Γ δ ( r )4 πr (cid:21) (cid:104) | T Φ( x, r )Φ † ( y, r (cid:48) ) | (cid:105) = iδ ( x − y ) δ ( r − r (cid:48) ) , (31)where V ( r ) is V ( r ) = δm − S (1) P ot − i S (2) P ot − i S (2) P ot . (32)By defining the radial component of Green function as G ( E,l ) , Eq. (31) becomes the followingform. (cid:20) − E − mr d dr r + V ( r ) − i Γ δ ( r )4 πr (cid:21) G ( E, ( r, r (cid:48) ) = δ ( r − r (cid:48) ) r . (33)To determine G ( E, , Eq. (33) must be solved in a proper boundary condition. For thispurpose we consider the terms involved Γ as in perturbation series and by using variable10ransformation g ( r, r (cid:48) ) = rr (cid:48) G ( E, ii ( r, r (cid:48) ). The leading order’s solution g ( r, r (cid:48) ) satisfies theequation − m d dr g ( r, r (cid:48) ) + V ( r ) g ( r, r (cid:48) ) − E g ( r, r (cid:48) ) = δ ( r − r (cid:48) ) . (34)We get g ( r, r (cid:48) ) in the following form: g ( r, r (cid:48) ) = m g > ( r ) g T < ( r (cid:48) ) θ ( r − r (cid:48) ) + m g < ( r ) g T > ( r (cid:48) ) θ ( r (cid:48) − r ) . (35)Here The solutions g > ( r ) and g < ( r ) satisfy the below boundary condition written in [27]. (i) g < (0) = , d g < (0) dr = , (ii) g > (0) = , g > ( r ) has only outgoing wave at r → ∞ .Also, g < ( > ) ( r ) satisfies, respectively, the following differential equation − m d dr g < ( > ) ( r ) + V ( r ) g < ( > ) ( r ) = E g < ( > ) ( r ) . (36)Furthermore, (cid:101) τ does not exist at r → ∞ when E < δm , so[ g > ( r )] ij | r →∞ = δ i d j ( E ) e i √ mEr . (37)When we calculate g > ( r ) by using the first-order perturbation, cross section for annihilationchannel f can be written as σ ( S )2 v | f = [ (cid:101) Γ f ] d ( mv / d ∗ ( mv / . (38)In addition, when we write the sum of each annihilation channel f as Γ = (cid:80) f (cid:101) Γ f , total crosssection satisfies σ ( S )2 v = Γ d ( mv / d ∗ ( mv / . (39)Therefore it is necessary to solve the equation about g > ( r ) to determine d j and cross section. C. DM signature
Next, we calculate gamma-ray flux from DM annihilation that occurred in our galacticcenter. The spectrum of gamma-ray has two types. One of these is the line gamma-ray11pectrum and the other is the continuum gamma-ray spectrum. Because the DM movesnon-relativistically, the line spectrum lies at the mass of DM. On the other hand, continuumgamma-ray signal comes from jets from the DM annihilation. For example, produced π mesons from DM annihilation decay into γγ . Such a signal is useful when the cosmicbackground is well known.The gamma-ray flux from DM annihilation used by indirect detection experiments [29, 30]is given by d Φ γ dE = 14 π m χ (cid:88) f dN f dE (cid:104) σv (cid:105) f × J, (40)where m χ is DM mass, and dN f /dE · dE is the numbers of photon derived from annihilationchannel f whose energy is between E and E + dE . (cid:104) σv (cid:105) is the DM annihilation cross sectionaveraged with the velocity. J is called “ J -factor” which is determined by an astrophysical parameter and is given as J = (cid:90) line of sight dl ( θ ) (cid:90) ∆Ω d Ω ρ , (41)where ∆Ω is the angular resolution and ρ is DM density in our galaxy. N-body simulationsshow some DM halo profiles. For example, NFW [31], Burkert [32], and Einasto [33, 34]profiles are widely used. ρ NFW = ρ s ( r/r s )(1 + r/r s ) , (42) ρ Burkert = ρ s (1 + r/r s )(1 + ( r/r s ) ) , (43) ρ Einasto = ρ s exp (cid:26) − α (cid:20)(cid:18) rr s (cid:19) α − (cid:21)(cid:27) . (44)Here ρ s , r s , and α are determined from observations [35]. Next, we discuss quantities deter-mined from particle physics. Cross section (cid:104) σv (cid:105) is determined from Eq. (38) as mentionedabove. We calculate energy spectrum dN f /dE with pythia [36] for each annihilation chan-nels.For the case Z -boson annihilate to gamma is shown in Fig. 3.12 IG. 3: Energy spectrum in the case where DM decays into Z -boson. dN Z dx = 0 . e − . x x . + 1 . × − , (45)where x = E/m χ .For the case W -boson and τ annihilate to gamma is respectively shown in Fig. 4. FIG. 4: The left graph corresponds to the case where DM decays into W -boson. The right graphcorresponds to the case where DM decays into τ . dN W dx = 0 . e − . x x . + 2 . × − , dN τ dx = 0 . e − . x x . × − − . . (46)For directly decaying into γγ case, the spectrum becomes dN γ dE = 2 δ ( E − m ) . (47)13 V. RESULT
In this section, we will discuss our results. First, we find that the peak of cross sectionappears with the fixed value of δm by using the parameters which are presented in thefollowing tables for numerical calculations [37]: The dimensionless parameters are shown inTable 1 and the dimensionful parameters are shown in GeV units in Table 2 and the mixingparameters are shown in Table 3, Table 4 and Table 5. These parameters are satisfied witha positive solution for Li problem in cosmology. We should keep in our mind the fact thatstau mass is limited up to about 430 GeV by ATLAS experiment [38]. Thus we have to setup the mass parameters of neutralino and stau to adjust peak position at near 430 GeV. Weassume, however, that the calculation results are not affected if the mass values are slightlydifferent from the values in the tables. parameter g g (cid:48) s w tan β cot α y y y value 0.6387 0.3623 0.23 24.21 -24.21 1 . × − . × − m m (cid:101) τ m C µ A m e × − m µ m τ v m w m z m h value 379.596576 379.606567 725.76 1.776 -3098.1 0 .
511 0.105 1.776 243.5786 80.2 91.19 125.18TABLE II: The mass parameters used for numerical calculations are presented in GeV unit [37]. charginoreal part imaginary partRe U ij j = 1 j = 2 Im U ij j = 1 j = 2 i = 1 − . × − . × − i = 1 − . × − . × − i = 2 7 . × − . × − i = 2 − . × − − . × − Re V ij j = 1 j = 2 Im V ij j = 1 j = 2 i = 1 − . × − . × − i = 1 7 . × − . × i = 2 3 . × − . × − i = 2 0 . × . × − TABLE III: The values of the unitary matrix that diagonalizes chargino. eutralinoreal partRe N b (cid:101) G a b = 1 b = 2 b = 3 b = 4 a = 1 9 . × − − . × − . × − − . × − a = 2 3 . × − . × − − . × − . × − a = 3 − . × − . × − . × − − . × − a = 4 2 . × − − . × − − . × − . × − imaginary partIm N (cid:101) Gab b = 1 b = 2 b = 3 b = 4 a = 1 − . × − . × − − . × − − . × − a = 2 − . × − − . × − . × − . × − a = 3 1 . × − − . × − − . × − − . × − a = 4 5 . × − − . × − − . × − − . × − TABLE IV: The values of the diagonalizing unitary matrix of neutralino. sleptonreal partRe N (cid:101) lAB B = 1 B = 2 B = 3 B = 4 B = 5 B = 6 A = 1 -6.31168044 × − . × − − . × − . × − − . × − − . × − A = 2 − . × − − . × − . × − − . × − − . × − − . × − A = 3 − . × − − . × − − . × − − . × − . × − − . × − A = 4 1 . × − − . × − . × − − . × − . × − − . × − A = 5 6 . × − . × − . × − − . × − − . × − − . × − A = 6 − . × − . × − . × − . × − − . × − − . × − imaginary partIm N (cid:101) lAB B = 1 B = 2 B = 3 B = 4 B = 5 B = 6 A = 1 − . × − . × − − . × − − . × − − . × − . × − A = 2 3 . × − − . × − − . × − . × − − . × − . × − A = 3 1 . × − − . × − . × − . × − . × − . × − A = 4 1 . × − − . × − . × − . × − . × − − . × − A = 5 − . × − . × − − . × − . × − − . × − . × − A = 6 5 . × − . × − . × − − . × − − . × − . × − TABLE V: The values of the real part and imaginary part of diagonalizing unitary matrix ofslepton.
15e calculate for several annihilation channels. (1) we show the result of the cross sectionsDM decaying directly into two gammas for the corresponding δm as displayed in Fig. 5. -32 -30 -28 -26 -24 -22 -20 -18 -16 < σ v > ( c m / s ) m (GeV) annihilation cross section 109876543unit : MeV FIG. 5: Annihilation cross section to photons per δm . Each graph name represents δm in MeVunits. From Fig. 5 we realize that annihilation cross sections reach the height point at differentDM mass, and the cross sections decrease as its δm increases. (2) cross section of DMdecaying directly into Z -boson, W -boson, and τ are shown in Fig. 6 and Fig. 7, respectively. -30 -28 -26 -24 -22 -20 -18 -16
300 400 500 < σ v > ( c m / s ) m (GeV) annihilation cross section Z Z -40 -38 -36 -34 -32 -30 -28 -26
300 400 500 < σ v > ( c m / s ) m (GeV) annihilation cross section W + W - FIG. 6: The left(right) plot correspond to annihilating Z Z ( W + W − ) case, where we set δm =3 MeV. x10 -34 -32 -30 -28 -26 -24 -22 -20
300 400 500 < σ v > ( c m / s ) m (GeV) annihilation cross section τ + τ - FIG. 7: Annihilation cross section to τ . Here we set δm = 3 MeV. As an example of continuum flux, the case where the DM decays into Z Z , W + W − isshown in Fig. 8. Here J -factor and angular resolution are referred from [39]. FIG. 8: Continuum gamma-ray flux in the unit of (cm − sec − GeV − ) when DM decays into Z Z , W + W − . The color of the graph describes the size of flux. Here, it is set to δm = 3 MeV, J = 1 × GeV / cm , ∆Ω = 1 × − . In Fig. 8, the horizontal axis represents DM mass and the vertical axis represents observedphoton energy. The color of the figure means the size of flux which is larger at the peak17osition of the cross section and low photon energy and becomes smaller as the distanceincreases.We compare our result to HESS experimental data of χχ → γγ focusing on coannihilationregion. The graph represented this is shown in Fig. 9. As we can see from this figure crosssection invades the prohibited area, therefore we can constrain the parameter in our model.We also show the sensitivity projected by CTA [40]. Using this restricted parameter, we areable to constrain the parameters even more than now. -32 -30 -28 -26 -24 -22 -20
300 400 500 < σ v > ( c m / s ) m(GeV)annihilation cross section HESSCTA γγ FIG. 9: A comparison of the cross section for δm = 3 MeV in the coannihilation region withthe HESS result [39] and projected CTA sensitivity [40]. The blue-solid line shows the calculationresult, and the purple region shows the HESS result. The green-solid line shows the CTA sensitivity. Next, we discuss the result of comparing DMs annihilation to χχ → τ + τ − channel withHESS and Fermi-LAT experimental data is shown in Fig. 10.18 x10 -32 -30 -28 -26 -24 -22 -20
420 425 430 < σ v > ( c m / s ) m(GeV)annihilation cross section FermiHESSCTA τ + τ - FIG. 10: A comparison of the cross section for δm = 3 MeV for τ + τ − channel in the coannihilationregion with the HESS result [20], Fermi-LAT result [30], and projected CTA sensitivity [40]. Thered-solid line shows the calculation result, and the purple dotted-line shows the HESS result. Thegreen area describes the upper limit by Fermi-LAT, and blue-solid line shows CTA sensitivity [40]. We also draw the projected CTA sensitivity line same as of χχ → γγ case. Clearly, wecannot limit in most parameter region for τ channel case, even for other channels. Thus weput the limitation of parameter region by using χχ → γγ channel.By applying the above-mentioned method to other δm cases as well, we can limit theparameter between δm and DM mass. The restricted parameter region is drawn in Fig. 11.In this figure, the blue-area describes the constrained region of our model.19 IG. 11: The parameter region is limited by HESS experimental data. Parameters in colored areasare restricted.
We see that the narrow band in Fig. 11, which presents the limit from the current exper-imental restrictions on δm . In Fig. 12. we show that the future planned sensitivity of CTAthat restricts about 100 times stronger than the present limit. It is found that the limit for δm can be set in a wide range in the future. FIG. 12: The left graph corresponds to the limited parameter region constrained by projectedCTA sensitivity. The right graph corresponds to the case where the limit is 100 times stronger. Asexperimental limits increase, more areas are restricted. . CONCLUSION We have investigated neutralino dark matter in the framework of MSSM. Hereby themechanism of Sommerfeld enhancement is taken into account for the calculation of darkmatter annihilation cross section and flux in the coannihilation region. In this region, iflepton flavor is violated, Lithium problem in cosmology is solved. The cross sections forseveral annihilation channels are shown in Fig. 5, Fig. 6 and Fig. 7: The dependence of δm on cross sections of χ (cid:101) χ → γγ channel is shown in Fig. 5. For (cid:101) χ (cid:101) χ → ZZ , (cid:101) χ (cid:101) χ → W + W − and (cid:101) χ (cid:101) χ → τ + τ − channel, cross sections are displayed in Fig. 6 and Fig. 7. It is revealedthat these cross sections increase significantly due to Sommerfeld Enhancement in thesefigures. In Fig. 9, we compare our calculation result with the limits of current experimentaldata. Clearly, we can constrain the range of prohibited dark matter mass with the valueof δm = 3 MeV. We vary the value of δm from 3 MeV to 4 MeV then of course weget a similar limit comparing with the current experimental result as in Fig. 9. Further,we continue the same processes to vary the value of δm up to 10 MeV. Then finally allconstraints on δm vs DM mass m are plotted in Fig. 11. On the left panel of Fig. 12, weshow the limited parameter region acquired by comparing our result with future plannedsensitivity of CTA [41, 42]. On the right panel of Fig. 12, we draw the disallowed parameterband which is obtained by comparison of the case of restricting about 100 times strongeraccording to the future planned experiments e.g. MAGIC [43].In conclusion the cross section calculated by our model has already reached an observablerange. Thus we can find signals from DM, and solve the problems such as dark matter andLi problems in the near future. In addition, even if no signal is detected, the parameters ofthe model can be limited and the validity of the supersymmetric particles can be verifiedsoon.
Acknowledgments
This work was supported by JSPS KAKENHI Grants No. JP18H01210 (J.S.), and MEXTKAKENHI Grant No. JP18H05543 (J.S.). 21
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12 ( c α v u + s α v d ) (cid:16) g N (cid:101) l i N † (cid:101) l i + g (cid:48) (cid:16) − N (cid:101) l i N † (cid:101) l i + 2 N (cid:101) l i +3 N † (cid:101) l i +31 (cid:17)(cid:17) + 2 s α v d (cid:16) N (cid:101) l i y † ik y kj N † (cid:101) l j + N (cid:101) l i +3 y ik y † kj N † (cid:101) l j +31 (cid:17) + c α (cid:16) µ ∗ N (cid:101) l i +3 y ij N † (cid:101) l j + h.c. (cid:17) + s α (cid:16) N (cid:101) l i +3 a ij N † (cid:101) l j + h.c. (cid:17)(cid:105) . (50) • Higgs-stau-stau 4-point interaction L h − h − (cid:101) τ − (cid:101) τ = (cid:101) τ ∗ (cid:101) τ h (cid:20) − g ( c α − s α ) N (cid:101) l i N † (cid:101) l i − g (cid:48) ( c α − s α ) (cid:16) − N (cid:101) l i N † (cid:101) l i + 2 N (cid:101) l i +3 N † (cid:101) l i +31 (cid:17) − s α (cid:16) N (cid:101) l i y † ik y kj N † (cid:101) l j + N (cid:101) l i +3 y ik y † kj N † (cid:101) l j +31 (cid:17)(cid:21) . (51) • Gaugino-interaction L gaugino = (cid:101) τ ∗ ¯ (cid:101) χ (cid:34) P L (cid:32) √ g (cid:48) N (cid:101) l i N (cid:101) G + √ gN (cid:101) l i N (cid:101) G + 12 N (cid:101) l j +3 y ji N (cid:101) G (cid:33) + P R (cid:18) −√ g (cid:48) N (cid:101) l i +3 N (cid:101) G + 12 N (cid:101) l i y † ij N (cid:101) G (cid:19)(cid:21) e Di + h.c. (52)24here, P L , P R is projection operetor given as P L = 1 − γ , P R = 1 + γ . (53)Also, four-component spinors e Di , (cid:101) ψ Da are defined as e Di = e Lαi e R ˙ αi , (cid:101) ψ Da = (cid:101) ψ αa (cid:101) ψ † ˙ αa . • Interaction between neutrino and chargino to release neutrino in the final state L chargino = − g (cid:101) e ∗ Li (cid:102) W − ν i − gν † i (cid:102) W −† (cid:101) e Li − (cid:101) e ∗ Ri y ij ν j (cid:101) H − d − ν † i y † ij (cid:101) e Rj (cid:101) H − d † = (cid:101) τ ∗ ¯ (cid:101) C α P L ν Di (cid:20) − gN (cid:101) l i U † α − N (cid:101) l j +3 y ji U † α (cid:21) + h.c., (54)where four-component spinor (cid:101) C α is (cid:101) C α = (cid:101) C + α (cid:101) C −† α . In Eq.(9), main terms are chosen. 25 ppendix B
We note the result of integrating out the fields other than A µ . In the following formula,the sums of i is taken from 1 to 3. • Effective action obtained by integrating out Z µ S Z =2 ie g z (cid:18) s w − N (cid:101) l i N † (cid:101) l i (cid:19) × tr (cid:90) d x d x | (cid:101) τ | ( x ) | (cid:101) τ | ( x ) D Zµν ( x − x ) D Aνρ ( x − x )+ ig z (cid:18) s w − N (cid:101) l i N † (cid:101) l i (cid:19) × tr (cid:90) d x d x | (cid:101) τ | ( x ) | (cid:101) τ | ( x ) D Zµν ( x − x ) D Zνρ ( x − x )+ i (cid:90) d xd yJ µZ ( x ) D Zµν ( x − y ) J νZ ( y ) , (55)where D Zµν ( x − y ) = − i (cid:90) d q (2 π ) g µν q − m Z + i(cid:15) e − iq ( x − y ) , (56) J µZ ( x ) = − ig z (cid:18) s w − N (cid:101) l i N † (cid:101) l i (cid:19) (cid:101) τ ∗ ←→ ∂ µ (cid:101) τ . (57) • Effective action obtained by integrating out W + , W − S W = i g (cid:16) N (cid:101) l i N † (cid:101) l i (cid:17) tr (cid:90) d x d x | (cid:101) τ | ( x ) | (cid:101) τ | ( x ) D Wµν ( x − x ) D W νρ ( x − x )+ i (cid:90) d xd yJ µW ( x ) D Wµν ( x − y ) J νW † ( y ) , (58)where D Wµν ( x − y ) = − i (cid:90) d q (2 π ) g µν q − m W + i(cid:15) e − iq ( x − y ) , (59) J µW ( x ) = − i √ gN (cid:101) l i (cid:101) τ ∗ ←→ ∂ µ (cid:101) ν i , (60) J µW † ( x ) = − i √ gN † (cid:101) l i (cid:101) ν ∗ i ←→ ∂ µ (cid:101) τ . (61)26 Effective action obtained by integrating out (cid:101) ν ∗ i , (cid:101) ν i S (cid:101) ν =2 ig N † (cid:101) l i N (cid:101) l j N † (cid:101) l k N (cid:101) l l tr (cid:90) d x d x d x d x × ∂ σ (cid:101) τ ∗ ( x ) ∂ µ (cid:101) τ ( x ) ∂ ν (cid:101) τ ∗ ( x ) ∂ ρ (cid:101) τ ( x ) × D (cid:101) ν ji ( x − x ) D Wµν ( x − x ) D (cid:101) ν lk ( x − x ) D Wρσ ( x − x ) , (62)where D (cid:101) ν ij ( x − y ) = − i (cid:90) d q (2 π ) δ ji q − m (cid:101) ν + i(cid:15) e − iq ( x − y ) . (63) • Effective action obtained by integrating out e D , ¯ e D S e = i (cid:90) d xd y (cid:101) τ ∗ ( x ) (cid:101) τ ( y )¯ (cid:101) χ ( x ) (cid:2) C i P L + C i P R (cid:3) S τ ( x − y ) ij (cid:2) C † j P R + C † j P L (cid:3) (cid:101) χ ( y ) , (64)where S τ ( x − y ) ij = i (cid:90) d q (2 π ) q/δ ji + ( m e ) ij q − m e + i(cid:15) e − iq ( x − y ) , (65) C i = (cid:32) √ g (cid:48) N (cid:101) l i N (cid:101) G + √ gN (cid:101) l i N (cid:101) G + 12 N (cid:101) l j +3 y ji N (cid:101) G (cid:33) , (66) C i = (cid:18) −√ g (cid:48) N (cid:101) l i +3 N (cid:101) G + 12 N (cid:101) l i y † i N (cid:101) G (cid:19) . (67) • Effective action obtained by integrating out ν D , ¯ ν D S (cid:101) ν = i (cid:90) d xd y (cid:101) τ ∗ ( x ) (cid:101) τ ( y ) ¯ (cid:101) C α ( x ) C iα P L S ν ( x − y ) ij C † jβ P R (cid:101) C β ( y ) , (68)where S ν ( x − y ) ij = i (cid:90) d q (2 π ) q/δ ji q + i(cid:15) e − iq ( x − y ) , (69) C iα = − gN (cid:101) l i U † α − N (cid:101) l j +3 y ji U † α . (70) • Effective action obtained by integrating out ¯ (cid:101) C, (cid:101) C S (cid:101) C = i (cid:90) d x d x d x d x (cid:101) τ ∗ ( x ) (cid:101) τ ( x ) (cid:101) τ ∗ ( x ) (cid:101) τ ( x ) × S (cid:101) C ( x − x ) αβ C iα P L S ν ( x − x ) ij C † jβ P R S (cid:101) C ( x − x ) γδ C kγ P L S ν ( x − x ) kl C † lδ P R , (71)27here S (cid:101) C ( x − y ) αβ = i (cid:90) d q (2 π ) q/ + m C α q − m C α + i(cid:15) δ βα e − iq ( x − y ) . (72) • Effective action obtained by integrating out h S h = iC (4) h tr (cid:90) d x d x | (cid:101) τ | ( x ) | (cid:101) τ | ( x ) D h ( x − x ) D h ( x − x ) − i (cid:90) d xd yJ h ( x ) D h ( x − y ) J h ( y ) , (73)where D h ( x − y ) = − i (cid:90) d q (2 π ) q − m h + i(cid:15) e − iq ( x − y ) ,J h ( x ) = | (cid:101) τ ( x ) | C h ,C (4) h = − g ( c α − s α ) N (cid:101) l i N † (cid:101) l i − g (cid:48) ( c α − s α ) (cid:16) − N (cid:101) l i N † (cid:101) l i + 2 N (cid:101) l i +3 N † (cid:101) l i +31 (cid:17) − s α (cid:16) N (cid:101) l i y † ik y kj N † (cid:101) l j + N (cid:101) l i +3 y ik y † kj N † (cid:101) l j +31 (cid:17) , (74) C h = 1 √ (cid:20) −
12 ( c α v u + s α v d ) (cid:16) g N (cid:101) l i N † (cid:101) l i + g (cid:48) (cid:16) − N (cid:101) l i N † (cid:101) l i + 2 N (cid:101) l i +3 N † (cid:101) l i +31 (cid:17)(cid:17) + 2 s α v d (cid:16) N (cid:101) l i y † ik y kj N † (cid:101) l j + N (cid:101) l i +3 y ik y † kj N † (cid:101) l j +31 (cid:17) + c α (cid:16) µ ∗ N (cid:101) l i +3 y ij N † (cid:101) l j + h.c. (cid:17) s α (cid:16) N (cid:101) l i +3 a ij N † (cid:101) l j + h.c. (cid:17)(cid:105) . (75)Furthermore, we note S (cid:48) A = ie tr (cid:90) d x d x | (cid:101) τ | ( x ) | (cid:101) τ | ( x ) D Aµν ( x − x ) D Aνρ ( x − x )+ i (cid:90) d xd yJ µA ( x ) D Aµν ( x − y ) J νA ( y ) , (76) S (cid:48) W = i g (cid:16) N (cid:101) l i N † (cid:101) l i (cid:17) tr (cid:90) d x d x | (cid:101) τ | ( x ) | (cid:101) τ | ( x ) D Wµν ( x − x ) D W νρ ( x − x ) , (77)where J µA ( x ) = ie (cid:101) τ ∗ ←→ ∂ µ (cid:101) τ . (78)28 ppendix C For the calculation of the imaginary part, the result is as follows for the fields other than S γ . For example, a diagram in which fermions mediate is shown the below diagram. (cid:1) ˜ τ ˜ τ f ˜ χ ˜ τf ˜ τ ˜ χ With optical theorem, effective action of this case becomes S e = i π (1 − m e /m ) / (2 m − m e ) (cid:2) ( | C | + | C | )( m e + m ( m c + m † c )) + m c m † c + m | C | | C | ) (cid:3) × (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) ≡ i Γ e i e i (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) . (79)For other fields, S Z = i g z (cid:16) s w − N (cid:101) l i N † (cid:101) li (cid:17) m πm z (cid:16) − m m (cid:17) ( m + m (cid:101) τ − m z ) + g z ( s w − N i (cid:101) l N † (cid:101) li ) πm (cid:114) − m z m × (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) ≡ i Γ Z Z (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) , (80)where the sums of i is taken from 1 to 3. S AZ = i e g z (cid:16) s w − N (cid:101) l i N † (cid:101) l i (cid:17) πm (cid:18) − m z m (cid:19) (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) ≡ Γ γZ (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) . (81)29 W = i g (cid:16) N (cid:101) l i N † (cid:101) li (cid:17) m πm W (cid:18) − m W m (cid:19) m + m (cid:101) ν − m W ) + g (cid:16) N i (cid:101) l N † (cid:101) li (cid:17) πm (cid:114) − m W m × (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) ≡ i Γ W + W − (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) . (82) S ν = i (cid:16) C iα C † iα (cid:17) π m C α (cid:0) m + m C α (cid:1) (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) ≡ i Γ ν i ν i (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) . (83) S h = i (cid:34) C h πm (cid:114) − m h m m + m (cid:101) τ − m h ) + C (4) h πm (cid:114) − m h m (cid:35) × (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) ≡ i Γ h h (cid:90) d xη ∗ ( x ) η ( x ) ξ ∗ ( x ) ξ ( x ) ..