Resonances in cosmology - revisitation -

Transcription

1 Resonances in cosmology - revisitation - Bohdan Grzadkowski University of Warsaw Osaka University, February 24, 2017

2 Outline Resonance beyond the Breit-Wigner (BW) approximation Early kinetic decoupling and coupled Boltzmann equations Generic conclusions on the BW approximation U(1) vector dark matter (VDM) model Resonant self-interaction in VDM model Gauge dependance and unitarity violation in VDM annihilation Numerical results Summary M. Duch, BG, Resonances in cosmology, in progress M. Duch, BG, M. McGarrie, A stable Higgs portal with vector dark matter, JHEP 1509 (2015) 162, arxiv:

3 Resonance beyond the B-W: σ self DM DM R DM DM η Breit-Wigner resonance (2M DM M) DM self-interaction. σ self = 32πω sβ 2 i σ self M DM vrel 0 M 2 Γ 2 i (s M 2 ) 2 + Γ 2 M 2, 8πω M 3 DM η 2 δ 2 + γ 2 Γ i Mβ i, δ 4M 2 DM M 2 1, β i ( δ ) 1/2, γ Γ M and ω = (2J + 1) (2S + 1) 2

4 Resonance beyond the B-W: σv rel (x) DM SM DM R SM Breit-Wigner resonance (2M DM M) annihilation. σv rel = 64πω M 2 β i σv rel (x) = x3/2 2 π γ i γ f (δ + vrel/4) γ 2 0 dvv 2 e xv2 /4 σv P. Gondolo and G. Gelmini, Nucl. Phys. B 360, 145 (1991), K. Griest and D. Seckel, Phys. Rev. D 43, 3191 (1991), M. Ibe, H. Murayama and T. Yanagida, Phys. Rev. D 79, (2009)

5 Resonance beyond the B-W: σv rel (x) DM SM DM R SM Is the BW approximation applicable? σ 1 (s M 2 ) 2 + Γ 2 M 2 s M 2

6 Resonance beyond the B-W: σv rel (x) DM SM DM R SM vrel 1 and 2M DM M = s 4M 2 DM + M 2 DMv 2 rel M 2 The BW propagator is an approximation that follows from re-summation of an innite series of 2-point Green's functions, so in general ΓM Γ(s)M IΣ(s) IΣ(s) = 1 dπ 2 f M(R f) 2 (2π) 4 δ (4) (k R q f ) Is the BW approximation applicable? f

7 Resonance beyond the B-W: σv rel (x) σv rel σv rel M 2 Γ i Γ f s M 2 + iγm 2 M 2 Γ i Γ f s M 2 + iiσ(s) 2 where η σv rel γ i γ f (δ + v 2 rel/4) 2 + [γ non DM + γ DM (v rel )] 2 γ non DM γ DM ( Γ i Mβ, β i i 1 4M 2 DM M 2 γ i γ f (δ + vrel/4) η 2 vrel/4 2 ) 1/2 and γ i,f = Γ i,f M

8 Resonance beyond the B-W: σv rel (x) R(x) = σv rel (x) σv rel 0 = x3/2 2 π dy dx = λ 0 x 2 R(x)(Y 2 YEQ) 2 0 dv rel v 2 rele xv2 rel/4 δ 2 (δ + v 2 rel/4) 2 + η 2 v 2 rel/ σv / σv x= δ =10-5, η=1 δ =10-5, η=1/2 δ =10-4, η=1 δ =10-4, η=1/ x Thermally averaged annihilation cross-section σv rel / σv rel x=20. σv rel saturates at x η 2 /δ 2 for temperatures orders of magnitude smaller than with the naive constant width Γ(M 2 ). The short-dashed lines refer to the constant width approximation γ DM (v rel ) = γ DM (2 δ 1/2 ) = η δ 1/2 (corresponding to Γ DM (M 2 )), which can be obtained for δ < 0, whereas for δ > 0 long-dashed curves were obtained for γ DM = 0.

9 Resonance beyond the B-W: σv rel (x) dy dx = λ 0 x 2 R(x)(Y 2 Y 2 EQ) with R(x) σv rel (x) At low x, σv rel (x) is larger for the naive constant width approximation than for velocity dependent one. Velocity dependent width implies higher asymptotic DM yield.

10 Early kinetic decoupling and coupled Boltzmann equations DM SM DM R SM Resonance enhancement of DM annihilation and unsuppressed σ self /M DM Suppressed DM-SM interactions (to get Ω DM 0.1) and tiny σ(dmsm DMSM) Possibility of DM early kinetic decoupling at Tkd T WIMP kd No problems with direct detection. MeV,

11 Early kinetic decoupling and coupled Boltzmann equations If dark matter decouples kinetically, when it is non-relativistic and its thermal distribution is maintained by self-scatterings, then the DM temperature T DM evolves according to T DM R 2, The temperature of the radiation-dominated SM thermal bath, scales as T R 1. { T, if T T T DM = kd T 2 /T kd, if T < T kd, where T stands for the SM temperature.

12 Early kinetic decoupling and coupled Boltzmann equations The DM relic density can be obtained by solving the Boltzmann equation dy (x) = λ [ ] 0 dx x 2 R(x DM) Y 2 (x) YEQ(x) 2, where x m T, x DM and m T DM R(x DM ) = σv rel (x DM ) = x3/2 DM σv rel 0 2 π Y EQ (x) = 45 π g 2π 4 x 3/2 e x 8 g 0 dvv 2 e x DM v 2 /4 δ 2 + γ(0) 2 (δ + v 2 /4) 2 + γ(v) 2 The thermal average of the annihilation cross-section is calculated at with x kd m/t kd. x DM = M DM T DM = x2 x kd

13 Early kinetic decoupling and coupled Boltzmann equations ( ) x x DM = x σv rel (x) x x kd ( x R(x DM ) x kd ) R(x) R(x). The asymptotic yield expected in the early decoupling scenario is substantially reduced by more ecient annihilation.

14 Early kinetic decoupling and coupled Boltzmann equations Y[x] 10-7 Y δ=-10-7, γ=10-5, M=1TeV Y kd Y EQ Y(0) Ykd x d x f kd x f x=m/t Evolution of the dark matter yield Y (x) (blue) for dark matter in kinetic equilibrium with the SM and for simultaneous chemical and kinetic decoupling Y kd at x kd = x d (red). The decoupling temperature x d adopted for the Y kd (x) (red) was determined by the decoupling of Y (x) (the blue curve).

15 Early kinetic decoupling and coupled Boltzmann equations Dene DM temperature: T DM 2 p 2 3 2M DM for O( p) 1 n DM d 3 p (2π) 3 O( p)f( p) T DM T for a situation close to thermal equilibrium and ɛ (T T DM )/T is a parameter that measures the deviation of f( p) from a thermal distribution. The Boltzmann equation: ˆL[f] = C[f] The second moment of the Boltzmann equation: d 3 2 p p ˆL[f] d 3 2 p p (2π) 3 p 0 = (2π) 3 p 0 C[f] T. Bringmann and S. Hofmann, Thermal decoupling of WIMPs from rst principles, JCAP 0704, 016 (2007), Erratum: [JCAP 1603, no. 03, E02 (2016)]

16 Early kinetic decoupling and coupled Boltzmann equations dy dx = s [ ] Y 2 σv Hx rel 2 x=m DM/(s 2/3 Y 2 y) EQ σv rel x dy dx = 1 { 2MDM c(t )(y y Hx EQ ) + [ sy Y ( σv rel σv rel 2 ) 2 x=m DM/(s 2/3 Y EQ 2 y) Y ( σv rel σv rel 2 ) x where the temperature parameter y is dened as ]} y M DMT DM s 2/3, for sharp splitting: y { x, if T Tkd M DM T kd const., if T < T kd, T. Bringmann and S. Hofmann, Thermal decoupling of WIMPs from rst principles, JCAP 0704, 016 (2007), Erratum: [JCAP 1603, no. 03, E02 (2016)]

17 Early kinetic decoupling and coupled Boltzmann equations dy dx = s [ ] Y 2 σv Hx rel 2 x=m DM/(s 2/3 Y 2 y) EQ σv rel x dy dx = 1 { 2MDM c(t )(y y Hx EQ ) + [ sy Y ( σv rel σv rel 2 ) 2 x=m DM/(s 2/3 Y EQ 2 y) Y ( σv rel σv rel 2 ) x where the temperature parameter y is dened as the scattering rate c(t ) as c(t ) = σv rel 2 = 1 12(2π) 3 )M 4 DMT 0 y M DMT DM s 2/3, f dv rel x 3/2 4 π σv rel ]} dkk 5 ω 1 g M 2 t=0;s=m 2 DM+2M DM ω+m 2 SM ( ) 6 v2 relx vrel 2 exp v2 relx/4

18 Early kinetic decoupling and coupled Boltzmann equations Y M DM =1 TeV, δ=10-5, α=10-4 YEQ 10-4 Y Yx KD= Yx KD= y M DM =1 TeV, δ=10-5, α=10-4 y yx 10 7 KD=90 yeq x x Dark matter yield Y (left panel) and corresponding DM temperatures (right panel) in dierent kinetic decoupling scenarios. The blue curves show the solution of the set of BE, whereas the green ones refer to the sharp splitting at x kd = 90. For the red curves dark matter remains in the kinetic equilibrium during its whole evolution. Dashed curves present the corresponding results for the standard Breit-Wigner approximation (with γ δ).

19 Generic conclusions on the BW approximation Remarks: The presence of velocity-dependent width implies that Y decouples at lower x (as compared to the case with constant width Γ(M 2 )) and the asymptotic DM yield is much larger. The asymptotic yield expected in the early decoupling scenario is substantially reduced by more ecient annihilation, R(x DM ) x x R(x) R(x). kd Both eects cancel to same extend, so that the increase by the velocity depended width is reduced by 50%.

20 U(1) VDM model The model: extra U(1) gauge symmetry (A µ X ), a complex scalar eld S, whose vev generates a mass for the U(1)'s vector eld, S = (0, 1, 1, 1) under U(1) Y SU(2) L SU(3) c U(1) SM elds neutral under U(1), to ensure stability of the new vector boson, a Z2 symmetry is assumed to forbid U(1)-kinetic mixing between U(1) and U(1) Y. The extra gauge boson A µ X and the scalar S eld transform under Z 2 as follows A µ X Aµ X, S S, where S = φe iσ, so φ φ, σ σ. T. Hambye, JHEP 0901 (2009) 028, O. Lebedev, H. M. Lee, and Y. Mambrini, Phys.Lett. B707 (2012) 570, A. Falkowski, C. Gross and O. Lebedev, JHEP 05 (2015) 057

21 U(1) VDM model The scalar potential V = µ 2 H H 2 + λ H H 4 µ 2 S S 2 + λ S S 4 + κ S 2 H 2. The vector bosons masses: M W = 1 2 gv, M Z = 1 2 g 2 + g 2 v and M Z = g x v x, where H = ( 0 v 2 Positivity of the potential implies ) and S = v x 2 λ H > 0, λ S > 0, κ > 2 λ H λ S

22 U(1) VDM model The mass squared matrix M 2 for the uctuations (φ H, φ S ) and their eigenvalues ( M 2 2λH v 2 ) κvv = x κvv x 2λ S vx 2 M± 2 = λ H v 2 + λ S vx 2 ± λ 2 Svx 4 2λ H λ S v 2 vx 2 + λ 2 Hv 4 + κ 2 v 2 vx 4 ( ) M 2 M 2 ( ) h diag = 1 0 cos α sin α 0 Mh 2, R = 2 sin α cos α ( ) ( ) h1 = R 1 φh h 2 φ S where M h1 = GeV is the mass of the observed Higgs particle.

23 U(1) VDM model sin 2α = sign(λ SM λ H ) 2M 2 12 (M 211 M 222) 2 + 4(M 212) 2, cos 2α =. There are 5 real parameters in the potential: µ H, µ S, λ H, λ S and κ. Adopting the minimization conditions µ H, µ S could be replaced by v and v x. The SM vev is xed at v = GeV. Using the condition M h1 = GeV, v 2 x could be eliminated in terms of v 2, λ H, κ, λ S, λ SM = M 2 h 1 /(2v 2 ): v 2 x = v 2 4λ SM (λ H λ SM ) 4λ S (λ H λ SM ) κ 2 Eventually there are 4 independent parameters: (λ H, κ, λ S, g x ), where g x is the U(1) coupling constant.

24 U(1) VDM model Bottom part of the plot (λh < λ SM = M 2 h 1 /(2v 2 ) = 0.13): the heavier Higgs is the currently observed one. Upper part (λh > λ SM ) the lighter state is the observed one. White regions in the upper and lower parts are disallowed by the positivity conditions for v 2 x and M 2 h 2, respectively.

25 U(1) VDM model Contour plots for the vacuum expectation value of the extra scalar v x 2 S (left panel) and of the mixing angle α (right panel) in the plane (λ H, κ).

26 Vacuum stability U(1) VDM model V = µ 2 H H 2 + λ H H 4 µ 2 S S 2 + λ S S 4 + κ S 2 H 2 2-loop running of parameters adopted λ H (Q) > 0, λ S (Q) > 0, κ(q) + 2 λ H (Q)λ S (Q) > 0 1- (solid) and 2- (dashed) loop, g x[m t]= 0.3, λ H[m t]= 0.14, λ S[m t]= 0.1,κ[m t]= (solid) and 2- (dashed) loop, gx[mt]= 0.3, λh[mt]= 0.14, λs[mt]= 0.1, κ[mt]= g1 g Running couplings g3 gx Yt Quartic couplings λh λs κ κ+2 λh * λs t = Log 10 (Q [GeV]) t = Log 10 (Q [GeV])

27 U(1) VDM model The mass of the Higgs boson is known experimentally therefore within the SM the initial condition for running of λ H (Q) is xed λ H (m t ) = M 2 h 1 /(2v 2 ) = λ SM = 0.13 For VDM this is not necessarily the case: Mh 2 1 = λ H v 2 + λ S vx 2 ± λ 2 Svx 4 2λ H λ S v 2 vx 2 + λ 2 Hv 4 + κ 2 v 2 vx. 4 VDM: Larger initial values of λh such that λ H (m t ) > λ SM are allowed delaying the instability (by shifting up the scale at which λ H (Q) < 0). Even if the initial λh is smaller than its SM value, λ H (m t ) < λ SM, still there is a chance to lift the instability scale if appropriate initial value of the portal coupling κ(m t ) is chosen. β (1) SM (1) λ H = βλ H + κ 2

28 U(1) VDM model 0.25 λh(mpl) > 2π g x (m t )=g 1 (m t ),λ S (m t )=λ SM (m t ) M h2 [GeV] [κ+2 λh λs ](Q) > 0 κv > 0.85 v x 2 < λh(mt) 0.15 λh(q) > Mh 2 > Mh Mh 2 < Mh 1 MZ',Mh- < Mh+/2 LEP excluded κ 2 > 4 λhλs κv > κ(m t ) M h2 [GeV]

29 Resonant self-interaction of VDM In the VDM model, the self-interaction cross-section (Z Z Z Z ) at relative velocity v 0 can be written in the vicinity of the resonance as σ self M Z g 8π v=v 0 4 xm Z R 4 2i [4M 2 Z (1 + v2 0/4) M 2 h i ] 2 + Γ 2 h i (v 0 )M 2 h i, where R is the scalar mixing matrix (R 21 = sin α and R 22 = cos α). The velocity-dependent width equals Γ hi (v 0 ) = R 2 1iΓ SM + ηm hi v 0 /2, where Γ SM is the SM value of the Higgs width.

30 Resonant self-interaction of VDM σ self M Z g 4 xm Z v=v 0 8πMh 4 2 cos 4 α (δ + v0/4) γ(v 0 ) 2, where δ = (4M 2 Z M h 2 2 )/Mh 2 2 and γ(v) = Γ h2 (v)/m h2. On the other hand the annihilation rate into the SM is proportional to sin α. σ self < M Z v=v 0 g 4 x 32πη 2 v 2 0M 3 Z = ( 10 4 v 0 ) 2 (100 ) 3 GeV GeV 3 M Z "cusp-core" problem, σ self M Z v=v GeV 3

31 Gauge dependance and unitarity violation in VDM annihilation Gauge dependance σ 1 (δ + v 2 /4) 2 + (γ non DM + γ DM (v)) 2, where γ non DM = IΣ non DM (m 2 h 2 )/m h2 (non-dm contributions to the width) and γ DM (v) = IΣ DM (s)/m h2 (DM contributions to the imaginary part of the self-energy) with s 4M 2 Z + M 2 Z v2. [( ) g 2 x Σ DM (s) = R π 2 s 2 4M 2 Z s + 3M 2 Z + m4 h 2 s 2 4M 2 Z B 0 (s, ξm 2 Z, ξm 2 Z ) ], B 0 (s, M 2 Z, M 2 Z )+ where B 0 (s, m 2, m 2 ) is a Passarino-Veltman function, while ξ is the gauge-xing parameter.

32 Gauge dependance and unitarity violation in VDM annihilation 10-5 M Z' =1TeV, δ =10-4 σv [GeV -2 ] NON-REL UG ξ=0 ξ=1/2 ξ= x Here we illustrate consequences of gauge dependence of the resonance propagator. Results shown correspond to selected values of ξ specied in the legend. The unitary gauge (ξ ) is denoted as UG, the NON-REL curve shows results obtained within a non-relativistic approximation. We show the thermal averaged annihilation cross-section for Z Z SMSM.

33 Gauge dependance and unitarity violation in VDM annihilation M Z' =1 TeV, δ =10-4, α= Y x Y EQ NON-REL UG ξ=0 ξ=1/2 ξ=1 Here we illustrate consequences of gauge dependence of the resonance propagator. Results shown correspond to selected values of ξ specied in the legend. The unitary gauge (ξ ) is denoted as UG, the NON-REL curve shows results obtained within a non-relativistic approximation. We plot numerical solution of the Boltzmann equations for the dark matter yield Y (x).

34 Gauge dependance and unitarity violation in VDM annihilation Unitarity Z LZ L W + L W L ɛ µ pµ L (p) + M V M(s, t) s 2 s M 2 + iiσ(s) + if IΣ(s) = 0 then M(s, t) O(s 2 ) 0 + O(s) 0 + const + if IΣ(s) 0 then M(s, t) O(s 2 ) 0 + O(s) + const + Uniartity is violated: M O(s)

35 Unitarity Gauge dependance and unitarity violation in VDM annihilation 10 8 M = 1 TeV, δ = 10-4, g = 2 π smax/4m Z' ξ=0 ξ=1/2 ξ=1 ξ= α The energy s max /(4M 2 Z ) at which amplitude of the process Z Z W + W violates unitarity presented as a function of the mixing angle α for dierent gauge parameters ξ.

36 Gauge dependance and unitarity violation in VDM annihilation Unitarity σv rel (x) 4M 2 DM ds ( ) s(s 4MDM)K 2 x s 1 σ(s) M DM σv rel (x) smax smax 4M 2 DM ds ( ) s(s 4MDM)K 2 x s 1 σ(s) M DM

37 Gauge dependance and unitarity violation in VDM annihilation M Z' = 1 TeV, δ = 10-4, α = 10-4 σv smax/ σv s MAX /4M 2 Z' =3 ξ=0 ξ=1/2 ξ= ξ= x The relative error for the thermally averaged cross-section σv rel (x), which is implied by choosing s max = 12M 2 Z as the cut-o.

38 Numerical results σ self /M Z' [cm 2 /g] σ self /M Z' [cm 2 /g] 10 3 σv v0 [cm 3 /s] σv v0 [cm 3 /s] M Z' [GeV] M Z' [GeV] Result of the scan in the parameter space over M Z, δ and sin α. For each point in the plot we t α to satisfy the relic abundance constraint and then calculate the annihilation σv rel v0 and self-interaction σ self /M Z cross-section at the dispersive velocity v 0 equal to 10 km/s (left panel) and 1 km/s (right panel). The maximal value of η in the VDM model, η = 3/16, was chosen.

39 Numerical results δ δ 10-3 σv v0 [cm 3 /s] σv v0 [cm 3 /s] M Z' [GeV] M Z' [GeV] Result of the scan in the parameter space over M Z, δ and sin α. Colouring with respect to δ the dispersive velocity v 0 equal to 10 km/s (left panel) and 1 km/s (right panel). The maximal value of η in the VDM model, η = 3/16, was chosen.

40 Numerical results α α 10-2 σv v0 [cm 3 /s] σv v0 [cm 3 /s] M Z' [GeV] M Z' [GeV] 10-5 Result of the scan in the parameter space over M Z, δ and sin α. Colouring with respect to α the dispersive velocity v 0 equal to 10 km/s (left panel) and 1 km/s (right panel). The maximal value of η in the VDM model, η = 3/16, was chosen.

41 Summary Breit-Wigner approximation was modied by adopting s-dependent width ( IΣ(s)), eects are large. If non-dm contribution to the resonance width is non-negligible, then the DM abundance implies early kinetic decoupling with important numerical consequences. A possibility of enhancing the dark-matter self-interaction cross-section (σ self /M DM ) by s-channel resonance was considered in a model independent way. To illustrate generic results a model of vector U(1) dark matter (VDM) was introduced and discussed (extra neutral Higgs boson h 2 ). Within VDM model a gauge dependence and unitarity violation of the resonance enhancement was discussed. When the Fermi-LAT limits are taken into account, heavy 1 TeV DM is favored and only very limited enhancement of σ self /M DM O(10 5 ) GeV 3 is possible.