1. Dan ToveyUniversity of Sheffield UKDMCDan Tovey Model-Independent Spin-Dependent Cross- Section Limits from Dark Matter Searches Dan Tovey, Rick Gaitskell, Paolo Gondolo, Yorck Ramachers and Leszek Roszkowski. The current techniqueThe current technique Why it is WIMP model-dependentWhy it is WIMP model-dependent A new model-independent techniqueA new model-independent technique Dan Tovey, Rick Gaitskell, Paolo Gondolo, Yorck Ramachers and Leszek Roszkowski. The current techniqueThe current technique Why it is WIMP model-dependentWhy it is WIMP model-dependent A new model-independent techniqueA new model-independent technique (hep-ph/0005041)

2. Dan ToveyUniversity of Sheffield UKDMCDan Tovey Background Weakly Interacting Massive Particles (WIMPs) one of the best motivated candidates for Dark Matter.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.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).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.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 spin- dependent.WIMP-nucleus couplings effectively of two types: spin-independent and spin- dependent. Weakly Interacting Massive Particles (WIMPs) one of the best motivated candidates for Dark Matter.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.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).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.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 spin- dependent.WIMP-nucleus couplings effectively of two types: spin-independent and spin- dependent.

3. Dan ToveyUniversity of Sheffield UKDMCDan Tovey Total WIMP-nucleus scattering cross-section  A can be written:Total WIMP-nucleus scattering cross-section  A can be written: where  A is the reduced mass and C A is the cross-section enhancement factor. Cross-section limits normalised to those for free nucleons (conventionally protons):Cross-section limits normalised to those for free nucleons (conventionally protons): Normalisation model-independent for spin- independent interactions as:Normalisation model-independent for spin- independent interactions as: Normalisation more complex in spin- dependent case where:Normalisation more complex in spin- dependent case where: and a p and a n are WIMP model-dependent effective couplings and J is the nuclear spin. Total WIMP-nucleus scattering cross-section  A can be written:Total WIMP-nucleus scattering cross-section  A can be written: where  A is the reduced mass and C A is the cross-section enhancement factor. Cross-section limits normalised to those for free nucleons (conventionally protons):Cross-section limits normalised to those for free nucleons (conventionally protons): Normalisation model-independent for spin- independent interactions as:Normalisation model-independent for spin- independent interactions as: Normalisation more complex in spin- dependent case where:Normalisation more complex in spin- dependent case where: and a p and a n are WIMP model-dependent effective couplings and J is the nuclear spin. Cross-Section Enhancement

4. Dan ToveyUniversity of Sheffield UKDMCDan Tovey WIMP Model-Dependence Assume model-dependencies in form-factor sufficiently small to be neglected (true for Na, I, F etc. at least).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 C A via a p and a n.All WIMP model-dependence therefore contained in normalisation C A via a p and a n. In early calculations (single-particle or odd- group models) C A dominated by either proton or neutron terms (but not both) => normalisation of odd-proton targets to WIMP- proton cross-section model-independent (a p factors cancel).In early calculations (single-particle or odd- group models) C A dominated by either proton or neutron terms (but not both) => normalisation of odd-proton targets to WIMP- proton cross-section model-independent (a p factors cancel). Normalisation of odd-neutron targets model- dependent however due to factor (a p /a n ) 2 =  p /  n remaining in  p lim(A).Normalisation of odd-neutron targets model- dependent however due to factor (a p /a n ) 2 =  p /  n remaining in  p lim(A). HOWEVER, for SUSY neutralino WIMPs early estimates of  q values (used to calculate a p and a n ) gave  p /  n ~ 1.5, independent of neutralino composition.......BUT......HOWEVER, for SUSY neutralino WIMPs early estimates of  q values (used to calculate a p and a n ) gave  p /  n ~ 1.5, independent of neutralino composition.......BUT...... Assume model-dependencies in form-factor sufficiently small to be neglected (true for Na, I, F etc. at least).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 C A via a p and a n.All WIMP model-dependence therefore contained in normalisation C A via a p and a n. In early calculations (single-particle or odd- group models) C A dominated by either proton or neutron terms (but not both) => normalisation of odd-proton targets to WIMP- proton cross-section model-independent (a p factors cancel).In early calculations (single-particle or odd- group models) C A dominated by either proton or neutron terms (but not both) => normalisation of odd-proton targets to WIMP- proton cross-section model-independent (a p factors cancel). Normalisation of odd-neutron targets model- dependent however due to factor (a p /a n ) 2 =  p /  n remaining in  p lim(A).Normalisation of odd-neutron targets model- dependent however due to factor (a p /a n ) 2 =  p /  n remaining in  p lim(A). HOWEVER, for SUSY neutralino WIMPs early estimates of  q values (used to calculate a p and a n ) gave  p /  n ~ 1.5, independent of neutralino composition.......BUT......HOWEVER, for SUSY neutralino WIMPs early estimates of  q values (used to calculate a p and a n ) gave  p /  n ~ 1.5, independent of neutralino composition.......BUT......

5. Dan ToveyUniversity of Sheffield UKDMCDan Tovey 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):Later estimates of  q show that although this remains true for higgsino neutralinos, it is certainly not true for gaugino neutralinos (best motivated theoretically): higgsinoshiggsinosgauginosgauginos

6. Dan ToveyUniversity of Sheffield UKDMCDan Tovey SUSY Model-Dependence Problem now more acute because shell-model calculations of and indicate both proton and neutron contributions to C A can be significant =>Problem now more acute because shell-model calculations of and indicate both proton and neutron contributions to C A can be significant => Model-dependent factor (a p /a n ) 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:WIMP-proton cross-section limits vary with assumed neutralino WIMP composition: Problem now more acute because shell-model calculations of and indicate both proton and neutron contributions to C A can be significant =>Problem now more acute because shell-model calculations of and indicate both proton and neutron contributions to C A can be significant => Model-dependent factor (a p /a n ) 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:WIMP-proton cross-section limits vary with assumed neutralino WIMP composition: WIMP Mass (GeV)  p lim(A) (pb)

7. Dan ToveyUniversity of Sheffield UKDMCDan Tovey Model-Independent Limits To solve this problem identify separate proton and neutron contributions to C A : The separate proton and neutron contributions to  A are thus: This then gives: Now assume independently that  A ~  A p and  A ~  A n. Then define WIMP-proton(neutron) cross-section limit set by protons(neutrons) in target A: To solve this problem identify separate proton and neutron contributions to C A : The separate proton and neutron contributions to  A are thus: This then gives: Now assume independently that  A ~  A p and  A ~  A n. Then define WIMP-proton(neutron) cross-section limit set by protons(neutrons) in target A:

8. Dan ToveyUniversity of Sheffield UKDMCDan Tovey Enhancement Factor Ratios Use of the cross-section enhancement factor ratiosUse of the cross-section enhancement factor ratios, ensures cancellation of WIMP model- dependent a p and a n terms in  A. Values of C A p /C p and C A n /C n obtained from shell-model calculations:Values of C A p /C p and C A n /C n obtained from shell-model calculations: Use of the cross-section enhancement factor ratiosUse of the cross-section enhancement factor ratios, ensures cancellation of WIMP model- dependent a p and a n terms in  A. Values of C A p /C p and C A n /C n obtained from shell-model calculations:Values of C A p /C p and C A n /C n obtained from shell-model calculations:

9. Dan ToveyUniversity of Sheffield UKDMCDan Tovey 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: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 A i limits are combined using:For a detector consisting of more than one target nucleus A i limits are combined using: Spin-independent (SI) and spin-dependent limits from protons and neutrons in several target nuclei A i can be combined using:Spin-independent (SI) and spin-dependent limits from protons and neutrons in several target nuclei A i can be combined using: 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: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 A i limits are combined using:For a detector consisting of more than one target nucleus A i limits are combined using: Spin-independent (SI) and spin-dependent limits from protons and neutrons in several target nuclei A i can be combined using:Spin-independent (SI) and spin-dependent limits from protons and neutrons in several target nuclei A i can be combined using:

10. Dan ToveyUniversity of Sheffield UKDMCDan Tovey NaI Example Consider limits from same hypothetical NaI(Tl) detector.Consider limits from same hypothetical NaI(Tl) detector. WIMP-proton cross-section limit using new definition:WIMP-proton cross-section limit using new definition: Consider limits from same hypothetical NaI(Tl) detector.Consider limits from same hypothetical NaI(Tl) detector. WIMP-proton cross-section limit using new definition:WIMP-proton cross-section limit using new definition:  p lim(A) (pb) WIMP Mass (GeV) Excluded Region

11. Dan ToveyUniversity of Sheffield UKDMCDan Tovey  n lim(A) (pb) WIMP Mass (GeV) NaI Example Similarly WIMP-proton cross-section limit using new definition.Similarly WIMP-proton cross-section limit using new definition. Limits less stringent due to lack of odd- neutron target.Limits less stringent due to lack of odd- neutron target. Similarly WIMP-proton cross-section limit using new definition.Similarly WIMP-proton cross-section limit using new definition. Limits less stringent due to lack of odd- neutron target.Limits less stringent due to lack of odd- neutron target. Excluded Region

12. Dan ToveyUniversity of Sheffield UKDMCDan Tovey Combined Limits Combine limits as above for M WIMP =100 GeV.Combine limits as above for M WIMP =100 GeV. Sign of a p /a n determines shape of allowed region: a p /a n /a n determines shape of allowed region: a p /a n < 0 gives poorer (and hence conservative) limit. Destructive interference prevents limit from being set for any one nucleus when  p /  p lim(A) =  n /  n lim(A). Combined limit OK.Destructive interference prevents limit from being set for any one nucleus when  p /  p lim(A) =  n /  n lim(A). Combined limit OK. Combine limits as above for M WIMP =100 GeV.Combine limits as above for M WIMP =100 GeV. Sign of a p /a n determines shape of allowed region: a p /a n /a n determines shape of allowed region: a p /a n < 0 gives poorer (and hence conservative) limit. Destructive interference prevents limit from being set for any one nucleus when  p /  p lim(A) =  n /  n lim(A). Combined limit OK.Destructive interference prevents limit from being set for any one nucleus when  p /  p lim(A) =  n /  n lim(A). Combined limit OK. Allowed Region

13. Dan ToveyUniversity of Sheffield UKDMCDan Tovey Combined Limits Limits for a p /a n > 0 (constructive interference) more stringent.Limits for a p /a n > 0 (constructive interference) more stringent. Sign of a p /a n will be known when trying to exclude a particular WIMP model (for which the signs of a p and a n are known).Sign of a p /a n will be known when trying to exclude a particular WIMP model (for which the signs of a p and a n are known). Limits for a p /a n > 0 (constructive interference) more stringent.Limits for a p /a n > 0 (constructive interference) more stringent. Sign of a p /a n will be known when trying to exclude a particular WIMP model (for which the signs of a p and a n are known).Sign of a p /a n will be known when trying to exclude a particular WIMP model (for which the signs of a p and a n are known). Allowed Region

14. Dan ToveyUniversity of Sheffield UKDMCDan Tovey Summary The current formalism for setting limits on the spin-dependent WIMP-nucleon cross- sections is inherently WIMP model- dependent.The current formalism for setting limits on the spin-dependent WIMP-nucleon cross- sections is inherently WIMP model- dependent. These model-dependencies arise from the existence of both proton and neutron contributions to the nuclear spin.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.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.The new limits can be combined in a model-independent way when attempting to constrain particular WIMP models. The current formalism for setting limits on the spin-dependent WIMP-nucleon cross- sections is inherently WIMP model- dependent.The current formalism for setting limits on the spin-dependent WIMP-nucleon cross- sections is inherently WIMP model- dependent. These model-dependencies arise from the existence of both proton and neutron contributions to the nuclear spin.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.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.The new limits can be combined in a model-independent way when attempting to constrain particular WIMP models.