Hidden Fermion as Dark Matter
!
in Stueckelberg Z Model
Tzu-Chiang Yuan
National Tsing Hua University, Hsinchu
(In Collaboration with Kingman Cheung)
(arXiv: hep-ph/0701107 to appear in JHEP)
March 29, 2007
NTHU HEP Seminar
Outline
• Introduction to Stueckelberg Z ! extension of Standard Model
(StSM)
• Hidden Fermions
• Collider Implication
• Astrophysical Implication
• Conclusions
Introduction
• Stueckelberg Lagrangian (1938)
2
m
1
1 µ
1
µν
µ
(Bµ − ∂µ σ)(B − ∂ σ)
L = − Bµν B +
4
2
m
m
• Gauge invariant
δL = 0 under δBµ = ∂µ $ , δσ = m$
• Rξ gauge: LRξ = −(∂µ B µ + ξmσ)2 /2ξ
L + LRξ
2
2
1
m
1
1
m
= − Bµν B µν +
Bµ B µ − (∂µ B µ )2 + ∂µ σ∂ µ σ − ξ
σ2
4
2
2ξ
2
2
!
• R∞ → Unitary gauge: Bµ = Bµ −
1
m ∂µ σ
2 !
!
!
1 !
m
µν
L + LR∞ =⇒ − Bµν B
+
Bµ B µ
4
2
• Massive QED. Unitarity and renormalizability are manifest!
4
• Stueckelberg mechanism only works for abelian group!
• However, Stueckelberg shows up in compactification and string
theory.
• Stueckelberg extension of SM [Kors and Nath (2004)]
SU (2)L
Wµa
LStSM
LSt
× U (1)Y × [U (1)X ]hidden sector
Bµ
Cµ
= LSM + LSt
1
1
µ
2
µν
= − Cµν C + (∂µ σ − M1 Cµ − M2 Bµ ) − gX Cµ JX
4
2
• JXµ is the matter (both visible and hidden sectors in general)
current that couples to the hidden gauge field Cµ . More later.
√
• After EW symmetry breaking by the Higgs mechanism !Φ" = v/ 2

Cµ



1
3
2 
(Cµ , Bµ , Wµ ) M  Bµ 

2
Wµ3
2

M12

M =
 M1 M2
0
M1 M2
M22
+
1 2
4 gY
0
v
− 41 g2 gY v 2
2
− 41 g2 gY v 2
1 2 2
4 g2 v




• Diagonalize the mass matrix




Cµ
Zµ!




2
T
2
2
2
 Bµ  = O  Zµ  ,
m
O
M
O
=
diag(m
,
m
,
!
γ = 0) .
Z
Z




Wµ3
Aµ
6
• The m2Z ! and m2Z are given by
"
!
1
1 2
2
2
2
mZ ! , Z =
M1 + M2 + (gY + g22 )v 2 ± ∆
2
4
#
1 2 2 1 2 2 2
2
2
∆ = (M1 + M2 + gY v + g2 v ) − (M12 (gY2 + g22 )v 2 + g22 M22 v 2 )
4
4
• The orthogonal matrix O is parameterized as

c ψ c φ − sθ sφ sψ
sψ c φ + sθ sφ c ψ

O=
sψ sφ − sθ c φ c ψ
 c ψ sφ + sθ c φ sψ
−cθ sψ
cθ cψ
−cθ sφ
cθ cφ
sθ




where sφ = sin φ, cφ = cos φ and similarly for the angles ψ and θ.
7
• The angles are related to the parameters in the Lagrangian LStSM
by
M2
δ ≡ tan φ =
M1
tan ψ
gY cos φ
, tan θ =
,
g2
tan θ tan φ m2W
,
=
2
2
2
cos θ[mZ ! − mW (1 + tan θ)]
where mW = g2 v/2.
• The Stueckelberg Z ! decouples from the SM when φ → 0, since
M2
tan φ =
→ 0 ⇒ tan ψ → 0 and
M1
where θw is the Weinberg angle.
tan θ → tan θw
Matter current JX :
• If SM fermion carries X charge, one can has
1 gX
2 gX
tan φ QX (u), Qd = − −
tan φ QX (d)
Qu = −
3 gY
3 gY
However, Qneutron = 0 implies Qu + 2Qd = 0 to high precision.
QX (SM particle) = 0
=⇒
SM
JX
=0
But, for the hiddden sector, one can has
QX (hidden particle) #= 0
=⇒
hidden sector
JX
#= 0
• Mixing effects in neutral current of SM fermions ψf
C
−LN
int
3
τ
3
µ
µY
= g2 Wµ ψ̄f γ
ψf + gY Bµ ψ̄f γ
ψf
2
2
#
!"
$fZL! PL + $fZR! PR Zµ!
#
$
"
fR
fL
+ $Z PL + $Z PR Zµ + eQem Aµ ψf
= ψ̄f γ µ
where
fL,R
$Z
fL,R
$Z !
=
=
cψ
%
%
g22 + gY2 c2φ
sψ
g22 + gY2 c2φ
&
&
−c2φ gY2
c2φ gY2
3
'
'
Y
+ cψ sφ gY
.
2
Y
2τ
+ g2
2
2
3
Y
2τ
− g2
2
2
Y
+ sψ sφ gY
,
2
• Constraints on StSM.
[Feldman, Liu, and Nath, PRL 97, 021801 (2006)]
• Z mass shift requires (mZ /M1 ! 1)
!
|δ| ≤ 0.061 1 − (mZ /M1 )2
• Drell-Yan data of Stueckelberg Z !
mZ ! > 250 GeV
for
δ ≈ 0.035 ,
mZ ! > 375 GeV
for
δ ≈ 0.06 .
• Z ! width is narrow, since Z ! → SM fermions are suppressed by
mixing angles!
• Constraints on StSM.
P
H
Y
S
I
C
A
L
R
E
V
I
E
W
L
E
T
97, Nath,
021801
(2006)
[Feldman,PRL
Liu, and
PRL
97, 021801 (2006)]
• Z mass shift requires (mZ /M1 ! 1)
Stueckelberg Z’!
Signals
0
10
• D0≤
Run
II 246−275
1−1pb
−−1(mZ /M1 )2
|δ|
0.061
• CDF Run II 200 pb
σ ⋅Br(Z’ → l+l−) [pb]
• Drell-Yan data of Stueckelberg Z !
−1
10
mZ ! > 250 GeV
for
δ ≈ 0.035 ,
mZ ! > 375 GeV
for
δ ≈ 0.06 .
StSM ε constrained by EW
StSM ε = 0.05
StSM ε = 0.04
StSM ε = 0.03
StSM ε = 0.02
+ −
CDF µ µ
95% C.L.
+ −
CDF e e
95% C.L.
+ −
95% C.L.
CDF l l
+ −
95% C.L.
D0 µ µ
D0 (e+ e− + γ γ) 95% C.L.
the region
1
2 # %1 #
sin
W
2
of Eq. (5) on
Next we obta
by using a fit
We follow cl
[14] [see als
vector (vf ) an
in the StSM.
plings of the
• Z ! width is narrow, since Z ! → SM fermions are suppressed by
mixing angles!
−2
10
−3
10
200
250
300
350
11
400
450
500
550
600
650
700
750
800
StSM Z’ Mass [GeV]
FIG. 1 (color online). Z0 signal in StSM using the CDF [1] and
D0 [2] data. The data put a lower limit of about 250 GeV on MZ0
for " ) 0:035 and 375 GeV for " ) 0:06.
v‘ %a‘ &
where %L;R a
(in general c
corrections f
Hidden Fermions
• Add a pair of Dirac fermion χ and χ̄ in the hidden sector. Then
JXµχ
C
−LN
int
= χ̄γ µ QχX χ
= · · · + gX Cµ JXµχ
"
!
χ
χ
!
µ χ
= · · · + χ̄γ #γ Aµ + #Z Zµ + #Z ! Zµ χ
#χγ
= gX QχX (−cθ sφ ) ,
#χZ
= gX QχX (sψ cφ + sθ sφ cψ ) , #χZ ! = gX QχX (cψ cφ − sθ sφ sψ )
• Z ! couples to χ is not suppressed. Its width needs not to be narrow.
Drell-Yan constraint may be relaxed, if Z ! → χχ̄ is kinematic allowed.
• Photon couples to χ can be milli-charged! (#χγ # e)
• χ is stable! In general, all hidden fermions are stable w.r.t. U (1)X .
[Feinberg, Kabir, and Weinberg, PRL 3, 527 (1957)]
12
Collider Phenomenology
• Z ! → invisible χχ̄ mode is dominant.
100
Z’ -> ! -!
branching ratio
10-1
10-2
10-3
u+d+s+c+b
e+µ
"
10
-4
200
t
#
300
400
500
600
700
mZ’ (GeV)
800
900
1000
• LEPII constraint (e+ e− → Z " γ → γ + missing energy).
0.3
45 < !" < 135
DELPHI data
Cross Section limits (pb)
0.25
0.2
0.15
0.1
0.05
0
100
120
140
160
mZ’ / Missing mass (GeV)
180
200
• CDF Drell-Yan constraint (pp̄ → Z ! → e+ e− )
10-1
!(Z’) * B(Z’ -> e+ e-) (pb)
10-2
CDF 95% C.L. upper limit
10-3
10-4
10-5
Stueckelberg Z’
10-6
10-7
200
300
400
500
600
mZ’ (GeV)
700
800
900
• LHC prediction: pp → Z ! + monojet
10
0
LHC: pp -> j+Z’
Cross Section (pb)
pTj > 20 GeV, |!j| < 2.5
10
-1
10
-2
10
-3
200
gX = g2, " = 0.03
400
600
800
1000
mZ’ (GeV)
1200
1400
• ILC prediction: e+ e− → Z " + γ
10
-1
| cos !" | < 0.95
Cross Sections (pb)
E" > 10 GeV
10
-2
Ecm = 0.5 TeV
10
-3
Ecm = 1 TeV
Ecm = 1.5 TeV
10
-4
200
400
600
800
mZ’ (GeV)
1000
1200
1400
Astrophysical Implication
• χ as milli-charged dark matter candidate.
[Goldberg and Hall (1986); Holdom (1986)]
• WMAP constraint
ΩCDM h2 = 0.105 ± 0.009 (WMAP)
0.1 pb
! "σv# ! 0.95 ± 0.08 pb
Ωχ h !
"σv#
2
• Relic density calculation
• χχ̄ → fSM f¯SM , γZ ! , ZZ ! are considered.
• Thermal average in σv is ignored and v 2 ! 0.1 is used.
0.1
0.14
(a)
(b)
"v = 0.95 +- 0.16 pb
0.08
0.12
"v = 0.95 +- 0.16 pb
mZ’ = 300 GeV
m# = 60 GeV
!
m# = 60 GeV
!
0.06
mZ’ = 600 GeV
0.1
0.04
0.08
0.06
0.04
0.02
0.02
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0
0.8
0.1
0.2
0.3
0.4
gX
0.5
0.6
0.7
0.8
gX
500
(d)
450
mZ’ (GeV)
400
350
300
"v = 0.95 +- 0.16 pb
250
gX = g2, ! = 0.03
200
150
50
100
150
m# (GeV)
200
250
• WMAP constraint =⇒ gX ∼ g2 and δ = tan φ = M2 /M1 ∼ O(10−2 )
• Indirect detection of χ
• Monochromatic line from χχ̄ → γγ, γZ, γZ ! could be “smoking
gun” signal of dark matter annihilation at Galaxy center.
• Photon flux
−12 dNγ
Φγ (∆Ω, E) ≈ 5.6 × 10
dEγ
!
σv
pb
"!
1 TeV
mχ
"2
J(∆Ω)∆Ω cm−2 s−1
with the quantity J(ψ) defined by
!
"2 #
1
1
2
J(ψ) =
dsρ
(r(s, ψ))
3
8.5 kpc 0.3 GeV/cm
line of sight
• J(ψ) depends on the halo profile ρ of the dark matter
• TeV gamma-rays from Sgr A* (hypothetical super-massive black
hole) near the Galactic center had been observed recently by
CANGAROO, Whipple, HESS.
• These may play the role of continuum background for dark matter
detection. Detectability of photon line above continuum background
at GLAST and HESS [Zaharijas and Hooper, PRD 73 (2006) 103501]
2
−14
−13
−2 −1
1.9
×
(TeV/m
)
×
(10
−
10
)
cm
s
Photon flux >
χ
∼
21
Fig. 1.—Broadband spectral energy distribution (SED) of Sgr A*. Radio
data are from Zylka et al. (1995), and the IR data for quiescent state and for
flare are from Genzel et al. (2003). X-ray fluxes measured by Chandra in the
quiescent state and during a flare are from Baganoff et al. (2001, 2003). XMMNewton measurements of the X-ray flux in a flaring state is from Porquet et al.
(2003). In the same plot we also show the recent INTEGRAL detection of a
hard X-ray flux; however, because of relatively poor angular resolution, the
relevance of this flux to Sgr A* hard X-ray emission (Bélanger et al. 2004)
is not yet •established.
The same is true also for the EGRET data ( MayerTeV gamma-rays from Sgr A* (hypothetical super-massive black
Hasselwander et al. 1998), which do not allow localization of the GeV source
hole) near the Galactic center had been observed recently by
with accuracy better than 1% . The very high energy gamma-ray fluxes are obCANGAROO, Whipple, HESS.
tained by the CANGAROO (Tsuchiya et al. 2004), Whipple ( Kosack et al.
These may
play the et
role
continuum
background
dark
matter
2004), and•HESS
(Aharonian
al.of2004)
groups.
Note thatforthe
GeV
and TeV
Detectability
line above
background
gamma-raydetection.
fluxes reported
from of
thephoton
direction
of thecontinuum
Galactic center
may originate in sources
different
from Sgr
A*; therefore,
strictly
should
at GLAST
and HESS
[Zaharijas
and Hooper,
PRDspeaking,
73 (2006) they
103501]
be considered as upper limits of radiation2 from −14
Sgr A*.−13[See −2
the −1
electronic
> 1.9 × (TeV/mχ ) × (10
Photon
flux
−
10
)
cm
s
edition of the Journal for∼a color version of this figure.]
• Aharonian and Neronov, Astrophys. Journal 619, 306 (2005)
#
Gamma Ray Fluxes from ΧΧ & ΓΓ,ΓZ,ΓZ,
$Γ !cm#2 s#1 "
1. " 10#12
#
ΧΧ & ΓZ,
#14
1. " 10
''''
J ( 100, )* ( 10#3
MZ' ( 300 GeV
1. " 10#16
#
ΧΧ & ΓΓ
#
ΧΧ & ΓZ
1. " 10#18
100
200
300
mΧ !GeV"
400
500
600
Conclusions
• Stueckelberg Z ! extension of SM is interesting.
• Phenomenology of Stueckelberg Z ! is different from traditional
Z ! . Mass limits can be much lower.
• Hidden fermion carries milli-charge.
• Hidden fermion is viable dark matter candidate.
• New invisible decay mode of Z ! → χχ̄ other than neutrinos has
great impact on phenomenology.
• Hidden fermion annihilation at Galactic center can give rise
“smoking gun” signal of monochromatic line that can be probed
by next generation of gamma-ray exps. However, it faces big
challenge from astrophysical background, e.g. gamma-ray from
Sgr A*. Perhaps continuum spectrum from secondary photons
due to processes like χχ̄ → fSM f¯SM , W + W − , ... are important!