Non-relativistic effective theory approach to dark matter direct and
indirect detection
Riccardo Catena
Chalmers University of Technology, Gothenburg (Sweden)
July 19, 2016
Outline
I Introduction
I Effective theory of dark matter-nucleon interactions
(Fitzpatrick et. al, 2013)
I Comparison with observations:
R. Catena, A. Ibarra, S. Wild, JCAP 1605, 05, 039 (2016)
R. Catena and P. Gondolo, JCAP 1508, 08, 022 (2015)
R. Catena, JCAP 1507 07, 026 (2015)
R. Catena, JCAP 1409, 09, 049 (2014)








R. Catena, JCAP 1407, 07, 055 (2014)







R. Catena, JCAP 1504 04, 052 (2015)
R. Catena and P. Gondolo, JCAP 1409, 09, 045 (2014)
R. Catena and B. Schwabe JCAP 1504 04, 042 (2015)
Direct Detection
Indirect Detection
Dark matter direct detection / physical observables
I Rate of dark matter-nucleus scattering events:
X
ρχ
dR
=
ξT
dER
mT 6
mχ
T
Z
f (v + ve (t)) v
1
v >vmin (q)
galactic distribution
dσT 2 2 3
(v , q ) d v
dER
@
I
@
@
particle nature
I Modulation amplitude:
A(E− , E+ ) =
1
1h
R(E− , E+ )
E+ − E− 2
June
1st
− R(E− , E+ )
i
Dec 1st
Dark matter-nucleus scattering cross-section
I Standard paradigm: spin-independent and spin-dependent dark
matter-nucleon interactions
A
2
X
X mT
dσT
e −iq·ri (HSI + HSD ) |I i
=
|
hF
2
1
dER
2πv [jχ ][JT ] spins
i=1
nucleus ⊗ DM state
6 one-body DM-nucleon interaction
Spin-independent interaction HSI
I Scalar/Scalar coupling: LSS =
1
Λ3
P
q
CqSS χ̄χmq q̄q
I S-matrix element:
hf |iS|ii = −i ūχ (p 0 )uχ (p)
Z
d 4 x e i qx hN 0 |
X
cq q̄(x)q(x)|Ni
q
' −i(2π)4 δ 4 (q − k 0 + k) ξχ0† ξχ ξN0† (b0 + b1 τ3 )ξN
I Underlying non-relativistic Hamiltonian
HSI =
X
τ =0,1
bτ 1χ 1N t τ ≡
X
τ =0,1
c1τ 1χN t τ
Spin-dependent interaction HSD
I Axial-Vector/Axial-Vector: LAA =
1
Λ2
P
q
CqAA χ̄γ µ γ5 χq̄γµ γ5 q
I S-matrix element:
hf |iS|ii = −i ūχ (p 0 )γµ γ5 uχ (p)
Z
d 4 x e i qx hN 0 |
X
cq q̄(x)γ µ γ5 q(x)|Ni
q
4 4
0
' −i(2π) δ (q − k +
k) ξχ0† σχ ξχ
·
ξN0† (a0
+ a1 τ3 )σN ξN
I Underlying non-relativistic Hamiltonian
HSD =
X
τ =0,1
aτ σχ · σN t τ ≡
X
τ =0,1
c4τ Ŝχ · ŜN t τ
Effective Theory (ET) of dark matter-nucleon interactions
I It relies on a separation of scales: |q|/mV 1, where mV is the mediator
mass
I The most general Hamiltonian for χ-N interactions is a power series in
q̂/mV . Each term in the series preserves 3 fundamental symmetries, and
is a combination of basic operators.
I Fundamental symmetries: Galilean, rotational and translational
invariance
I Basic operators:
I
I
I
I
I
Consider the scattering χ(p) + N(k) → χ(p0 ) + N(k0 )
Momentum conservation → M(p, k, q)
Galilean invariance → M(v = p/mχ − k/mN , q)
In general, M = M(v, q, Sχ ,SN )
We therefore identify five basic operators
1χN
i q̂
v̂⊥ = v̂ +
q̂
2µN
Ŝχ
ŜN
Effective Hamiltonian for dark matter-nucleon interactions
I 14 linearly independent operators can be constructed from the basic
operators, if we demand that they are at most quadratic in q̂ (and arise
from the exchange of a mediator of spin ≤ 1)
I The most general Hamiltonian density is therefore
Ĥ(r) =
X X
τ =0,1
ckτ Ôk (r) t τ
k
• t 0 = 1, t 1 = τ3
• ckp = (ck0 + ck1 )/2 and ckn = (ck0 − ck1 )/2
Dark matter-nucleon interaction operators
Ô9 = i Ŝχ · ŜN ×
Ô1 = 1χN
Ô3 = i ŜN ·
q̂
mN
× v̂⊥
Ô4 = Ŝχ · ŜN
Ô5 = i Ŝχ · mq̂N × v̂⊥
Ô6 = Ŝχ · mq̂N
ŜN · mq̂N
Ô7 = ŜN · v̂⊥
Ô8 = Ŝχ · v̂⊥
q̂
mN
q̂
mN
i Ŝχ · mq̂N
Ô10 = i ŜN ·
Ô11 =
Ô12 = Ŝχ · ŜN × v̂⊥
Ô13 = i Ŝχ · v̂⊥ ŜN · mq̂N
Ô14 = i Ŝχ · mq̂N
ŜN · v̂⊥
h
Ô15 = − Ŝχ · mq̂N
ŜN × v̂⊥ ·
q̂
mN
i
Dark matter-nucleus scattering cross-section
I In the ET framework, the dark matter-nucleus scattering cross-section is
A Z
2
X
X dσT
mT
=
|
dr e −iq·r Ĥi (r)|I i
hF
2
dER
2πv [jχ ][JT ] spins
i=1
I This expression depends on:
- 28 coupling constants
- 8 nuclear response functions
I Available nuclear response functions, e.g.:
- For Xe, Ge, I, Na, F: Anand et al. 2013
- For 16 elements in the Sun: R. Catena & B. Schwabe 2015
Global limits: mass vs coupling constants
R. Catena and P. Gondolo, JCAP 1409 (2014) 045
3
0
2
0.4
v
−5
−2
4
4
χ
0
1
2
3
log (m /GeV)
10
2
1
2
3
log10(mχ/GeV)
−2
−4
4
2
v
0
10
10
0
−2
−4
1
2
3
log (m /GeV)
10
4
−4
1
2
3
log (m /GeV)
χ
0
−2
10
4
χ
4
4
2
2
2
−4
1
2
3
log10(mχ/GeV)
4
2
v
0
11
0
−2
−4
1
2
3
log10(mχ/GeV)
4
1
2
3
log10(mχ/GeV)
4
1
2
3
log10(mχ/GeV)
4
0
10
10
log (c
v
10
2
0
8
−2
0
m )
4
2
log (c0 m2)
4
0
1
0
7
v
6
2
0
−2
log (c m )
4
2
log (c0 m2)
4
2
log10(c5 mv )
4
2
0
0.5
4
χ
4
log10(c9 mv )
10
1
2
3
log (m /GeV)
10
3
v
log (c0 m2)
Profile likelihood color code:
0
−4
v
0
−1
χ
1D marginal pdf
1D profile Likelihood
99% CR
95% CL
1
−4
10
10
−3
0.2
1
2
3
log (m /GeV)
log (c0 m2)
−2
10
0.6
4
−1
0
0.8
1
log (c0 m2)
2
log10(c1 mv )
1
−2
−4
−2
−4
1
2
3
log10(mχ/GeV)
4
Operator interference
R. Catena and P. Gondolo, JCAP 1508, 08, 022 (2015)
1
0
1
1
1
1
c1 − c1 (all data)
c1 − c3 (all data)
0
c11 (LUX)
0.8
log10(c11 m2v )
−1
0.6
−2
0.4
−3
0.2
−4
−5
0.5
1
1.5
2
2.5
3
log10(mχ/GeV)
3.5
4
0
I Pairs of operators, or isoscalar and isovector components of the same
operator can interfere
I For instance, the operators Ô1 and Ô3 generate a transition probability
proportional to
"
c c
P 1 3 ∝
X
τ τ0
ττ0
c1 c1 WM
ττ0
+
q2
2
mN
q2
(q) +
1 q2
2
8 mN
⊥2 τ τ 0
ττ0
vT c3 c3 WΣ0
τ τ0
ττ0
c c WΦ00 (q)
2 3 3
4mN
τ τ0
(q)
ττ0
+ c1 c3 WΦ00 M (q)
!#
DAMA confronts null searches I
Is there a linear combination of Ôk such that DAMA can be reconciled with
null searches?
95% -u.l.
cT Aexp-1
(mχ ) c < 1
95% -u.l.
cT Aexp-1
(mχ ) c < 1
0
-u.l.
cT A95%
(mχ ) c < 1
exp-2
(0, 0)
cτj
cτj
0
95% -u.l.
cT Aexp-2
(mχ ) c < 1
σ -l.l.
cT AnDAMA
[E− ,E+ ] (mχ ) c > 1
cτi
R. Catena, A. Ibarra, S. Wild, JCAP 1605, 05, 039 (2016)
increasing nσ
up to nmax
σ
(0, 0)
nmax -l.l.
σ
cT ADAMA
[E− ,E+ ] (mχ ) c > 1
cτi
DAMA confronts null searches II
DM with non-zero spin
DM with zero spin
7σ
7σ
DD + Super-K
(τ + τ − /W + W − )
DD + Super-K
(τ + τ − /W + W − )
6σ
Only DD
5σ
Nσmax
Nσmax
6σ
4σ
Only DD
5σ
4σ
QNa = 0.3, QI = 0.09
3σ
3
10
QNa = 0.3, QI = 0.09
3σ
100
mχ
3
10
DM with non-zero spin
[GeV]
DM with zero spin
7σ
7σ
DD + Super-K
(τ + τ − /W + W − )
DD + Super-K
(τ + τ − /W + W − )
6σ
6σ
Only DD
5σ
Nσmax
Nσmax
100
mχ
[GeV]
4σ
Only DD
5σ
4σ
QNa = 0.4, QI = 0.05
3σ
3
10
100
mχ
[GeV]
QNa = 0.4, QI = 0.05
3σ
3
10
100
mχ
[GeV]
Neutrino telescopes: highlights
R. Catena, JCAP 1504 04, 052 (2015)
R. Catena and B. Schwabe JCAP 1504 04, 042 (2015)
c04 vs. c011
22
c07 ≠ 0
7
10
0
c13 ≠ 0
7
10
c10≠0
10
IceCube (hard)
IceCube (soft)
SUPER−K (hard)
SUPER−K (soft)
LUX
COUPP
c40≠0
c011≠0
6
10
20
10
H
4
He
3
He
6
12
C
10
14
N
16
O
20
5
Ne
5
10
23
10
Na
24
c013 [m2V]
c07 [m2V]
C [1/s]
Mg
18
10
4
10
27
Al
28
Si
4
10
32
S
40
Ar
40
Ca
3
56
Fe
3
10
10
59
Ni
Total
LUX
16
10
2
2
10
14
10
10
1
1
10
2
10
mχ [GeV]
3
10
10
1
1
2
10
10
mχ [GeV]
• Ô4 = Ŝχ · ŜN , Ô11 = i Ŝχ · q̂/mN
• Ô7 = ŜN · v̂⊥
• Ô13 = i Ŝχ · v̂⊥ ŜN ·
q̂
mN
3
10
4
10
10
1
10
2
10
mχ [GeV]
3
10
4
10
Conclusions
I A complete classification of all one-body dark matter-nucleon interaction
operators has recently been performed
I Current direct detection data place interesting constraints on dark
matter-nucleon interaction operators commonly neglected
I Destructive interference effects can weaken standard direct detection
exclusion limits by up to 1 order of magnitude in the coupling constants
I Even within this general theoretical framework, it seems to be difficult to
reconcile DAMA with null searches