stefano profumo split supersymmetry & dark matter florida state university università di...

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Stefano Profumo Stefano Profumo Split Supersymmetry Split Supersymmetry & & Dark Matter Dark Matter Florida State University Florida State University Università di Torino, March 15th, 2005 Università di Torino, March 15th, 2005 Based on Based on A.Masiero, S.P. and P.Ullio, hep-ph/0412058 A.Masiero, S.P. and P.Ullio, hep-ph/0412058 and on and on H.Baer, T.Krupovnickas, S.P. and P.Ullio, in preparation H.Baer, T.Krupovnickas, S.P. and P.Ullio, in preparation

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Stefano ProfumoStefano Profumo

Split Supersymmetry Split Supersymmetry & & Dark MatterDark Matter

Florida State UniversityFlorida State University

Università di Torino, March 15th, 2005Università di Torino, March 15th, 2005

Based on Based on A.Masiero, S.P. and P.Ullio, hep-ph/0412058 A.Masiero, S.P. and P.Ullio, hep-ph/0412058

and on and on H.Baer, T.Krupovnickas, S.P. and P.Ullio, in preparation H.Baer, T.Krupovnickas, S.P. and P.Ullio, in preparation

PLAN OF THE TALKPLAN OF THE TALK

Split Supersymmetry: Introduction and Motivations

Neutralino Relic Abundance

Dark Matter

• Current Constraints

• Halo Insensitive Future Experiments

• Halo Sensitive Future Experiments

A Split SUSY example: mSUGRA Focus Point Region

The role of future accelerators

PRPRÆLUDIUM: WHY SUPERSYMMETRY?ÆLUDIUM: WHY SUPERSYMMETRY?

Tracing back the early motivations for low energy Supersymmetrylow energy Supersymmetry...

Item 1. Item 1. (~1979)

FINE-TUNING PROBLEMFINE-TUNING PROBLEM GAUGE-COUPLING GAUGE-COUPLING UNIFICATIONUNIFICATION

Item 2. Item 2. (~1981)

DARK MATTER DARK MATTER CANDIDATECANDIDATE

Item 3. Item 3. (~1982)

*See: L.Maiani (1979); S.Dimopoulos,S.Raby,F.Wilczek (1981); H.Pagels, J.R.Primack (1982)

m2H~ 2

UV

m2H~ log(UV / m)

+

SM

SUSY

Lightest Neutralino is a suitable WIMP

DM candidate

INTRODUCING SPLIT SUPERSYMMETRYINTRODUCING SPLIT SUPERSYMMETRY

SUSY may be irrelevant for the FINE-TUNING PROBLEMFINE-TUNING PROBLEM (Item 1.Item 1.) (as it is for the cosmological costant fine-tuning problem, which is even worse!)

*See: N.Arkani-Hamed and S.Dimopoulos, hep-ph/0405159

TeV-scale

SSM Fermions+ SM-like Higgs

SSM Scalars+ Heavy Higgses

GUT-scale

...MSSM GAUGE-COUPLING UNIFICATION GAUGE-COUPLING UNIFICATION (Item 2.Item 2.), and...

...the MSSM prediction of a good DARK MATTER CANDIDATEDARK MATTER CANDIDATE (Item 3.Item 3.)

SUSY SCALARS MAY BE (VERY) HEAVY !!SUSY SCALARS MAY BE (VERY) HEAVY !! ...without spoiling...

MOTIVATIONS FOR SPLIT SUPERSYMMETRYMOTIVATIONS FOR SPLIT SUPERSYMMETRY

*See: N.Arkani-Hamed et al, hep-ph/0405159 & hep-ph/0409232

Absence of light sparticles & Higgs

Dimension-five proton decay

SUSY flavor & CP problems

Cosmological gravitino problem

If SUSY is broken at a high scale, this is 1 out of 2 possibilities!

Chiral symmetries or R-sym can protect fermion masses

Generic situation with “D-breaking”

Natural situation in many SUSY-bkg scenarios (cfr. FP/HB or mAMSB)

PHENOMENOLOGICAL THEORETICAL

String theory “landscape” scenarios (10125 quantized vacua)

CONSEQUENCES OF SPLIT SUPERSYMMETRYCONSEQUENCES OF SPLIT SUPERSYMMETRY

*Plot from Kilian et al, hep-ph/0408088; see also: Arkani-Hamed et al, hep-ph/0409232, Romanino et al, hep-ph/0406088, Anchordoqui et al, hep-ph/0408284

Relatively heavy Higgsheavy Higgs, 120-160 GeV

Long-lived GluinoLong-lived Gluino (accelerators + cosmic rays)

e- and n EDMEDM @ 2-loops (without small phases)

Dark matterDark matter phenomenology (pure “fermionic SUSY-DMfermionic SUSY-DM” benchmark)

Small, manageable parameter space

Get hints on general facts about SUSY DM

FROM THE MSSM LAGRANGIAN...FROM THE MSSM LAGRANGIAN...

ggMWWWWMBBMLsoftMSSM

~~2

1~~2

~~

2

1~~

2

13

3321

212*21

*1 21

HHBHmHHmH HH

2*

1*

1* QHUYAQHDYALHEYA UUDDEE

SCALARS HEAVY, DECOPULED

FERMIONS LIGHT

HIGGS SECTOR FINE TUNED

EMEDMDUMULMLQMQ EDULQ

~~~~~~~~~~ 2*2*2*2*2*

...TO THE SpS PARAMETER SPACE...TO THE SpS PARAMETER SPACE

M1 tan mh mSM2 M3

FERMIONS HIGGS SECTOR SCALARS

SpS PARAMETER SPACE: NEUTRALINO PHYSICSSpS PARAMETER SPACE: NEUTRALINO PHYSICS

M1

-

M2

-mZ cos sin W mZ cos cos W

mZ sin sin W -mZ sin sin W 0

0

0

0

-

-mZ cos sin W mZ cos cos W

mZ sin sin W -mZ sin sin WMN =

B W HdHu

BINO WINO HIGGSINO

Spin-independentNeutralino-Nucleon

N

h

THE SpS PARAMETER SPACETHE SpS PARAMETER SPACE

M1 tan mh mSM2 M3

Gluino PhenomenologyGluino PhenomenologyDirect DM DetectionDirect DM Detection

Relic Abundance &Relic Abundance &Indirect DM DetectionIndirect DM Detection

No GUT relationsNo GUT relations

assumed!!assumed!!

NEUTRALINO RELIC ABUNDANCENEUTRALINO RELIC ABUNDANCE

*See: A.Masiero, S.Profumo, P.Ullio, hep-ph/0412058

mmmin (M1,M2)

WMAPWMAP-compatible relic abundance (0.094< 0.094< hh22 <0.129 <0.129) produced with:

B-inoB-ino like LSP+ (W or H) chargino-neutralino coannihilations (M1M2)

Large Mass W-inoW-ino ( 2 TeV) or H-inoH-ino ( 1 TeV) like LSP

• slice along the M1 direction• work on the (M2) plane

hh22 depends on (M1,M2):

cii /330

2

108

3 cmgG

Hc

MpcskmHh //100/0

THE WMAP HYPERSURFACETHE WMAP HYPERSURFACE

*See: A.Masiero, S.Profumo, P.Ullio, hep-ph/0412058; M.Kamionkowski, M.Turner, PRD (1990); S.Profumo, P.Ullio, JHEP (2003); R.Catena, N.Fornengo, A.Masiero, M.Pietroni, F.Rosati, PRD (2004)

hh22=0.11 =0.11 points define a surface in the (M1,M2) space

M2 (GeV)M1 (GeV)

(GeV)

hh22>0.129 >0.129

Cosmologically excludedCosmologically excluded

hh22<0.094 <0.094

Low Relic Density ModelsLow Relic Density Models• Account for all CDM through

- Non-thermal production (moduli, gravitinos, Q-balls, cosmic strings, ...)- Cosmological enhancement (Quintessence, Scalar-tensor, ...)

(caveat: post-freeze-out entropy release compatible with BBN)

• SUSY-DM a subdominant DM component Rescale DM

SPLITTING mSUGRA: THE FOCUS POINT REGIONSPLITTING mSUGRA: THE FOCUS POINT REGION

*See J.Feng,K.Matchev,T.Moroi, PRL (2000)

Focus Point: one of the few mSUGRA regions survivng the hh2 2 constraintconstraint

Large values of the universal scalar mass (mm00) drive mm11

All scalars (but the lightest Higgs) take multi-TeV masses (SplitSUSY-like setupSplitSUSY-like setup)

The lightest neutralino is a bino-higgsinobino-higgsino mixture

Contrary to the previous general setup, the GUT relation MM11=0.5M=0.5M22 holds [slice in the (M1,M2) space!]

Focus Point Focus Point RegionRegion

MSSMMSSM(105 param.)

SplitSUSYSplitSUSY(7 param.)

mSUGRAmSUGRA(5 param.)

SPLITTING mSUGRA: THE FOCUS POINT REGIONSPLITTING mSUGRA: THE FOCUS POINT REGION

UNFOLDING THE FOCUS POINT REGIONUNFOLDING THE FOCUS POINT REGION

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

The standard GUT-scale parameterization (mm00,M,M1/21/2) is AMBIGUOUSAMBIGUOUS

(m0) critically depends on• the top quark mass

• the RG numerical evolution

It is impossible to read out • the Neutralino Mass

• the Neutralino Composition

Trade mm00 for and work on the physical plane (mm11)

UNFOLDING THE FOCUS POINT REGIONUNFOLDING THE FOCUS POINT REGION

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

The standard GUT-scale parameterization (mm00,M,M1/21/2) is AMBIGUOUSAMBIGUOUS

(m0) critically depends on• the top quark mass

• the RG numerical evolution

It is impossible to read out • the Neutralino Mass

• the Neutralino Composition

Trade mm00 for and work on the physical plane (mm11)

FOCUS POINT: THE DM PARAMETER SPACEFOCUS POINT: THE DM PARAMETER SPACE

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

2 - WMAP range

Low-relic densityModels

* Source: XENON, IceCube, Pamela and HESS Home Pages

DM SEARCHES & CONSTRAINTS: OVERVIEW DM SEARCHES & CONSTRAINTS: OVERVIEW

DIRECT DETECTIONDIRECT DETECTION

Observe scatteringof ’s off nucleiin low bckg.environments

NEUTRINO FLUXESNEUTRINO FLUXES

Look for neutrinosproduced in ann.in the core of gravitational dips,like the center of the Sun or of the Earth

ANTIMATTER SEARCHESANTIMATTER SEARCHES

Disentangle antimatterproduced in ann.in the galactic halo fromstandard antimattersources

GAMMA RAYSGAMMA RAYS

Observe gamma rays produced by ann. in the galactic center

CURRENT BOUNDS: THE EGRET DATACURRENT BOUNDS: THE EGRET DATA

*Plot from P.Ullio et al., Astrop.Phys., see also: A.Cesarini et al., Astrop.Phys., M.Pohl, Astron.Astrophys.,

The EGRET experiment, on board the C.G.R.O.

EGRET identified a -ray source close to the GCGC (*)

The data cannot be explained by the diffuse -raysdue to cosmic-rays-interstellar medium interactions

SUSYSUSY is a possible expanation (*)

If the source is identified with the GC, given a halo profile one gets constraintsconstraints on the SUSY model (*)

The source is 1.5°off the GC (is it the GC?)

Alternative explanations exist (Central BH arc...)

The GLAST exp. will shed light...

CURRENT BOUNDS: THE H.E.S.S. DATACURRENT BOUNDS: THE H.E.S.S. DATA

*See: Aharonian et al, Astron.Astrophys. Plots from Horns, astro-ph/0408192

The ACT H.E.S.S. experiment, Namibian Desert (*)

The ACTACT accuracy & angularresolution keeps increasing!

Again, standard sources do not fit the data

A heavy heavy might explain the H.E.S.S. data

The data can be regarded as a constraintconstraint

ANTIMATTER SEARCHES: ANTIMATTER SEARCHES: POSITRONS & ANTIPROTONSPOSITRONS & ANTIPROTONS

Balloon-borne experiments (HEAT,CAPRICE,BESS) measured Positrons & Anti-protons fluxes

Independent data allow to predictpredict the standard secondary background

The (primary) SUSY contribution can be constrained by a analysis analysis

Data are statistically consistent with backround alonebackround alone

Positrons Antiprotons

A NEWCOMER: THE A NEWCOMER: THE 66LI ABUNDANCELI ABUNDANCE

*See: Jedamzik, Phys.Rev.D (2004); plot from Rollinde et al., astro-ph/0412426

6Li is poorly sinthesized during BBN (1000 times less abundantly than observed)

Residual post-freeze-out Neutralino annihilationsNeutralino annihilations sinthesize 6Li

The plateau of 6Li in low-metallicity stars indicates its primordial originprimordial origin it is substantially neitherneither producedproduced nor depletednor depleted in stars’ nuclear reactions!

Alternative mechanisms might also produce 6Li

The 6Li abundance constrainsconstrains, in any case, the Neutralino annihilation rate!

3

6

,

3,

2

2

6

n

n

n

n

Ann

n

m

vdt

n

n

np

np

H

Li

CONSTRAINTS AT WORK! CONSTRAINTS AT WORK! PRPRÆÆLUDIUM LUDIUM

*See: S.Profumo & P.Ullio, JCAP (2003)

-rays & AM fluxes critically depend on the DM Halo choiceDM Halo choice

Current data on the Milky Way still allow both CUSPYCUSPY and COREDCORED profiles

Pick two extremeextreme instances:

Other consistent DM Halos would give intermediateintermediate detection rates

Self-consistently computed velocity distributionsvelocity distributions

• Adiabatically contracted N-03 Profile (cuspy)

• Burkert Profile (cored)maximal vs minimal DM-Baryonsangular momentum transfer

DARK MATTER IN THE FOCUS POINT REGIONDARK MATTER IN THE FOCUS POINT REGION

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

First Example: The Adiabatically contracted-N03

(cuspycuspy) Halo Profile

DARK MATTER IN THE FOCUS POINT REGIONDARK MATTER IN THE FOCUS POINT REGION

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

Second Example: The BURKERT

(coredcored) Halo Profile

FUTURE: DIRECT DM DETECTIONFUTURE: DIRECT DM DETECTION[1- HALO-INSENSITIVE QUANTITIES][1- HALO-INSENSITIVE QUANTITIES]

*See: A.Masiero, S.Profumo, P.Ullio, hep-ph/0412058

CDMS-II(Stage-2Stage-2)

XENON-1t(Stage-3Stage-3)

Shaded region: points on the

WMAP HypersurfaceWMAP Hypersurface

CDMS current constraint

CDMS-II

XENON-1t

• scintillation• photons

• scintillation• ionization

FUTURE: KmFUTURE: Km22–SIZE NEUTRINO TELESCOPES–SIZE NEUTRINO TELESCOPES[1- HALO-INSENSITIVE QUANTITIES][1- HALO-INSENSITIVE QUANTITIES]

As for Direct Det., current limitscurrent limits still far from required sensitivityfar from required sensitivity

*See: A.Masiero, S.Profumo, P.Ullio, hep-ph/0412058

Shaded region: points on the

WMAP HypersurfaceWMAP Hypersurface

SUPER-K current constraint

IceCube proj. sens.

Candidate for IceCube:

mm < 600 GeV < 600 GeV

MM1,21,2 (large SDP)

FUTURE: SPACE-BASED ANTIMATTER SEARCHESFUTURE: SPACE-BASED ANTIMATTER SEARCHES[2- HALO-SENSITIVE QUANTITIES][2- HALO-SENSITIVE QUANTITIES]

Pamela Soyuz Rocket AMS-02 (onboard ISS)

Total Antiprotons: 3x104

Total Positrons: 105

Launch: April 2005

Acceptance: 20.5 cm2sr

Launch: 2008 (?)

Total Antiprotons: 3x105

Total Positrons: 106

Acceptance: 450 cm2sr

Assume a subdominant SUSY contribution DiscriminateDiscriminate against a bckg. only scenario Determine the experimental sensitivityexperimental sensitivity to SUSY

*See: S.Profumo, P.Ullio, JCAP (2004)

ANTIMATTER SEARCHES: ANTIDEUTERONSANTIMATTER SEARCHES: ANTIDEUTERONS[2- HALO-SENSITIVE QUANTITIES][2- HALO-SENSITIVE QUANTITIES]

*See: Donato et al, PRD; Mori et al, Astroph.J.

Low energy antideuterons have a very low backgroundlow background from Standard Sources

The discovery of a singlesingle low-energy antideuteron can be regarded as an evidence for new physics

AMS-02 can look at moderately energetic antideuterons it is not sensitivenot sensitive to most SUSY models

ANTIDEUTERONS: THE G.A.P.S. PROPOSALANTIDEUTERONS: THE G.A.P.S. PROPOSAL[2- HALO-SENSITIVE QUANTITIES][2- HALO-SENSITIVE QUANTITIES]

*See: Mori et al, Astroph.J. (2000)

G.A.P.S.: Gaseous Anti-Particles SpectrometerG.A.P.S.: Gaseous Anti-Particles Spectrometer

Despite being much smallersmaller and cheapercheaper than AMS-02, GAPS would be 20 times more sensitive20 times more sensitive!

PrototypePrototype currently under construction & testing

X-ray de-excitation ofExotic atoms

SUSY (DM) SEARCHES: TENTATIVE ROADMAPSUSY (DM) SEARCHES: TENTATIVE ROADMAP

LHCLHCPRE-LHCPRE-LHCDM SEARCHESDM SEARCHES

Stage-2 Direct Detectors

Pamela (Antiprotons)

Pamela (Positrons)

GLAST

POST-LHCPOST-LHC SEARCHES SEARCHES

IceCube

Stage-3 Direct Detectors

AMS (Antipr. & Positrons)

GAPS (Antideuterons)

e+e- TeV Linear Collider

2005 2006 2007 2008 2009 2010 2011 2012

DARK MATTER IN THE FOCUS POINT REGIONDARK MATTER IN THE FOCUS POINT REGION

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

PRE-LHCPRE-LHCDM SEARCHESDM SEARCHES

[CUSPY HALOCUSPY HALO]

DARK MATTER IN THE FOCUS POINT REGIONDARK MATTER IN THE FOCUS POINT REGION

*See: H.Baer et al, JHEP (2003)

Ordinary missing- transverse-energy

No specialized cuts

LHCLHCSEARCHESSEARCHES

DARK MATTER IN THE FOCUS POINT REGIONDARK MATTER IN THE FOCUS POINT REGION

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

POST-LHCPOST-LHCDM SEARCHES DM SEARCHES

[CUSPY HALOCUSPY HALO]

DM SEARCHES DM SEARCHES COMPLEMENTARITY!!COMPLEMENTARITY!!

DARK MATTER IN THE FOCUS POINT REGIONDARK MATTER IN THE FOCUS POINT REGION

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

POST-LHCPOST-LHCDM SEARCHES DM SEARCHES

[CUSPY HALOCUSPY HALO]

HALO MODEL SENSITIVITYHALO MODEL SENSITIVITY

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

GAMMA RAYS FROM THE GALACTIC CENTER ON GLASTGAMMA RAYS FROM THE GALACTIC CENTER ON GLAST

Adiabatically contracted N03 (cuspycuspy) Halo Profile

Burkert(coredcored) Halo Profile

HALO MODEL SENSITIVITYHALO MODEL SENSITIVITY

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

POSITRONS ON PAMELAPOSITRONS ON PAMELA

Adiabatically contracted N03 (cuspycuspy) Halo Profile

Burkert(coredcored) Halo Profile

ONGOING WORK: LHC SEARCH STRATEGIESONGOING WORK: LHC SEARCH STRATEGIES

*Fig. from H.Baer et al, Phys.Rev.D (1999); H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

Standard LHCLHC search strategies failfail with heavy squarks-gluinos

EW-produced neutralinos/charginos can be detected through channels with

suppressed background, e.g. the isolated trilepton signalisolated trilepton signal

Work out optimized sets of cutssets of cuts

Assess the maximal LHC reachmaximal LHC reach

Compute the dominant backgrounddominant background (from W*Z*, W*

ONGOING WORK: LHC SEARCH STRATEGIESONGOING WORK: LHC SEARCH STRATEGIES

*H.Baer, T.Krupovinckas, S.Profumo, P.Ullio, in preparation

Other strategies can be based on long-lived particleslong-lived particles

If the LSP is higgsino-like, charginoscharginos are long lived

(new promising data from D-zero on CHAMP’s, good detection prospects at the LHC)

The gluinogluino can be long-lived, depending on m0

(displaced vertexes or gluino hadronization)

CONCLUSIONS: SUSY DM + HEAVY SCALARSCONCLUSIONS: SUSY DM + HEAVY SCALARS

*See: A.Masiero, S.Profumo, P.Ullio, hep-ph/0412058 and H.Baer, T.Krupovinckas, S.Profumo, P.Ullio

1. The current sensitivitycurrent sensitivity of Direct SearchesDirect Searches and Neutrino TelescopesNeutrino Telescopes is typically far below what needed to detect SUSY DM

2. Antimatter searchesAntimatter searches and the 66LiLi abundance already put constraints on low relic density SUSY DM models

3. -rays-rays constraints highly depend on the assumed Dark Halo structure in the Galactic Center

4. Stage-3 Direct DetectorsStage-3 Direct Detectors will (typically) do better than IceCubeIceCube

5. Antimatter Searches & Direct detection are often complementarycomplementary techniquestechniques

6. The LHCLHC reach can compete with that of DM searches only with suitable search strategiessuitable search strategies

BACKUPS

““D-BREAKING” AND “F-BREAKING”D-BREAKING” AND “F-BREAKING”

R-symmetric soft terms correspond to dimension 2-operators

R-breaking soft terms correspond to dimension 3-operators

D-breaking F-breaking

24 ~1 mY 22 ~1 mX

mM g~

~ 22

~~mm

Q22

~~mm

Q

*

2

~

~

M

mM g

Spurion Superfield

Soft Terms Scale

COMPLEMENTARITY (1/2)COMPLEMENTARITY (1/2)

*See: A.Masiero, S.Profumo, P.Ullio, to appear on hep-ph

Burkert cored Halo Model

Set M1 = 10 TeV

Role of Wino-Higgsino mix

DD/AM complementarity

COMPLEMENTARITY (2/2)COMPLEMENTARITY (2/2)

Only SI Direct Detection survives

Accelerators only handle to pure wino/higgsino neutralinos

Long –lived gluino detection promsing but model-dependent

Apply rescaling procedure:

CDMCDM

,1min

*See: A.Masiero, S.Profumo, P.Ullio, hep-ph/0412058

COMPUTING ANTIMATTER YIELDSCOMPUTING ANTIMATTER YIELDS

Solar Modulation Solar Modulation effects computedeffects computed with the one parameter analytical with the one parameter analytical

Gleeson-Axford force-field approx.Gleeson-Axford force-field approx.

BackgroundBackground calculated with the calculated with the GalpropGalprop package with the same package with the same propagation parameterspropagation parameters

PropagationPropagation in Galactic magnetic fields in Galactic magnetic fields accounted for with a two-dimensional diffusion accounted for with a two-dimensional diffusion model in the steady state approximationmodel in the steady state approximation

DISCRIMINATION OF SUSY DM DISCRIMINATION OF SUSY DM AT FUTURE AM SEARCHESAT FUTURE AM SEARCHES

Si

Bi

Pi NNN

OiNNO

i

Bi

Oi NN

dEI

E

E B

S

max

min

2

TALCX

2..%

bins

Oi

n

i N

Oi

Pi NN

12

2

2Exclusion limits from Exclusion limits from Current DataCurrent Datacome from a statistical analysiscome from a statistical analysis

For future perspectives, assume For future perspectives, assume Gaussian errorsGaussian errors......

......limiting caseslimiting cases for discrimination, for discrimination,

...and, for a ...and, for a large number of binslarge number of bins, get:, get:

tipically very weakly depending on tipically very weakly depending on extremaextrema and and independent of the apparatusindependent of the apparatus

DiscriminationDiscrimination is performed against the quantity is performed against the quantity