Model-Independent
Spin-Dependent CrossSection Limits from
Dark Matter Searches
Dan Tovey, Rick Gaitskell,
Paolo Gondolo, Yorck Ramachers and Leszek
Roszkowski.
• The current technique
• Why it is WIMP model-dependent
• A new model-independent technique
(hep-ph/0005041)
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
Background
• Weakly Interacting Massive Particles
(WIMPs) one of the best motivated
candidates for Dark Matter.
• Predicted to exist by many supersymmetric
(SUSY) models, where the Lightest
Supersymmetric Particle (LSP) is stable
and massive and hence a WIMP candidate.
• In many SUSY models LSP is the lightest
neutralino: superposition of SUSY partners
of the electroweak gauge bosons
(gauginos) and Higgs bosons (higgsinos).
• Evidence for WIMPs (SUSY or otherwise)
sought through their elastic scattering from
atomic nuclei in detector materials.
• WIMP-nucleus couplings effectively of
two types: spin-independent and spindependent.
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
Cross-Section Enhancement
• Total WIMP-nucleus scattering cross-section
A can be written:
where A is the reduced mass and CA is the
cross-section enhancement factor.
• Cross-section limits normalised to those for
free nucleons (conventionally protons):
• Normalisation model-independent for spinindependent interactions as:
• Normalisation more complex in spindependent case where:
and ap and an are WIMP model-dependent
effective couplings and J is the nuclear spin.
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
WIMP Model-Dependence
• Assume model-dependencies in form-factor
sufficiently small to be neglected (true for Na,
I, F etc. at least).
• All WIMP model-dependence therefore
contained in normalisation CA via ap and an .
• In early calculations (single-particle or oddgroup models) CA dominated by either proton
or neutron terms (but not both) =>
normalisation of odd-proton targets to WIMPproton cross-section model-independent (ap
factors cancel).
• Normalisation of odd-neutron targets modeldependent however due to factor (ap/an )2 =
p/n remaining in plim(A).
• HOWEVER, for SUSY neutralino WIMPs
early estimates of q values (used to calculate
ap and an ) gave p/n ~ 1.5, independent of
neutralino composition.......BUT......
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
SUSY Model-Dependence
• Later estimates of q show that although
this remains true for higgsino neutralinos,
it is certainly not true for gaugino
neutralinos (best motivated theoretically):
higgsinos
Dan Tovey
Dan
Tovey
gauginos
University
of Sheffield
UKDMC
SUSY Model-Dependence
• Problem now more acute because shell-model
calculations of <Sp> and <Sn> indicate both
proton and neutron contributions to CA can be
significant =>
Model-dependent factor (ap/an )2 = p/n
present even in limits from odd-proton targets
(Na, I, F etc.).
• WIMP-proton cross-section limits vary with
assumed neutralino WIMP composition:
10
0.1
p
lim(A)
plim(A)(pb)
(pb)
1
0.01
Na (higgsino)
I (higgsino)
NaI (higgsino)
NaI (gaugino)
(a) Current Technique
0.001
10
100
1000
10000
WIMP Mass (GeV/c 2)
WIMP Mass (GeV)
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
Model-Independent Limits
To solve this problem identify separate proton
and neutron contributions to CA :
The separate proton and neutron contributions
to A are thus:
This then gives:
Now assume independently that A ~ Ap and
A ~ An . Then define WIMP-proton(neutron)
cross-section limit set by protons(neutrons) in
target A:
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
Enhancement Factor Ratios
• Use of the cross-section enhancement
factor ratios
,
,
ensures cancellation of WIMP modeldependent ap and an terms in A .
• Values of CAp/Cp and CAn/Cn obtained from
shell-model calculations:
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
Combining Limits
• Key point of new procedure is that limits from
different nucleons in the same nucleus can still
be combined in a model-independent fashion
using:
• For a detector consisting of more than one
target nucleus Ai limits are combined using:
• Spin-independent (SI) and spin-dependent
limits from protons and neutrons in several
target nuclei Ai can be combined using:
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
NaI Example
• Consider limits from same hypothetical
NaI(Tl) detector.
• WIMP-proton cross-section limit using
new definition:
10
Excluded Region
plim(A)
plim(A) (pb)
(pb)
1
0.1
0.01
Na
I
NaI
(b) New Technique
0.001
10
100
1000
10000
WIMP Mass (GeV/c 2)
WIMP Mass (GeV)
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
NaI Example
• Similarly WIMP-proton cross-section limit
using new definition.
• Limits less stringent due to lack of oddneutron target.
10
nlim(A)
nlim(A) (pb)
(pb)
1
Excluded Region
0.1
0.01
Na
I
NaI
(c) New Technique
0.001
10
100
1000
10000
WIMP Mass (GeV/c 2)
WIMP Mass (GeV)
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC
Combined Limits
• Combine limits as above for MWIMP=100 GeV.
• Sign of ap<Sp>/an<Sn> determines shape of
allowed region: ap<Sp>/an<Sn> < 0 gives
poorer (and hence conservative) limit.
• Destructive interference prevents limit from
being set for any one nucleus when p/ plim(A)
= n/ nlim(A) . Combined limit OK.
100
Na
I
NaI
n (pb)
10
1
NaI - Allowed
Allowed
Region
0.1
0.01
m = 100 GeV/c 2
ap<Sp>/a n<Sn> < 0
(a)
0.001
0.001
Dan Tovey
Dan
Tovey
0.01
0.1
p (pb)
1
10
University
of Sheffield
UKDMC
Combined Limits
• Limits for ap<Sp>/an<Sn> > 0
(constructive interference) more stringent.
• Sign of ap<Sp>/an<Sn> will be known
when trying to exclude a particular WIMP
model (for which the signs of ap and an are
known).
100
Na
I
NaI
n (pb)
10
1
0.1
0.01
Allowed
Region
NaI - Allowed
m = 100 GeV/c 2
(b)
0.001
0.001
Dan Tovey
Dan
Tovey
ap<Sp>/a n<Sn> > 0
0.01
0.1
p (pb)
1
10
University
of Sheffield
UKDMC
Summary
• The current formalism for setting limits on
the spin-dependent WIMP-nucleon crosssections is inherently WIMP modeldependent.
• These model-dependencies arise from the
existence of both proton and neutron
contributions to the nuclear spin.
• Model-dependencies can be eliminated by
quoting separate limits on the proton and
neutron contributions to the WIMP-proton
and WIMP-neutron cross-sections
respectively.
• The new limits can be combined in a
model-independent way when attempting
to constrain particular WIMP models.
Dan Tovey
Dan
Tovey
University
of Sheffield
UKDMC