A Masses, mixings, and couplings

# Prospects for Higgs- and Z-resonant Neutralino Dark Matter

UT-15-35

IPMU-15-0183

Prospects for Higgs- and -resonant

Neutralino Dark Matter
Koichi Hamaguchi and Kazuya Ishikawa
Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113–0033, Japan

Kavli Institute for the Physics and Mathematics of the Universe (Kavli IPMU),

University of Tokyo, Kashiwa 277–8583, Japan

## 1 Introduction

The identity of the dark matter (DM) is one of the biggest mysteries in particle physics, astrophysics, and cosmology. Among various DM candidates, the lightest neutralino in the supersymmetric (SUSY) extension of the Standard Model (SM) is particularly attractive, since it can have the desired thermal relic abundance with a weak scale mass.

The discovery of the 125 GeV Higgs boson [1, 2] and no signal of physics beyond the SM so far at the LHC run I might imply that the SUSY particles, in particular the scalar partners of SM fermions (sfermions), are much heavier than . After the Higgs discovery, therefore, such heavy sfermion scenarios have attracted attentions (see, e.g., Refs. [3]).

If all sfermions are heavy, the lightest neutralino can have the correct thermal relic abundance,  [4, 5], only in limited cases. For instance, the pure Wino DM with a mass of TeV can explain the DM density [6, 7], which may be probed by indirect detections [8]. The (almost) pure Higgsino with a mass of TeV is also an attractive candidate [7, 9]. In addition, the coannihilations among gauginos [10, 11] and the well-tempered Bino-Higgsino mixng [11, 12, 13] can lead to the desired DM density, with various phenomenological implications (see, e.g., Refs. [14, 15] and references therein).

In this paper, we study another viable corner of the neutralino parameter space in the heavy sfermion scenario, i.e., the Higgs- and -reonant neutralino DM. When the mass of the lightest neutralino is close to the Higgs- or -resonance, or , it can have the correct thermal relic abundance while avoiding the constraints from the direct detection and other experiments. Aspects of such Higgs- and -resonant neutralino DM have been investigated e.g., in Refs. [16, 13, 17, 18].1

We revisit this Higgs- and -reonant neutralino DM scenario and extend previous studies by comprehensively investigating the current constraints and future prospects. We include the following constraints and prospects.

• relic abundance  [4, 5].

• DM direct detection.

• constraints on the spin-independent (SI) scattering cross section from the LUX [20], and on the spin-dependent (SD) scattering cross section from the XENON 100 [21].

• prospects of the XENON 1T for the SI [22] and SD [23] scattering cross sections.

• Higgs invisible decay.

• current constraint from global fit [24, 25].

• expected sensitivity of the HL-LHC [26, 27], and of the ILC [28].

• chargino/neutralino search at the LHC.

• expected sensitivity at 14 TeV for 300 fb and 3000 fb [29].

Other constraints are also briefly discussed. Constraints from the LHC run I [30] is discussed in Appendix B. We use a simplified model with only three parameters, the Bino mass , the Higgsino mass , and , assuming that all other supersymmetric particles and heavy Higgs bosons are decoupled. As we will see, the “blind spot” [15] of the Higgs-neutralino coupling plays an important role also in the Higgs- and -resonant neutralino DM scenario. It is shown that there is still a large viable parameter space, and almost all the parameter space of the scenario will be covered complementarily by the future experiments. Our results are summarized in Figs. 15.

## 2 Model

We assume that all SUSY particles but Bino and Higgsino multiplets are (much) heavier than 1 TeV, and the only scalar particle at the electroweak scale is the SM-like Higgs boson with the mass of 125 GeV.2 In this limit, the Wino component is decoupled, and the neutralino mass matrix becomes a matrix,

 Mχ =⎛⎜⎝M1−mZsWcosβmZsWsinβ−mZsWcosβ0−μmZsWsinβ−μ0⎞⎟⎠, (1)

in the basis of Bino and down-type and up-type Higgsinos, . Here, and are the Bino and Higgsino masses, respectively, is the ratio of the vacuum expectation values of the up- and down-type Higgs, and . In this work, we assume that there is no CP-violation in the neutralino sector, and take and real. Then, the mass matrix is diagonalized by a real orthogonal matrix as

 OχMχOTχ =diag(ϵ1mχ1,ϵ2mχ2,ϵ3mχ3), (2)

where and . Analytical and approximate formulae for the masses and the mixing matrix are given in Appendix A. The lightest neutralino is the DM candidate. In the chargino sector, there is only one light chargino with mass . We assume to avoid the LEP bound [4].

In the following, we only consider a light Bino,

 M1<80 GeV. (3)

The lightest neutralino then becomes Bino-like, and its coupling to the SM is given by the following Lagrangian,

 Lχ1-SM =12λhh¯¯¯¯¯¯ψ1ψ1−12λZZμ¯¯¯¯¯¯ψ1γμγ5ψ1, (4)

where , and denote the fields of the the SM-like Higgs boson, -boson, and the lightest neutralino, respectively. The DM thermal relic abundance, its direct and indirect detections, and the Higgs and invisible decay rates are all determined by the DM mass and the two couplings and . The couplings are given by3

 λh =g′ϵ1(Oχ)1,˜B((Oχ)1,˜Hdcosβ−(Oχ)1,˜Husinβ), (5) λZ =12gZ(−[(Oχ)1,˜Hd]2+[(Oχ)1,˜Hu]2), (6)

where , and and are the SU(2) and U(1) gauge couplings, respectively.

In terms of expansion, they are approximately given by (cf. Appendix A)

 λh ≃g′ϵ1(μsin2β+M1μ2−M21mZsW+O(mZsWμ)3), (7) λZ ≃12gZ(cos2βm2Zs2Wμ2−M21+O(mZsWμ)4). (8)

From Eq. (7), the DM- coupling vanishes when

 M1≃−μsin2β. (9)

This leads to a “blind spot” [15], where the Higgs resonant annihilation, the spin-independent DM scattering for the direct detection, and the Higgs invisible decay are all suppressed.4 This suppression of results in a parameter region that is not probed by the direct detection nor the Higgs invisible decay, which is important especially for the -resonant region . On the other hand, the DM- coupling is almost independent of for , and has only a mild dependence on as far as .

At the LHC, searches for charginos and neutralinos can probe this scenario, which will be discussed in Sec. 3.4. The relevant interaction Lagrangian is given by

 L =W−μ3∑i=1¯¯¯¯¯ψiγμ(λWLiCPL+λWRiCPR)ψC+h.c. +Zμ∑i

where

 λWLiC =−1√2gηi(Oχ)i,˜Hu, (11) λWRiC =sign(μ)1√2gη∗i(Oχ)i,˜Hd, (12) λZLij=−(λZRij)∗ =12gZηiη∗j(−(Oχ)i,˜Hd(Oχ)j,˜Hd+(Oχ)i,˜Hu(Oχ)j,˜Hu), (13) λhLij=(λhRij)∗ =12g′η∗iη∗j[(Oχ)i,˜B((Oχ)j,˜Hdcosβ−(Oχ)j,˜Husinβ)+(i↔j)]. (14)

Here, , and denotes the chargino field which is defined to have a mass term and to have a positive charge.

## 3 Constraints and Prospects

Our main results are shown in Figs. 15, where the constraints and prospects listed in Sec. 1 are presented in the -planes for . In the figures, we show only the region with , because the relic density is always too large outside this region for and . In the following subsections, we explain each of the constraints and prospects in turn. We also briefly mention other possible constraints in Sec. 3.5.

### 3.1 Thermal relic abundance

We assume that the present energy density of DM is dominantly given by that of the thermal relic of the lightest neutralino. In the present scenario, the lightest neutralino can only annihilate into a pair of SM fermions, and the annihilation cross section is given by

 σ(χ1χ1→f¯f)=σ(χ1χ1→h∗→f¯f)+σ(χ1χ1→Z∗→f¯f), (15)

where

 σ(χ1χ1→h∗→f¯f) ≃12(λh)2√1−4m2χ1s1(s−m2h)2+(mhΓh)2smhΓ(h→f¯f), (16) σ(χ1χ1→Z∗→f¯f) ≃(λZ)2√1−4m2χ1s1(s−m2Z)2+(mZΓZ)2smZΓ(Z→f¯f), (17)

and there is no interference term. Here, we have neglected the terms proportional to , where is the mass of final state fermion. The DM abundance is calculated by solving the Boltzmann equation

 dYχ1dt=−nS⟨σvrel⟩(Y2χ1−Y2χ1,eq), (18)

where the thermal average of the annihilation cross section times the relative velocity (with Maxwell-Boltzmann equilibrium distribution) is given by [33],

 ⟨σvrel⟩(T) =∫d3p1d3p2e−E1/Te−E2/Tσvrel∫d3p1d3p2e−E1/Te−E2/T =18m4χ1T[K2(mχ1/T)]2∫∞4m2χ1σ(s)√s(s−4m2χ1)K1(√s/T)ds. (19)

Here, , is the entropy density, is the DM number density, is its equilibrium value, and and are the cosmic time and the temperature, respecitvely.5 are the modified Bessel functions of the first and second kind. The final relic abundance is given by , where is the critical density divided by the entropy density at present, with being the scale factor for Hubble constant [4].6

In Figs. 15, the contours of relic DM density is shown in black lines. The contours have clear peaks at the - and Higgs-resonances, and . In these regions, the chargino mass can take a large value, corresponding to small DM- and DM-Higgs couplings, and .

In the -resonant region, , the relic abundance shows a universal behavior for all . This is because the DM- coupling is almost independent of for , as shown in Eq. (8). In this region, the chargino mass is always bounded from above as  [18], which corresponds to [cf. Eq. (8)].7 This upper bound on the chargino mass is crucial for the LHC search discussed in Sec. 3.4.

In the Higgs resonant region, , the behavior of the relic abundance in the -planes strongly depends on as well as the sign. This is also understood in terms of the DM-Higgs coupling, , in Eq. (7). As can be seen in Figs. 2, for and , there are two regions corresponding to . This is because of the blind-spot behavior discussed in Sec. 2. The coupling has opposite signs in the two separate regions, and it becomes zero in between. For and , the region of large disappears because a sufficiently large can no longer be obtained there. For both and and for all , the maximal chargino mass corresponds to . For , the upper bound on the chargino mass is as small as for and for . As we shall see in Sec. 3.4, these regions can be probed by the 14 TeV LHC. For small , however, a much larger chargino mass is allowed, e.g., for and . Although such a heavy chargino is out of the 14 TeV LHC reach, the direct detection experiments can cover most of the region, as we will see in the next subseciton.

### 3.2 Direct detection

In the present scenario, the spin-independent (SI) and spin-dependent (SD) scatterings between the DM and nuclei are induced by Higgs-exchange and -exchange, respectively. As we shall see, the former gives a strong bound and high future sensitivity, while the latter plays a complementary role in the blind spot regions for .

#### spin-independent scattering

The SI scattering cross section of DM per nucleon is given by

 σSIN =4πλ2Nm2N(1+mNmχ1)−2, (20)

where is the nucleon mass and is the effective coupling between the DM and nucleons, . In our scenario, the coupling is induced by the Higgs exchange, and given by (cf. [37, 35])

 λN =λh2m2h⋅mNfN2mW/g,fN=∑q=u,d,sf(N)Tq+29(1−∑q=u,d,sf(N)Tq), (21)

where . In our analysis, we use the default values of adopted in micrOMEGAs 4.1 [35], , , and , which leads to .8 Therefore, the SI scattering cross section is given by

 σSIN≃5.2×10−43⋅(λh)2(1+mNmχ1)−2 cm2. (22)

In Fig. 15, we show the constraint obtained by the LUX [20] (90% CL limit) and the future prospects for the XENON 1T [22] with blue dashed and solid lines, respectively. The results are understood in terms of the coupling in Eqs. (5) and (7). As can be seen in the figures, for , the region with –400 GeV (150–400 GeV) are excluded by the LUX, for (). The XENON 1T can cover most of the viable parameter space for , except for the peak of the Higgs-resonance, where (cf. Sec. 3.1) and , and the peaks for . Note that the SI cross sections in these peak regions are just below the sensitivity shown in Ref. [22]. Therefore, it is expected that future experiments with higher sensitivity [39] can cover the whole parameter region for .

For , because of the cancellation in Eq. (7), the constraint and the sensitivity are significantly reduced in terms of the chargino mass . In the Higgs-resonant region, the parameter regions with the correct thermal relic abundance, , will still be mostly covered by the XENON 1T. This is because both of and are determined by the same coupling, . The correlation is clearly seen, e.g., in Fig. 2, for . In the -resonant region, however, and are determined by different couplings, and , respectively. This results in a large parameter region which gives correct but very small , as can be seen in Figs. 35. Since can be zero at a certain value of , there always remains a region which cannot be probed by the SI scattering. Some of these regions are probed by SD scattering discussed in the next subsection, and the search for the chargino and the neutralino at the LHC, discussed in Sec. 3.4, will be a very sensitive probe in these regions.

#### spin-dependent scattering

Now let us discuss the SD scattering. The SD nucleon-DM scattering cross section is given by [40, 35]

 σSDN =12πξ2Nm2N(1+mNmχ1)−2, (23)

where is the axial-vector effective coupling between the DM and the nucleons, . In our scenario, the coupling is induced by the exchange, and given by (cf. [40, 35])

 ξN=λZgZ4m2Z∑q=u,d,sT3qΔ(N)q, (24)

where parametrize the quark spin content of the nucleon. In our analysis, we use the default values adopted in micrOMEGAs [35], , , and . Therefore, the SD nucleon-DM scattering cross section is given by9

 σSDn(p) ≃2.3(3.0)×10−37⋅(λZ)2(1+mn(p)mχ1)−2cm2. (25)

In Fig. 15, we show the constraint on the SD neutron-DM scattering cross section from the XENON100 [21] with green dashed lines.10 As can be seen from the figures, this bound gives the strongest current constraint in some of the small regions for . The bound on the chargino is about –140 GeV, and it has only very mild dependences on , , and sign. This can be understood from Eqs. (8) and (25), which lead to the following approximate formula,

 σSDn(p) ≃1.4(1.8)×10−41(300 GeV|μ|)4(1−m2χ1μ2)−2(cos2β)2cm2. (26)

In the figures, we also show the prospect of the XENON 1T for the , studied in Ref. [23], with green solid lines. It reaches the chargino mass of about 280–350 GeV, which will cover a large part of the blind spot for .

### 3.3 Higgs and Z invisible decays

For the Higgs boson can decay into a pair of DMs, which lead to a invisible decay. The branching ratio is given by

 Br(h→χ1χ1) =Γ(h→χ1χ1)Γ(h→SM)+Γ(h→χ1χ1), (27)

where

 Γ(h→χ1χ1)=(λh)216πmh(1−4m2χ1m2h)3/2. (28)

We have used  [4] in our calculation. The constraint on the Higgs invisible decay has been obtained by global fits to Higgs data [24, 25]. In our numerical calculation, we adopt the one in Ref. [25], (95% CL).

As for the future prospects, we consider the high-luminosity (HL) LHC and the ILC. The sensitivity of the HL-LHC depends on the systematic uncertainties. Here, we use the estimated future sensitivity of searches for Higgs decaying invisibly using channel in Ref. [26], (95% CL) for 3000 fb, adopting the value of the “realistic scenario” for the size of systematics. In the “conservative scenario” the estimated sensitivity becomes . In the Higgs working group report of the 2013 Snowmass [27], the 95% CL limit (for 3000 fb) is estimated as –0.16 for ATLAS and 0.06–0.17 for CMS. For the ILC, we use the value in [28], (1150 fb at ).

The constraint and prospects for the Higgs invisible decay are shown in Fig. 15 with magenta lines. As can be seen in the figures, a large parameter space is covered by the Higgs invisible decay search. For , the whole -resonant region will be covered by the ILC. The blind spots are again clearly seen for , where Higgs invisible decay becomes very small due to the suppression of in Eq. (7).

Before closing this subsection, let us comment on the invisible decay. For , the can decay into a pair of DMs, with a partial decay rate

 Γ(Z→χ1χ1)=(λZ)224πmZ(1−4m2χ1m2Z)3/2. (29)

The bound from the LEP, (95% CL) [42], corresponds to . In our setup, this is always weaker than the bound on from the XENON100 [21] for  (see Sec. 3.2), and hence we do not show the bound in the figures.

### 3.4 Search for the chargino and neutralinos at the LHC

As we have seen in Sec. 3.1, the requirement that the thermal relic abundance of explains the observed DM density, , gives upper bounds on the chargino mass in the present scenario. The chargino mass is except for the Higgs resonance peak for small . In particular, in the -resonant region, the chargino mass is always bounded from above as  [18]. The heavier neutralinos, and , also have masses . These chargino and neutralinos are within the reach of the LHC experiments.

In this work, we consider the following production and decay channels at the LHC,

 pp→χ2,3χ±→Zχ1W±χ1→ℓℓχ1ℓνχ1, (30)

which leads to a signal with three leptons and missing energy, and gives a high sensitivity in the present scenario [43]. The sensitivities of the SUSY searches at 14 TeV, including the high luminosity run of , is studied by ATLAS [29] and CMS [44]. In this work, we reinterpret the ATLAS study [29] of the search for the channel in Eq. (30), for and . The constraints from the LHC run I [30] are discussed in Appendix B.

In the ATLAS analysis [29], the results for the three-lepton process (30) is presented assuming a simplified “pure Wino” model with three states; two neutralinos , and one chargino , with degenerate masses of the chargino and the heavier neutralino, and 100% branching ratios of and . There are several differences in the present setup. (i) There are two heavier neutralinos and . (ii) The sum of the production cross sections of chargino-neturalino pair is a factor smaller, since the ATLAS analysis assumes Wino-like chargino-neutralino pair production. (iii) The neutralinos and have generically sizable branching fractions to the Higgs as well, and . (iv) Although their masses are close to the chargino mass , the difference can have a non-negligible effect on the production cross section, especially for small . Therefore, the results in Ref. [29] cannot be directly applied to the present model and reinterpretations are necessary.

In the ATLAS analysis, several signal regions (SRs) are defined by various kinematical cuts. We calculate the expected number of events in each signal region (SR) X as

 NSR-X=∑j=2,3 ∑χ±σNLO(pp→χ±χj)⋅Br(χ±→χ1W(∗)→χ1ℓν)⋅Br(χj→χ1Z(∗)→χ1ℓℓ) ×ASR-X⋅∫Ldt, (31)

where denote the integrated luminosity, and is defined by

 ASR-X =\# of events which pass the cuts of SR-X\# % of generated events in pp→χ±χi→W(∗)χ1Z(∗)χ1→ℓνχ1ℓℓχ1, (32)

where denotes , and . For simplicity, we discard the hadronic decays of and . The branching fractions are given by (when kinematically allowed)

 Br(χ±→χ1W→ℓν) =Br(χ±→χ1W)⋅Br(W→ℓν), (33) Br(χj→χ1Z→ℓℓ) =Br(χj→χ1Z)⋅Br(Z→ℓℓ), (34)

where the chargino branching fraction is , while the neutralino branching is given by

 Br(χj→Zχ1) =Γj(χj→Zχ)Γj(χj→Zχ)+Γj(χj→hχ), (35) Γ(χj→Zχ) =116πmχj|λZL1j|2(1+6ϵ1ϵjr1j+(r1j)2−2(rzj)2+(1−(r1j)2)2(rzj)2) ×(1−(r1j−rzj)2)1/2(1−(r1j+rzj)2)1/2, (36) Γ(χj→hχ) =116πmχj|λhL1j|2(1+2ϵ1ϵjr1j+(r1j)2−(rhj)2) ×(1−(r1j−rhj)2)1/2(1−(r1j+rhj)2)1/2, (37)

where , and the couplings are given by Eqs. (13) and (14).

In the numerical calculations, we generate the events with MadGraph5_aMC@NLO 2.2.3 [45] in combination with PYTHIA 6.4 [46]. We generate the events at LO and rescale the acceptance with the NLO cross section calculated by Prospino 2.1 [47] with CTEQ6L1 [48] parton distribution functions (PDFs). Delphes 3 [49] is used with the ATLAS parameters card11 given in the MadGraph5_aMC@NLO package for the fast detector simulation.

There are three (four) SRs considered to probe the signal of Eq. (30) for 300 (3000) fb, denoted as SRA–SRC (SRA–SRD). In our analysis, after electrons, muons, and jets are selected following the ATLAS analysis [29], the following cuts are applied.

• There should be exactly three leptons in each event, and at least one SFOS lepton pair is required to have invariant mass .

• Events with -tagged jets are discarded.

• The three lepton should be larger than 50 GeV.

• Then, the events are divided into SRs depending on the missing transverse energy and the transverse mass , where is calculated with the missing transverse energy and the lepton which does not form the SFOS lepton pair whose mass is closest to the -boson mass.

As a validation of our analysis, we have calculated the expected numbers of events in each SRs for the “pure Wino” model points with (400,0), (600,0), (800,0), (1000,0) GeV, which are studied in the ATLAS analysis [29]. They are in good agreement with the ATLAS analysis.

In the ATLAS analysis [29], the expected 95% exclusion limit is shown in the -plane by combining disjoint versions of SRs. We have analyzed the same parameter space as a validation. In our analysis, the expected exclusion line is obtained as follows. (i) For each SR, the expected upper limit on the number of beyond-the-SM events is calculated from the number of the background events given in [29], using =1.64 [50] for 95% CL exclusion. (ii) At each model point, the expected number of signal events in each SR is calculated. (iii) The model point is excluded if and only if it is excluded in at least one of the SRs. The obtained expected exclusion line, 800 (1100) GeV for at 300 (3000) fb, agrees with the ATLAS result within an error of .

Next, we apply the same analysis in the present scenario, i.e., the Higgs- and -resonant neutralino DM. The cross sections and acceptances are calculated as follows. For , the cross sections are calculated at and .12 Then, the cross sections normalized by the coupling, , are interpolated. The acceptances are calculated by varying , and by for 100 and for 300, while the couplings are fixed as the ones of , , for simplicity.13 Here, we do not consider the region of , for simplicity.

Results are shown in Fig. 15 with red lines. The expected exclusion region at 300 fb is shown in light orange region with red dotted lines. One can see that the -peaks in the whole parameter space, including the blind spot, will be probed at 300 fb. For , the Higgs peaks can also be covered. The small region is not covered because of the small mass differences between and .

At 3000 fb, much larger parameter space will be probed, up to . The Higgs-resonant regions are covered for ( 6) for (). Though the small region can not be covered even at 3000 fb, combination with other experiments such as the direct searches can probe almost all the parameter region of the present scenario.

As can be seen in the figures, the expected reach for the chargino mass, , is almost independent of and . This can be understood as follows. In the large region, the cross section is mainly determined by because the masses are almost degenerate as , and the coupling with -boson is universal for [cf. Eqs. (11), (12), (55), (56)]. In addition, because of the large mass hierarchy , the acceptance is determined almost only by . Thus, from Eq. (31), becomes

 NSR-X≃ ∑χ±σNLO(pp→χ±χ2)⋅Br(χ±→χ1W→χ1ℓν)⋅ASR-X⋅∫Ldt (38) ×∑j=2,3Br(χj→χ1Z→χ1ℓℓ),

The first line of this equation is determined almost only by . The second line can be expanded in terms of as

 Br(χ2→Zχ1) = 12(1+sin2β)+M1μ(1−sin22β)+O(mZsWμ)2, (39) Br(χ3→Zχ1) = 12(1−sin2β)−M1μ(1−sin22β)+O(mZsWμ)2. (40)

From this expression, for , and it is almost independent of and .

### 3.5 Other constraints

Let us briefly comment on other possible constraints on the present scenario.

• indirect search.
The DM annihilation in the present Universe can lead to cosmic rays such as photons, positrons, and anti-protons. In the present model, however, the annihilation cross section in the present Universe is suppressed by the velocity, , as shown in Eqs. (16) and (17). In the limit of , the leading term in the amplitude comes from the -exchange diagram and is proportional to the mass of the final state fermion. The annihilation cross section is approximately given by

 ⟨σvrel⟩0 ≃⟨σvrel(χ1χ1→b¯b,τ¯τ)⟩0 ≃g2Z32π(λZ)23m2b+m2τm4Z(vrel→0) ∼2.8×10−26⋅(λZ)2cm3/s Missing or unrecognized delimiter for \right (41)

where we have used running bottom quark mass (cf. [51]) in the third line, and Eq. (8) in the last line.14 Therefore, it is at most , and much smaller in most of the parameter space, which is smaller than the constraints such as the Fermi-LAT bounds in Ref. [52].

• neutrinos from DM annihilation in the Sun.

Pair annihilations of DMs which are captured in the Sun generate neutrinos, and there have been searches for such neutrinos. In the present scenario, the DM annihilation rate in the Sun is proportional to the following effective SD scattering cross section [53]

 σSD(eff)p=σSDptanh2(√Γcap.Γann.⋅t