Dark matter spin e ects - fuw.edu.pl

Dark matter spin effects

Bohdan GRZADKOWSKI

University of Warsaw

• The goal• Pseudo-Goldstone dark matter (pGDM)• Vector dark matter (VDM)• Fermion dark matter (FDM)• Parameters• Constraints• Dark matter at e+e− colliders• Summary

BG, M.Iglicki, K.Mekala and A. F.Zarnecki, ”Dark-matter-spin effects at future e+e− colliders”,

JHEP 08 (2020) 052, e-Print: 2003.06719 ,

D.Azevedo, M.Duch, BG, D.Huang, M.Iglicki, R.Santos, ”One-loop contribution to dark matter-

nucleon scattering in the pseudoscalar dark matter model”, JHEP 1901 (2019) 138,

D.Azevedo, M.Duch, BG, D.Huang, M.Iglicki, R.Santos, ”Testing scalar versus vector dark matter”,

Phys.Rev. D99 (2019) no.1, 015017,

M.Duch, BG, M.McGarrie, ”A stable Higgs portal with vector dark matter”, JHEP 1509 (2015) 162

Bohdan Grzadkowski, University of Warsaw, FCC Physics and Experiments Workshop, November 10th 2020

The goal

• Find simple (not simplified) consistent and renormaizable models of Dark Mater(DM) of spin 0, 1/2 and 1.

• Impose experimental and theoretical constraints.

• Maximize signals and differences between cross-sections to compare the models ina meaningful way.


Pseudo-Goldstone Dark Matter (pGDM)

• J. Cline, T. Toma, “Pseudo-Goldstone dark matter confronts cosmic ray and collider anomalies”,

Phys.Rev.D 100 (2019) no.3, 035023,

• D. Karamitros, “Pseudo Nambu-Goldstone Dark Matter: Examples of Vanishing Direct Detection

Cross Section”, Phys.Rev. D99 (2019) no.9, 095036,

• K. Kannike, M. Raidal, “Phase Transitions and Gravitational Wave Tests of Pseudo-Goldstone Dark

Matter in the Softly Broken U(1) Scalar Singlet Model”, Phys.Rev. D99 (2019) no.11, 115010,

• T. Alanne, M. Heikinheimo, V. Keus, N. Koivunen, K. Tuominen, “Direct and indirect probes of

Goldstone dark matter”, Phys.Rev. D99 (2019) no.7, 075028,

• K Huitu, N. Koivunen, O. Lebedev, S. Mondal, T. Toma, “Probing pseudo-Goldstone dark matter

at the LHC”, arXiv:1812.05952,

• K. Ghorbani, P. Hossein Ghorbani, “Leading Loop Effects in Pseudoscalar-Higgs Portal Dark

Matter”, JHEP 1905 (2019) 096,

• K. Ishiwata, T. Toma, “Probing pseudo Nambu-Goldstone boson dark matter at loop level”, JHEP

1812 (2018) 089,

• D. Azevedo, M. Duch, BG, D. Huang, M. Iglicki, R. Santos, ”One-loop contribution to dark

matter-nucleon scattering in the pseudoscalar dark matter model”, JHEP 1901 (2019) 138,

• D. Azevedo, M. Duch, BG, D. Huang, M. Iglicki, R. Santos, ”Testing scalar versus vector dark

matter”, Phys.Rev. D99 (2019) no.1, 015017,

• C. Gross, O. Lebedev, and T. Toma, “Cancellation Mechanism for Dark-Matter–Nucleon Interaction”,

Phys. Rev. Lett. 119 (2017), no. 19 191801.


VpGDM(H,S) = −µ2H|H|2 + λH|H|4 − µ2

S|S|2 + λS|S|4 + κ|S|2|H|2 + (µ2S2 + H.c.)

S =1√2

(vS + φ+ iA) , and H =

(π+

1√2(v + h+ iπ0)

).

Positivity: λH > 0, λS > 0, κ > −2√λHλS

Symmetries:

• µ2 6= 0 breaks global U(1) softly to residual Z2 : S → −S,

• rephase S such that Imµ2 = 0 (basis choice),

• V 3 µ2(S2 + S∗ 2), so SC→ S∗ (φ→ φ and A→ −A) is a symmetry,

• global minimum at 〈S〉 = vS√2

with vS being real, so C is unbroken and A is a

stable DM candidate, m2A ∝ µ2, if µ2 → 0 then A becomes a Goldstone boson of

broken U(1),


S =1√2

(vS + φ+ iA) , and H =

(π+

1√2(v + h+ iπ0)

).

M2 =

2λHv2 κvvS 0

κvvS 2λSv2S 0

0 0 −4µ2

M2

diag =

m21 0 0

0 m22 0

0 0 m2A

, R =

(cosα − sinαsinα cosα

),

(h1h2

)= R−1

(hφ

),

VpGDM 3 λS|S|4 + κ|S|2|H|2 3 A2

2(2λSvSφ+ κvφ) =

A2

2vS(sinαm2

1h1 + cosαm22h2)

−im2i

vsR2ihi

A

A

−i(R2iR2jλS −R1iR1jκ)

hi

hj

A

A


N N

A A

h1,2

The DM direct detection signals are naturally suppressed in the pGDM model:

iM = −isinα cosαfNmN

vvS

(m2

1

q2 −m21

− m22

q2 −m22

)uN(p4)uN(p2)

≈ −isinα cosαfNmN

vvS

(m2

1 −m22

m21m

22

)q2uN(p4)uN(p2) .


N N

A A

h1,2

The total cross section σAN :

σ(tree)AN ∝ sin2α cos2α

v2S(m2

1 −m22)

2 × v4A ,

where vA is the A velocity in the lab frame. Since vA ∼ 200 km/s, the total DMnuclear recoil cross section σAN is greatly suppressed by the factor v4A ∼ 10−13:

σ(tree)AN ∼ 10−70 cm2 σ

(XENON1T )AN ∼ 10−46 cm2

⇓1-loop effects are leading

• if q2 → 0 then loop corrections are expected to be UV finite,

• if m2A ∝ µ2 → 0 then loop corrections should vanish.


N N

A A

h1, h2

h1, h2, A h1, h2, A

N N

A A

h1, h2

A

h1, h2

N N

A A

h1, h2

A

h1, h2

N N

A A

h1, h2

A A

h1, h2

N N

A A

h1, h2

h1, h2 h1, h2

A

N N

A A

h1, h2

h1, h2

h1, h2, A

N N

A A

h1, h2

h1, h2

h1, h2

h1, h2, A

N N

A A

h1, h2

h1, h2h1, h2, A

N N

A A

h1, h2

h1, h2

h1, h2, A h1, h2, A

Figure 1: 1-loop diagrams contributing to A-nucleon scattering.

σ(1)AN =

f2Nπv2

m2Nµ

2AN

m2A

F2 ,

where the one-loop function F is defined as

F =V

(1)AA1cαm2

1

− V(1)AA2sαm2

2

with V(1)AA1 ,AA2 as one-loop corrections to the vertices h1A

2 and h2A2.


σAN =µ2m2

DM

π· m

2N

v2f2N

m41m

42

· sin2α cos2α

v2S(m2

1 −m22)

2 ·[ A

64π2vv2S

]2,

A =a1 · C2(0,m2DM,m

2DM,m

21,m

22,m

2DM)+

a2 ·D3(0, 0,m2DM,m

2DM, 0,m

2DM,m

21,m

21,m

22,m

2DM)+

a3 ·D3(0, 0,m2DM,m

2DM, 0,m

2DM,m

21,m

22,m

22,m

2DM)

with a1,2,3 being coefficientsComments:

• The one loop amplitude F is UV finite in the limit of zero momentum transferq2 → 0,

• F → 0 for mA → 0,

•V ⊃ M3

√2

(S + S∗) + µ2(S2 + S∗2)

LA ⊃1

2(∂µA∂µA−m2

AA2)− R2i

2vs

(m2i +

M3

vS

)hiA

2 .


The Vector Dark Matter (VDM) model

• T. Hambye, “Hidden vector dark matter”, JHEP 0901 (2009) 028,

• O. Lebedev, H. M. Lee, and Y. Mambrini, “Vector Higgs-portal dark matter andthe invisible Higgs”, Phys.Lett. B707 (2012) 570,

• Y. Farzan and A. R. Akbarieh, “VDM: A model for Vector Dark Matter”, JCAP1210 (2012) 026,

• S. Baek, P. Ko, W.-I. Park, and E. Senaha, “Higgs Portal Vector Dark Matter :Revisited”, JHEP 1305 (2013) 036,

• Ch. Gross, O. Lebedev, Y. Mambrini, “Non-Abelian gauge fields as dark matter”,arXiv:1505.07480,

• · · ·


The model:

• extra U(1)X gauge symmetry (AµX), DM candidate: AµX,

• a complex scalar field S, whose vev generates a mass for the U(1)’s vector field,S = (0,1,1, 1) under U(1)Y × SU(2)L × SU(3)c × U(1)X.

• SM fields neutral under U(1)X,

• in order to ensure stability of the new vector boson a Z2 symmetry is assumed toforbid U(1)-kinetic mixing between U(1)X and U(1)Y . The extra gauge boson Aµand the scalar S transform under Z2 (dark charge conjugation) as follows

AµXC→ −AµX , S

C→ S∗

The scalar potential

VVDM(H,S) = −µ2H|H|2 + λH|H|4 − µ2

S|S|2 + λS|S|4 + κ|S|2|H|2.


For κ2 < 4λHλS the global minimum is

v2 =4λSµ

2H − 2κµ2

S

4λHλS − κ2and v2S =

4λHµ2S − 2κµ2

H

4λHλS − κ2

The mass squared matrix M2 for the fluctuations (h, φ) and their eigenvalues read

M2 =

(2λHv

2 κvvSκvvS 2λSv

2S

)m2± = λHv

2 + λSv2S ±

√λ2Sv

4S − 2λHλSv2v2S + λ2Hv

4 + κ2v2v4S

M2diag =

(m2

1 00 m2

2

), R =

(cosα − sinαsinα cosα

),

(h1h2

)= R−1

(hφ

),

2imXgxR2ihi

X

X

2ig2xR2iR2j

hi

hj

X

X


Fermion Dark Matter (FDM)

• DM: χ - a left-handed Dirac fermion,

• spin 0 mediator: S - a real field.

Z4 : S → −S , χ→ iχ

L = LSM + iχ/∂χ+1

2∂µS ∂µS −

yx2

(χcχ+ χχc)S − V (H,S) ,

VFDM(H,S) = −µ2H|H|2 + λH|H|4 −

µ2S

2S2 +

λS4S4 +

κ

2|H|2S2 ,

where χc ≡ −iγ2χ∗ and

S = vS + φ , H =

(π+

v+h+iπ0√2

)


After SSB relevant parts of the Lagrangian take the following form:

iχ/∂χ+1

2∂µS ∂µS −

yx2

(χcχ+ χχc)S → i

2ψ /∂ψ +

1

2∂µφ ∂µφ−

yxvS2

ψψ − yx2ψψφ

where ψ = ψc ≡ χ+ χc is a Majorana mass eigenstate with mψ = yxvS.

(h1h2

)= R−1

(hφ

),R =

[cosα − sinαsinα cosα

], α ∈

[−π

4,π

4

], tan 2α =

κvvSλHv2 − λSv2S

.

−iyxR2ihi

ψ

ψ

Figure 2: The vertex relevant for the FDM model.


Parameters

VpGDM(H,S) = −µ2H|H|2 + λH|H|4 − µ2

S|S|2 + λS|S|4 + κ|S|2|H|2 + (µ2S2 + H.c.)

VVDM(H,S) = −µ2H|H|2 + λH|H|4 − µ2

S|S|2 + λS|S|4 + κ|S|2|H|2

VFDM(H,S) = −µ2H|H|2 + λH|H|4 −

µ2S

2S2 +

λS4S4 +

κ

2|H|2S2

S = vS + φ (+iA) , H =

(π+

v+h+iπ0√2

)(h1h2

)= R−1

(hφ

),R =

[cosα − sinαsinα cosα

]It is convenient to use the same input parameters for all the models:

m2, sinα, mDM ≡ (mA,mX,mψ) and vS

• ILC:√s = 250 GeV =⇒ mDM <∼ 77.5 GeV


Constraints

1. Perturbativity: yx < 4π(mψ

vS< 4π

), gX < 4π

(mXvS

< 4π), κ < 4π, mi

vS< 4π,

2. Vacuum stability,

3. Minimum globality,

4. Higgs invisible decays (1809.05937):

BR(h1 → inv) = BR(h1 → DM DM) < 19%

5. The mixing angle: 0 < sinα < 0.3 (1501.02234, 1604.04552).


https://arxiv.org/pdf/1809.05937.pdf



6. The Planck data (1807.06209): h2 ΩDM = 0.12± 0.0012. The thermally averagedcross section (DM DM → ff) reads:

〈σv〉 =nc3

mDMm2f

πv2· sin

2α cos2α

v2S(m2

1 −m22)

2 ··(m2

DM −m2f

)3/2(4m2

DM −m21)

2(4m2DM −m2

2)2·

×

12 +O

[(mDMT

)−1](pGDM)

1 +O[(mDM

T

)−1](VDM)

94

(mDMT

)−1+O

[(mDMT

)−2](FDM)

〈σv〉 = σ0x−n =⇒ h2ΩDM ∝

(n+ 1)xn+1f

σ0(x ≡ mDM

T)

⇓sin2α cos2α

v2S(m2

1 −m22)

2 =

= 2.1 · 10−5 GeV−2(4m2

DM −m21)

2 (4m2DM −m2

2)2

mDM(m2DM −m2

b)3/2

·

112 (pGDM)

1 (VDM)4mDM9Tf

(FDM)



7. Indirect-detection (1611.03184) limits.

101 102 103 104

DM Mass (GeV)

10−27

10−26

10−25

10−24

10−23

〈σv〉(

cm3

s−1)

bb

Ackermann et al. (2015)

Nominal sample

Median Expected

68% Containment

95% Containment

Thermal Relic Cross Section(Steigman et al. 2012)

101 102 103 104

DM Mass (GeV)

10−27

10−26

10−25

10−24

10−23

〈σv〉(

cm3

s−1)

τ+τ−

Ackermann et al. (2015)

Nominal sample

Median Expected

68% Containment

95% Containment

Thermal Relic Cross Section(Steigman et al. 2012)

Figure 3: Relic-density and indirect-detection limits. (1611.03184)

• For pGDM and VDM: mA,mX & 30 GeV, since otherwise the proper value ofannihilation cross section is forbidden.• For FDM due to the T0/Tf factor, current annihilation cross-section corresponding

to the correct value of relic density is orders of magnitude times smaller thanthe one for the pGDM and VDM, hence it satisfies the ID limit.




8. Direct-detecion (1805.12562) constraint the spin-independent nucleon-scatteringcross section.

101 102 103

WIMP mass [GeV/c2]

10−47

10−46

10−45

10−44

10−43

WIM

P-nu

cleo

nσ S

I[c

m2 ]

LUX (2017)PandaX-II (2017)

XENON1T (1 t×yr, this work)

101 102 103

WIMP mass [GeV/c2]

10−1

100

101

Nor

mal

ized

Figure 4: Direct-detection limits. (1805.12562)

For VDM and FDM:

σSI 'µ2m2

DM

π· m

2N

v2f2N

m41m

42

· sin2α cos2α

v2S(m2

1 −m22)

2 ,




For pGDM 1-loop calculations are needed (1810.06105).

σSI =µ2m2

DM

π· m

2N

v2f2N

m41m

42

· sin2α cos2α

v2S(m2

1 −m22)

2 ·[ A

64π2vv2S

]2,

A =a1 · C2(0,m2DM,m

2DM,m

21,m

22,m

2DM)+

a2 ·D3(0, 0,m2DM,m

2DM, 0,m

2DM,m

21,m

21,m

22,m

2DM)+

a3 ·D3(0, 0,m2DM,m

2DM, 0,m

2DM,m

21,m

22,m

22,m

2DM)

with [ A64π2vv2S

]2<∼ 10−5

In considered range of parameters, in the case of pGDM, the DD upper bound on

the value of sin2 α cos2 αv2S

(m21 −m2

2)2 is always higher than the value corresponding

to the correct relic density (for sinα that maximize the cross section for given m2

and mDM).



σSI =µ2m2

DM

π· m

2N

v2f2N

m41m

42

· sin2α cos2α

v2S(m2

1 −m22)

2 ·[ A

64π2vv2S

]2,

The XENON1T limit for mDM & 100 GeV can be parametrized as:

σmaxSI

1 cm2=

mDM

1 GeV· 10−48.05 .

Hence, the strictest possible DD limit reads

sin2α cos2α

v2S(m2

1 −m22)

2 <m4

2

mDM

v2

m2N

m41

f2N

π

µ2

1 cm2

1 GeV· 10−48.05 ·

[

A64π2vv2

S

]−2(pGDM)

1 (FDM,VDM)=

=m4

2

mDM· 1.5 · 10−6 GeV−1 ·

105 (pGDM)

1 (FDM,VDM).


Dark matter at e+e− colliders

e−

e+ Z

χ

χZ

Q

hi

Figure 5: Feynman diagram for e+e− → Zχχ, χ denotes the dark particle (χ =A,X,ψ).

• P. Ko, H. Yokoya, “Search for Higgs portal DM at the ILC”, JHEP 1608 (2016)109,

• T. Kamon, P. Ko, J. Li “Characterizing Higgs portal dark matter models at theILC”, Eur.Phys.J. C77 (2017) no.9, 652


e−

e+ Z

χ

χZ

Q

hi

Figure 6: Feynman diagram for e+e− → Zχχ, χ denotes the dark particle (χ =A,X,ψ).

dσ

dQ2(Q2) =

f(s,√Q2)

32π2

(Q2)2 · sin2α cos2α ·

[(m2

1 −m22)

2 + (m1Γ1 −m2Γ2)2]

[(Q2 −m21)

2 + (m1Γ1)2] [(Q2 −m22)

2 + (m2Γ2)2]×

×√

1− 4m2

DM

Q2· 1

v2S·

1 (pGDM)

1− 4m2

DMQ2 + 12

(m2

DMQ2

)2(VDM)

2[m2

DMQ2 − 4(

m2DMQ2 )2

](FDM)


100 200 300 400 500 600 700EZ[GeV]

10-14

10-11

10-8

dσ

dEZ[pb·GeV-1]

s = 1.5 TeV, m2 = 700 GeV, vS = 5.54 TeV

two-pole case: mDM = 60 GeV, sinα = 0.01

one-pole case: mDM = 200 GeV, sinα = 0.05

no-pole case: mDM = 500 GeV, sinα = 0.3

Q2=m12

Q2=m22Figure 7: dσ

dEZfor the pGDM model.

EZ(Q2 = m2i ) = Ei ≡

s−m2i +m2

Z

2√s

, Emax =s− 4m2

DM +m2Z

2√s


Background

e−

e+

νe

Z

νe

νe

W

e−

e+

νe, νµ, ντ

νe, νµ, ντ

Z

Z

Figure 8: Exemplary diagrams of the Standard Model background processes. Neutrinoscontribute to missing energy and can therefore mimic dark particles. The backgroundcross-section could be reduced by polarizing the initial e+ and e− beams.


Strategy

• Fix√s e.g. at 240 GeV for the CEPC, or

√s = 250 GeV for the ILC,

• Parameters: m2, mDM, sinα and vS,

• For given (m2,mDM) and sinα, the value of vS is derived by solving the relic-densitycondition, so vS is no longer independent:

sin2α cos2α

v2S(m2

1 −m22)

2 =

= 2.1 · 10−5 GeV−2(m2

1 − 4m2DM)2 (m2

2 − 4m2DM)2

mDM(m2DM −m2

b)3/2

·

112 (pGDM)

1 (VDM)4mDM9Tf

(FDM)

• For each point (m2,mDM) of the plot, we choose such value of sinα ≤ 0.3 thatmaximizes the total cross section.


Extra scalars at 500 ILC → Exclusion Limits

Prelimiary results for 500 GeV.

Yan Wang | Searching for new extra scalars at the ILC | October 25, 2018 | 17/18

Figure 9: Expected sensitivity for the measurement of the cross-section for e+e− →DM DM Z at the ILC, from Yan Wang (DESY, IHEP) at LCWS 2018, Arlington,October 25, 2018. κ95 ≡ σ(e+e− → · · ·+ Z)/σSM(e+e− → hSMZ)


benchmark point for pGDM

m2 = 120.8 GeV , mDM = 58.9 GeV ,

sinα = 0.30 , vS = 646 GeV ,

Γ1 = 7.4 · 10−3 GeV , Γ2 = 9.8 · 10−3 GeV ,

BR(h1 → DM) = 19% , BR(h2 → DM) = 95% ,

σ = 62 fb

Figure 10: For the pGDM, the allowed region (greenish), the region forbidden by theinvisible BR of h1 (cyan) where BR(h1 → DM) > 19% and the gray region wherethe cross section falls below its expected precision at the 95% CL. Coloring of thegreenish area shows the value of the normalized total cross section σ/σSM . The stardenotes the chosen benchmark point, characterized by relatively high cross section.


benchmark point for VDM

m2 = 118.4 GeV , mDM = 58.5 GeV ,

sinα = 0.30 , vS = 561 GeV ,

Γ1 = 7.4 · 10−3 GeV , Γ2 = 6.4 · 10−3 GeV ,

BR(h1 → DM) = 18% , BR(h2 → DM) = 92% ,

σ = 61 fb

Figure 11: As in Fig. 10 for the VDM model. The region forbidden by the DDconstraint is denoted in black.


benchmark point for FDM

m2 = 123.6 GeV , mDM = 61.1 GeV ,

sinα = 0.30 , vS = 76 GeV ,

Γ1 = 7.4 · 10−3 GeV , Γ2 = 5.9 · 10−3 GeV ,

BR(h1 → DM) = 18% , BR(h2 → DM) = 91% ,

σ = 59 fb

Figure 12: As in Fig. 10 for the FDM model. The region forbidden by the DDconstraint is denoted by black.


Figure 13: Top-left: pGDM model, top-right: FDM model, bottom-left: VDM model,bottom-right: the three models combined. Light- and dark-gray regions denoteviolation of perturbativity conditions. Note that the |κ| < 4π condition is not violatedin any place of the considered range of parameters.


Figure 14: The difference between predictions for the pGDM and the VDM. The grayregion denotes parameter space for which the difference is smaller than the limit ofFig. 9. The models are compared in the region where both of them are consistentwith the data, see Fig. 13.


Figure 15: The difference between predictions for the pGDM and the FDM (leftpanel), and the FDM and the VDM (right panel). The gray region denotes parameterspace for which the difference is smaller than the limit of Fig. 9. The models arecompared in the region where both of them are consistent with the data, see Fig. 13.


Summary

1. The following models were discussed and compared:

• pGDM with complex scalar field S and U(1) global symmetry softly broken byµ2(S2 + S∗ 2),• VDM with gauged U(1)X and complex S,• FDM with chiral χ and real S.

2. Perturbativity, vacuum stability and globality of the vacuum ensured.

3. • Limits for invisible Higgs boson decays BR(h1 → DM DM) < 19% and LHClimits on sinα < 0.30 imposed.• Limits from ID and DD satisfied.• DM abundance constraint satisfied.

4. Direct detection efficiently suppressed in the pGDM model, σSI ∝ v4A, as aconsequence of A being a pseudo-Goldstone boson, 1-loop calculations wereperformed and adopted. For q2 = 0 the 1-loop results are UV finite and vanish inthe limit mA = mDM → 0.

5. In some regions of (m2,mDM) space e+e− colliders might be useful to disentanglethe models.


Backup slides: Vacuum stability

VVDM(H,S) = −µ2H|H|2 + λH|H|4 − µ2

S|S|2 + λS|S|4 + κ|S|2|H|2

λH(Q) > 0, λS(Q) > 0, κ(Q) + 2√λH(Q)λS(Q) > 0

5 10 150.2

0.4

0.6

0.8

1.0

1.2

t = Log10(Q [GeV])

Runningcouplings

g1g2g3gxYt

1- (solid) and 2- (dashed) loop, gx[mt]= 0.3, λH[mt]= 0.14, λS[mt]= 0.1,κ[mt]=-0.06

5 10 15-0.10

-0.05

0.00

0.05

0.10

0.15

0.20

t = Log10(Q [GeV])

Quarticcouplings

λH

λS

κ

κ+2 λH *λS

1- (solid) and 2- (dashed) loop, gx[mt]= 0.3, λH[mt]= 0.14, λS[mt]= 0.1, κ[mt]=-0.06

Figure 16: Running of various parameters at 1- and 2-loop, in solid and dashed linesrespectively. For this choice of parameters λH(Q) > 0 at 2-loop (right panel blue)but not at 1-loop. λS(Q) is always positive (right panel red), running of κ(Q) isvery limited, however the third positivity condition κ(Q) + 2

√λH(Q)λS(Q) > 0 is

violated at higher scales even at 2-loops (right panel green).


The mass of the Higgs boson is known experimentally therefore within the SM theinitial condition for running of λH(Q) is fixed

λH(mt) = M2h1/(2v2) = λSM = 0.13

For VDM this is not necessarily the case:

M2h1

= λHv2 + λSv

2S ±

√λ2Sv

4S − 2λHλSv2v2S + λ2Hv

4 + κ2v2v4S.

VDM:

• Larger initial values of λH such that λH(mt) > λSM are allowed delaying theinstability (by shifting up the scale at which λH(Q) < 0).

• Even if the initial λH is smaller than its SM value, λH(mt) < λSM , still there is achance to lift the instability scale if appropriate initial value of the portal couplingκ(mt) is chosen.

β(1)λH

= βSM (1)λH

+ κ2


f(s,√Q2) ≡ g2V + g2A

24π

(g2

cos θ2W

1

s−m2Z

)2λ1/2(s,Q2,m2

Z)[12 sm2

Z + λ(s,Q2,m2Z)]

8s2

Q2 = Q2(s, EZ) ≡ s− 2EZ√s+m2

Z

EZ(Q2 = m2i ) = Ei ≡

s−m2i +m2

Z

2√s

.

Emax =s− 4m2

DM +m2Z

2√s

,


Dark matter spin e ects - fuw.edu.pl

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