Effective Field Theory methods and
direct detection of Dark Matter
Massimiliano Procura
Albert Einstein Center for Fundamental Physics, University of Bern
Theory Seminar, NIKHEF, June 5, 2014
Outline
Introduction: Dark Matter and direct detection
Effective field theories describing interactions between DM and SM fields
New constraints on DM effective theories from SM loops and direct detection
Improved evaluation of nucleon matrix elements for scalar quark currents
Conclusions
A. Crivellin, F. D’Eramo, M.P., arXiv: 1402.1173, Phys. Rev. Lett. 112 (2014) 191304
A. Crivellin, M. Hoferichter, M.P., arXiv: 1312.4951, Phys. Rev. D 89 (2014) 054021
Dark Matter
Astrophysical observations: most of our Universe is non-baryonic and dark
Establishing the nature of DM is crucial for astrophysics and particle physics
Widely discussed DM candidate: Weakly Interacting Massive Particle (WIMP),
characterized by weak scale mass and weak scale cross sections with SM fields
ubiquitous in New Physics models addressing the SM hierarchy problem
a stable WIMP with mass ranging between about 10 GeV and 1 TeV has
the right thermal relic abundance to account for DM energy density
Interactions between WIMPs and SM fields: direct, indirect and collider searches
In this talk focus on DIRECT SEARCHES: measure (or set limits on) nuclear
recoil spectra for elastic scattering of WIMPs off target nuclei
(typically ERecoil ≈ 1-100 keV for vWIMP/c ≈ 10-3)
Spin-independent DD cross section
The zero-momentum WIMP-nucleus cross section has a spin-independent (SI) and
a spin-dependent part (momentum dependence encoded in form factors)
SI
σWIMP−nucleus
�2
µ2A �
=
Zfp + (A − Z)fn
π
WIMP-nucleus reduced mass
SI
Coherence makes σWIMP−nucleus
scale like A2 if fp = fn (no “isospin violation”)
higher sensitivity than spin-dependent case
negative searches put stringent bounds on UV-complete models and on
effective theories for DM-SM interactions
(in this talk we assume that DM is the only non-SM experimentally
accessible particle)
DM and effective operators
Integrate out heavy mediators, match onto an EFT whose d.o.f. are DM field χ and
SM fields ( mχ smaller than the New Physics scale Λ )
Supplement the SM Lagrangian with new operators allowed by symmetries
LSMχ =
�
d>4
(d)
LSMχ
(d)
LSMχ
=
�
(d)
Cα
(d)
Oα
α
dimensionless Wilson coefficients
∼
1
Λd−4
Allows us to:
argue independently of details of specific UV completions
constrain UV models via bounds on Wilson coefficients from different
experiments (complementarity of searches)
properly connect operators at different scales, in terms of the effective
d.o.f. at each scale: matching, running, mixing effects
DM and effective operators
EFT provides systematic framework to connect different scales
Λ (� 1 TeV)
Heavy messengers are integrated out: in our scenario,
only DM and SM fields are effective d.o.f. for scales ≤ Λ
EW and QCD running and mixing effects (due to SM loops)
EWSB scale
Top, Higgs, W and Z get integrated out (matching)
QCD and QED evolution
Direct detection scale
(hadronic scale)
u-, d-, s-quark and gluons appear in nucleon matrix
elements (heavier quarks integrated out)
DM and effective operators
EFT provides systematic framework to connect different scales
Λ (� 1 TeV)
Heavy messengers are integrated out: in our scenario,
only DM and SM fields are effective d.o.f. for scales ≤ Λ
EW and QCD running and mixing effects (due to SM loops)
EWSB scale
Top, Higgs, W and Z get integrated out (matching)
QCD and QED evolution
Direct detection scale
(hadronic scale)
u-, d-, s-quark and gluons appear in nucleon matrix
elements (heavier quarks integrated out)
Found interesting new effects from EW corrections and a better way to estimate
hadronic uncertainties in nucleon matrix elements for DD: both with EFT methods
Assumptions on the DM field
In this talk:
DM is a Dirac fermion and a SM gauge singlet
SI DM-nucleon scattering in EFT
Operators with SM d.o.f. (light q and g) appearing in SI nucleon matrix elements:
VV
Oqq
1
= 2 χ̄γ µ χ q̄ γµ q
Λ
SS
Oqq
mq
= 3 χ̄χ q̄q
Λ
S
Ogg
αs
= 3 χ̄χ Gµν Gµν
Λ
Correspondingly, the SI DM-nucleon cross section, up to dim. 7
SI
σN
=
m2χ
(mχ +
m2N
mN )2
� �
� �
�� 2
�
�
m
N
V
V
N
SS
N
S
N
�
�
C
f
+
C
f
−
12π
C
f
qq
q
gg
Q
qq
V
q
�
π Λ4 �
Λ
q=u,d
p
q=u,d,s
p
vector couplings: fV = fVnd = 2fV = 2fVnu = 2
u
d
N
N
scalar couplings: �N |mq q̄q|N � = fq mN , for heavy quarks fQ
S
The Wilson coefficient Cgg
encodes matching corrections from
integrating out t-, b- and c-quarks
SI DM-nucleon scattering in EFT
Operators with SM d.o.f. (light q and g) appearing in SI nucleon matrix elements:
VV
Oqq
1
= 2 χ̄γ µ χ q̄ γµ q
Λ
SS
Oqq
mq
= 3 χ̄χ q̄q
Λ
S
Ogg
αs
= 3 χ̄χ Gµν Gµν
Λ
Correspondingly, the SI DM-nucleon cross section, up to dim. 7
SI
σN
=
m2χ
(mχ +
m2N
mN )2
� �
� �
�� 2
�
�
m
N
V
V
N
SS
N
S
N
�
�
C
f
+
C
f
−
12π
C
f
qq
q
gg
Q
qq
V
q
�
π Λ4 �
Λ
q=u,d
q=u,d,s
Wilson coefficients of operators at the scale Λ are related to these
Wilson coefficients through matching corrections, running and mixing
In the vector sector: interesting EW loop effects that lead to new bounds
SI
on operators at the matching scale Λ from experimental bounds on σN
Relevant operators at Λ up to dim. 6
Consider operators at the New Physics scale Λ (> EWSB scale) contributing to this
set of low-scale effective operators via matching corrections, running and mixing
At dim.5 :
T
OM
1 µν
= χ̄σ χ Bµν
Λ
S
OHH
1
= χ̄χ H † H
Λ
U (1)Y field strength tensor
SM Higgs doublet
T
VV
VA
OM
contributes to the dim. 6 Oqq
and Oqq
(matching corrections in the
Wilson coefficients by integrating out the Z boson)
SS
S
OHH
contributes to the dim. 7 Oqq
after EWSB (tree-level Higgs exchange) and
S
after integrating out heavy quarks also to Ogg
finite matching corrections
v
g
χ̄
t
S
OHH
h0
χ
g
Relevant operators at Λ up to dim. 6
Consider operators at the New Physics scale Λ (> EWSB scale) contributing to this
set of low-scale effective operators via matching corrections, running and mixing
At dim.5 :
T
OM
1 µν
= χ̄σ χ Bµν
Λ
S
OHH
1
= χ̄χ H † H
Λ
U (1)Y field strength tensor
SM Higgs doublet
T
VV
VA
OM
contributes to the dim. 6 Oqq
and Oqq
(matching corrections in the
Wilson coefficients by integrating out the Z boson)
SS
S
OHH
contributes to the dim. 7 Oqq
after EWSB (tree-level Higgs exchange) and
S
after integrating out heavy quarks also to Ogg
finite matching corrections
q̄
S
SS
Mixing of Ogg
onto Oqq
has negligible impact on
SI DD cross section
χ̄
g
Well known in the literature
S
Ogg
g
χ
q
Relevant operators at Λ up to dim. 6
At dim. 6,
VA
Oqq
1
= 2 χ̄γ µ χ q̄ γµ γ5 q
Λ
was assumed not to contribute to SI WIMP-nucleon scattering
However, due to mixing with
V
OHHD
i
= 2 χ̄ γ µ χ [H † (Dµ H) − (Dµ H)† H]
Λ
VA
Cqq
enters the SI direct detection cross section, and this leads to improved
bounds on this Wilson coefficient
Loops should be systematically taken into account when connecting different scales
EW mixing at dim. 6
Mixing at one loop is due to 6 diagrams like
H
χ
H
χ
q
VA
Oqq
q
VA
Oqq
W i, B
χ
H
W i, B
χ
H
Evolution between Λ and the EWSB scale :
2
Y
µ
q
V
V
3
VA
CHHD (µ) = CHHD (Λ) − 2 Tq Nc 2 Cqq ln
4π
Λ
Tu3 = 1/2 ,
Td3 = −1/2
dominated by the top Yukawa
EW mixing at dim. 6
V
(µEWSB ) leads to matching corrections at EWSB scale to VV
A non-vanishing CHHD
Wilson coefficients by replacing both Higgs fields with VEVs and attaching a quark
pair to the Z boson that gets integrated out :
VV
Cuu
→
VV
Cuu
�
�
1 4 2
V
+
− sw CHHD
,
2 3
VV
Cdd
→
VV
Cdd
�
�
1 2 2
V
+ − + sw CHHD
2 3
Combining these results,
VV
Cuu
VV
Cdd
�
�
1 4 2
V
(µEWSB ) =
(Λ) +
− sw CHHD
(Λ)
2 3
�
��
�
2
1 4 2
Yt Nc V A µEWSB
+
− sw −
Ctt ln
− (t → b) + . . .
2
2 3
4π
Λ
VV
Cuu
�
�
1 2 2
V
(µEWSB ) =
(Λ) + − + sw CHHD
(Λ)
2 3
�
��
�
2
1 2 2
Yt Nc V A µEWSB
+ − + sw −
Ctt ln
− (t → b) + . . .
2
2 3
4π
Λ
VV
Cdd
EW mixing at dim. 6
EW radiative corrections generate:
quark vector currents from quark axial-vector currents
operators with light quarks from operators with heavy quarks
(Dark Matter bilinear here not affected: DM is SM gauge singlet)
Constraints on dim.6 operators
We can put constraints on Wilson coefficients that have not been bounded yet
from direct searches
VA
First scenario: Cqq
is the only non-vanishing Wilson coefficient at the scale Λ
VA
= 1 one can constrain Λ as a function of mχ from
setting Cqq
experimental bounds on the SI WIMP-nucleon cross section
VA
VV
notice that the contribution of Cqq
to Cqq
is isospin-violating, which
implies that the SI WIMP-nucleon cross section does not scale as A2
SI WIMP-nucleon cross section bounds
olumbia University, on behalf of the X ENON collaboration
For example:
10-39
WIMP-Nucleon Cross Section [cm2]
DAMA/Na
-40
10
CoGeNT
DAMA/I
CDMS (2011)
10-41
CDMS (2010)
10-42
CRESST (2011)
10-43
EDELWEISS (2011)
ZEPLIN-III
XENON100 (2010)
10-44
XENON100 (2011)
10-45
Trotta
et al.
XENON100 (2012)
Buchmueller
et al.
10-46
XENON1T (2017)
-47
10
6 7 8 9 10
20
30
40 50 60
100
200
300 400
1000
WIMP Mass [GeV/c2]
E. Aprile
et al.
collaboration),
ining WIMP mass and σW IMP−N
from
2.2(XENON1T
ton-years
with 1206.6288
00, 200, 500 GeV/c2 and σW IMP−N of 10−45 cm2 . Right:
Numerical analysis: bounds on Λ
Before our analysis, best bound from LHC limits on the cross section pp → χ̄χ + X
where X is an ISR mono-jet, mono-photon or mono-Z (Zhou, Berge, Whiteson, 1302.3619)
VA
Cqq
=1
XENON1T (proj.)
"!TeV"
100.0
50.0
SuperCDMS (proj.)
LUX
XENON 100
10.0
5.0
Thermal relic
abundance
1.0
0.5
10
LHC mono-everything
20
50
100
200
m Χ !GeV"
500
1000
BOUNDS GET SIGNIFICANTLY IMPROVED !
Numerical analysis:
VA
C
and
VV
C
VA
VV
Second scenario: both Cqq
and Cqq
are non-vanishing at the scale Λ
In the SI DM-nucleon cross section, if we assume flavor-universal DM coupling
�
��
�
1 4 2
3
µEWSB
VV
VV
VA
Cuu = Cuu (Λ) +
− sw
− 2 ln
Cqq
(Λ)
2 3
4π
Λ
VV
Cdd
�
��
�
1 2 2
3
µEWSB
VV
VA
= Cuu (Λ) + − + sw
− 2 ln
Cqq
(Λ)
2 3
4π
Λ
Then UV-complete models with only vector and axial vector quark currents at the
VV
VV
VV
(Λ) = Cdd
(Λ) ≡ Cqq
(Λ)
NP scale must fulfill the following constraints for Cuu
1.0
CVA
qq !""
0.5
LUX bounds, for mχ = 100 GeV
and Λ equal to 10, 20, 30 TeV
0.0
!0.5
!1.0
!0.06 !0.04 !0.02
0.00
CVV
qq !""
0.02
0.04
0.06
DM EFT constraints from SM loops
Our case study shows the importance of a systematic analysis which takes into
account both QCD and EW loop corrections when connecting operators at NP scale
with those at DD scale: a complete analysis for operators at dim. 6 is in progress
We plan to study also the case of non-universal DM-quark and DM-lepton
couplings, and the case of DM not being a SM gauge singlet
Dim. 7: one mixing effect for tensor operators has already been pointed out by
Haisch and Kahlhoefer 1302.4454. Dim. 7 operators at the scale Λ lead to
interesting effects involving EW field strength tensors
Systematic and complete EFT analysis up to dim. 7 at the scale Λ is desirable
SI direct detection cross section
Operators with SM d.o.f. (light q and g) appearing in SI nucleon matrix elements:
VV
Oqq
1
= 2 χ̄γ µ χ q̄ γµ q
Λ
SS
Oqq
mq
= 3 χ̄χ q̄q
Λ
S
Ogg
αs
= 3 χ̄χ Gµν Gµν
Λ
Correspondingly, the SI DM-nucleon cross section, up to dim. 7
SI
σN
=
m2χ
(mχ +
m2N
mN )2
� �
� �
�� 2
�
�
m
N
V
V
N
SS
N
S
N
�
�
C
f
+
C
f
−
12π
C
f
qq
q
gg
Q
qq
V
q
�
π Λ4 �
Λ
q=u,d
q=u,d,s
N
we now concentrate on the scalar couplings: �N |mq q̄q|N � = fq mN
fqN are sources of hadronic uncertainties: how to quantify them reliably?
Scalar couplings: traditional approach
p,n
Usually two-flavor couplings fu,d are extracted from three-flavor quantities
Ellis et al. (2000, 2008); micrOMEGAs
2�N |s̄s|N �
y=
¯
�N |ūu + dd|N
�
fdp
2z − (z − 1)y
with α =
2 + (z − 1)y
and
�N |ūu − s̄s|N �
z=
¯ − s̄s|N �
�N |dd
2 σπN
=
(1 + mu /md ) mp (1 + α)
and
¯
σπN = 1/2 �N |(mu + md )(ūu + dd)|N
�
“pion-nucleon sigma term”
y determined with large uncertainties from SU(3) Chiral Perturbation Theory
Borasoy, Meissner (1997)
z extracted from baryon octet mass relations derived in the SU(3) symmetric limit
Cheng (1989)
impossible within this approach to assign reliable theory uncertainties
Chiral Perturbation Theory
Effective theory of QCD for energies below 1 GeV
Effective d.o.f. are matter fields (nucleons, etc.) and Goldstone bosons associated
with spontaneous chiral symmetry breaking (pions in two-flavor case, also kaons
and eta in the three-flavor case)
Explicit chiral symmetry breaking due to small u-, d- (and s)-quark masses
m2π = B (mu + md ) + O(m2u,d )
Chiral expansion in small external momenta and quark (i.e. pion-, kaon-) masses
�
p
Λχ
�n
,
�
mGB
Λχ
�n
where
Λχ � 1 GeV
SU(2) ChPT better convergence in the quark mass expansion than SU(3) ChPT
Scalar couplings: our approach
Unnecessary assumptions about soft SU(3) flavor symmetry breaking can be
avoided by evaluating two-flavor matrix elements in SU(2) ChPT: better
convergence properties and reliable uncertainty estimates
From the quark mass expansion of mN with strong isospin breaking (mu �= md ) ,
fuN
σπN
=
(1 − ξ) + ∆fuN ,
2mN
with
σπN
=
(1 + ξ) + ∆fdN
2mN
md − mu
ξ=
= 0.36 ± 0.04
md + mu
∆fup = (1.0 ± 0.2) · 10−3 ,
∆fdp
fdN
= (−2.1 ± 0.4) · 10
−3
,
∆fun = (−1.0 ± 0.2) · 10−3 ,
∆fdn = (2.0 ± 0.4) · 10−3
Crivellin, Hoferichter, M.P. (2014)
(fup
fun )
p
and (fd
Isospin violation effects
−
− fdn ) are overestimated by a factor
of 2 in the traditional approach by Ellis et al. and micrOMEGAs
Scalar couplings: our approach
Unnecessary assumptions about soft SU(3) flavor symmetry breaking can be
avoided by evaluating two-flavor matrix elements in SU(2) ChPT : better
convergence properties and reliable uncertainty estimates
From the quark mass expansion of mN with strong isospin breaking (mu �= md ) ,
fuN
σπN
=
(1 − ξ) + ∆fuN ,
2mN
σπN
=
(1 + ξ) + ∆fdN
2mN
md − mu
ξ=
= 0.36 ± 0.04
md + mu
with
∆fup = (1.0 ± 0.2) · 10−3 ,
∆fdp
fdN
= (−2.1 ± 0.4) · 10
−3
,
∆fun = (−1.0 ± 0.2) · 10−3 ,
∆fdn = (2.0 ± 0.4) · 10−3
Coupling with s-quark: average of lattice QCD results, fsN = 0.043 ± 0.011
Junnarkar, Walker-Loud (2013)
Heavy quark coupling :
N
fQ
�
2�
N
N
N
=
1 − fu − fd − fs
27
Comparison between two approaches
0.05
0.06
0.05
0.04
0.04
p
n
fu,d
0.03
fu,d 0.03
0.02
0.02
0.01
0.01
35
40
45
50
55
ΣΠN !MeV"
60
65
0.00
35
40
45
50
55
ΣΠN !MeV"
60
Red bands: our approach
Yellow bands: traditional approach with y estimated from SU(3) ChPT and
z supplemented with a 30% error
Blue bands: traditional approach with y estimated from lattice value for fsN
and z supplemented with a 30% error
65
Isospin violation in the scalar sector
SS
SS
Assume that Cuu
and Cdd
for the DM-quark scalar operators at dim. 7 are
the only non-vanishing Wilson coefficients contributing to the SI cross section
2.0
p
ΣSI
n
ΣSI
1.5
1.0
0.5
0.0
0.5
1.0
1.5
SS
Cdd !Cuu
2.0
2.5
3.0
SS
responsible for isospin violation
Uncertainties get drastically reduced in our approach, hint at smaller isospinviolating effects than previously thought
Conclusions
EFTs are useful tools in the context of direct detection of Dark Matter.
They allow us to argue independently of details of specific UV completions and to
properly connect operators at different scales, involving different d.o.f.
A complete, systematic analysis for operators up to dimension 7 at the New Physics
scale is motivated and desirable. Work in progress in this direction.
Very stringent bounds on spin-independent DM-nucleon cross sections are expected
from forthcoming experiments.
Chiral effective field theory allows us to estimate hadronic uncertainties in nucleon
matrix elements contributing to direct detection. Ongoing efforts to pin down the
crucial pion-nucleon sigma term (based on lattice QCD, dispersion relations and
ChPT...) and to extend the chiral approach to incorporate nuclear effects (see for
example Cirigliano, Graesser, Ovanesyan, 1205.2695).