Strange quark content of the nucleon and dark matter searches

3.9 MB·English

by  R. D. Young

Strange quark content of the nucleon and dark
matter searches
3 R. D. Young∗
1
SpecialResearchCentrefortheSubatomicStructureofMatter(CSSM)
0
2 andARCCentreofExcellenceinParticlePhysicsattheTerascale(CoEPP),
n SchoolofChemistryandPhysics,UniversityofAdelaide,SA5005,Australia.
a E-mail: [email protected]
J

9
Thestrangequarkscalarcontentplaysanimportantroleinboththedescriptionofnucleonstruc-

]
t tureandinthedeterminationofdarkmatterdirectdetectioncrosssections. Asameasureofthe
a
l strange-quarkcontributiontothenucleonmass, thestrange-quarksigmaterm(σs)providesim-
-
p portantinsightintothenatureofmassgenerationinQCD.Thephenomenologicaldetermination
e
h ofσsexhibitsawiderangeofvariation,withvaluessuggestingthatthestrangequarkcontributes
[ anywhere between 0 and more than 30% of the nucleon mass. In the context of dark matter

1 searches, coupled with relatively large Higgs coupling to strangeness, this variation dominates
v
theuncertaintyinpredictedcrosssectionsforalargeclassofdarkmattermodels. Herewereport
5
6 on the recent results in lattice QCD, which are now giving a far more precise determination of
7
σ than can be inferred from phenomenology. As a consequence, the lattice determinations of
1 s
. σ cannowdramaticallyreducetheuncertaintyindarkmattercrosssectionsassociatedwiththe
1 s
0 hadronicmatrixelements.
3
1
:
v
i
X
r
a
The30InternationalSymposiumonLatticeFieldTheory-Lattice2012,
June24-29,2012
Cairns,Australia
∗Speaker.
(cid:13)c Copyrightownedbytheauthor(s)underthetermsoftheCreativeCommonsAttribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
Strangequarkcontentofthenucleonanddarkmattersearches R.D.Young
1. Introduction
Themodernpictureoftheuniversesuggeststhattheordinarymattercomponentintheenergy
composition of the universe is only about 4% — with the remainder comprised of some form
of cold, “dark matter”, which clusters around ordinary matter and an even larger “dark energy”
component. Whilelittleisknownaboutthephysicalnatureofdarkenergy,thereisstrongevidence
that suggests we are nearing the discovery phase in the identification of dark matter. Supposing
darkmattertotakeaparticle-likeform,therelevantmassscaleforsuchparticlesaremostlikelyto
bewithinreachoftheLHC.
For a general spin-independent interaction of a WIMP with a nucleus, the low-energy limit
reduces to a scalar contact interaction, and hence sensitive to the qq matrix elements within a nu-
cleon. And for Higgs-dominated exchange, these couplings are proportional to the corresponding
quarkmassandarehencesensitivetothenucleonsigmaterms. Thesigmatermsthereforebecome
of principal uncertainty in the predicting the cross sections associated with any candidate model
of dark matter. The significance of the uncertainty in these hadronic matrix elements has been
highlightedinarangeofdarkmattermodels,seeforexampleRefs.[1,2,3,4,5].
Before going on to discuss the sigma terms, and the strangeness scalar content, it is worth
recappingsomerecentadvancesinresolvingthestrangequarkvectorformfactorsinthenucleon.
Earlyestimatesofthestrangenesselectromagneticcurrentshadsuggestedthattheycouldberela-
tivelylarge[6]. Afteradedicatedexperimentaleffort,parity-violatingelectronscatteringmeasure-
ments[7,8,9]haverevealedthatthestrangequarkscontributemuchlessthanoriginallysuggested.
Thisfindingisalsosupportedbylattice-basedphenomenologicalestimates[10,11]andrecentdi-
rect lattice QCD simulation results [12, 25]. For a recent review of these latest revelations, see
Ref.[13].
After setting some general notation, the phenomenological determination of the sigma terms
are reviewed in Section 2; a summary of the latest lattice QCD results are reported in Section 3;
the significance of these results in the context of dark matter searches are discussed in Section 4;
follwedwithasummaryinSection5.
1.1 Notation
Of primary interest here are the scalar nucleon matrix elements, where we’ll use the notation
forthelight-andstrange-quarkmatrixelements
σ ≡m (cid:104)N|uu+dd|N(cid:105), σ ≡m (cid:104)N|ss|N(cid:105), (1.1)
l l s s
withtheaveragelight-quarkmassgivenbym ≡(m +m )/2.
l u d
Acommonlyreportedmeasureofthestrangenesssigmatermisthroughthey-parameter,
2(cid:104)N|ss|N(cid:105) 2m σ
l s
y≡ = . (1.2)
(cid:104)N|uu+dd|N(cid:105) ms σl
2. PhenomenologicalDetermination
BytheverynatureoftheweakcouplingoftheHiggstothelow-energysectoroftheStandard
Model, the sigma terms are essentially impossible to measure directly. Fortunately, σ can be in-
l
ferredthroughachirallow-energyrelation,wheretheamountofexplicitchiralsymmetrybreaking
2
Strangequarkcontentofthenucleonanddarkmattersearches R.D.Young
canberelatedtopion–nucleonscattering. Inparticular,thelight-quarksigmatermcanbeextracted
throughmeasurementoftheBorn-subtracted,isoscalaramplitudeΣ (t).
πN
BothΣ andσ vanishwiththequarkmass, butimportantlytheybecomeequalasthechiral
πN
limitisapproached. Thatis,theleadingdependenceofthequarkmassisthesame,withtheleading
difference being O(m3/2). The limited knowledge of this difference term can further be reduced
l
by moving to the unphysical kinematic point t =2m2 (the Cheng-Dashen point [14]), where the
π
remainderisO(m2)[15],beingdefinedby
l
∆ ≡Σ (t =2m2)−σ (t =2m2)=O(m2). (2.1)
R πN π l π l
Here,thescalarmatrixelementhasbeenextendedtonon-zeromomentumtransfer,withtheusual
sigmatermcorrespondingtothet →0limit,σ =σ (t =0). Anearlycalculationoftheremainder
l l
term has determined an estimate ∆ (cid:39)0.35MeV [16], later followed by an updated value ∆ (cid:39)
R R
2MeV[17].
Extraction of the sigma term at t =0 then requires the determination of the form factor cor-
rection,definedby
∆ ≡σ (t =2m2)−σ (t =0). (2.2)
σ l π l
Through dispersion relations, this form factor correction has been estimated to be ∆ (cid:39)15MeV
σ
[18].
With the required theoretical corrections under reasonable control, the pion–nucleon sigma
termcanthenbeextractedfromtheexperimentallydeterminedΣ ,asextrapolatedtotheunphys-
πN
icalCheng-Dashenpoint. Insummary,σ isdeterminedby
l
σ =Σ (t =2m2)−∆ −∆ . (2.3)
l πN π R σ
Following this outlined technique, analysis of πN scattering data gives the Gasser–Leutwyler–
Sainio(GLS)valueσ =45±8MeV[16],orasomewhatlargervaluefromtheGeorgeWashington
l
University/TRIUMF(GWU)groupanalysisσ =64±7MeV[19].
l
InadditiontothesebenchmarkcalculationsofGLSandGWU,ananalysisbasedonacovariant
baryonchiralperturbationtheoryhasrecentlybeenreportedbyAlarcón,Martin-Camalich&Oller
(AMO) [20]. Here the low-energy πN scattering phase shifts are fit directly within the effective
field theory framework. The analysis determines the relevant low-energy constants necessary for
the extraction of the sigma term by the Hellmann–Feynman theorem (discussed below). Together
withestimatesforthehigher-orderterms,thesigmatermisreportedtobeσ =59±7MeV,lying
l
betweenthetwovaluesreportedabove.
Estimating the strange-quark sigma term is significantly more challenging. In this case, the
strange quark is too heavy to reliably truncate the low-energy relation at low order. As an alter-
native, the conventional approach has been to estimate σ by studying the SU(3) breaking among
s
thebaryonoctet[21,22]. Here,thebaryonmasssplittingscanbeusedtoconstrainthenon-singlet
combination
σ ≡m (cid:104)N|uu+dd−2ss|N(cid:105). (2.4)
0 l
Toleading-orderinthequarkmasses, σ canbeestimatedfromthephysicallyobservedspectrum
0
by
m
l
σ (cid:39) (M +M −2M )(cid:39)24MeV. (2.5)
0 Σ Ξ N
m −m
s l
3
Strangequarkcontentofthenucleonanddarkmattersearches R.D.Young
500
Σ
0
400
(cid:76) 300
V
e
M
(cid:72)
Σs200
100
GLS GWU
AMO
0
0 20 40 60 80 100
Σ (cid:72)MeV(cid:76)
l
Figure1: Thedeterminationofσ , fromthebest-estimateofσ =36±7MeV[22], hasahighdegreeof
s 0
sensitivitytothedeterminationofpion–nucleonsigmaterm. Thethreevaluesforσ areasdiscussedinthe
l
text.
By incorporating the higher-order terms in the quark mass expansion, Borasoy and Meißner have
determinedanimprovedestimateσ =36±7MeV[22]. Withanestimateforσ ,thestrangeness
0 0
sigmatermisthengivenby
m
s
σ = (σ −σ ). (2.6)
s l 0
2m
l
Being multiplied by the large quark mass ratio, this method leads to a value for σ that is acutely
s
sensitivetothedifferencebetweenσ andσ . Further, giventhelimitedprecisionavailabletoσ ,
l 0 0
evenaperfectdeterminationofσ leavesaresidualuncertaintyinσ oforder90MeV. Forthethree
l s
determinationsofσ discussedabove,Figure1displaysthebroadrangeofpossibleσ values.
l s
Giventhedifficultyinextractingaprecisedeterminationofσ fromphenomenology, thereis
s
significantscopeforlatticeQCDtoprovideameaningfulconstraintonthisnucleonmatrixelement.
Inaddition,thereisalsothepotentialforlatticesimulationstoshedlightonthephenomenological
determinationofσ .
l
3. LatticeQCD
Within the framework of lattice QCD, there are two main methods used in the extraction of
the sigma terms. These divide into the explicit evaluation of the scalar qq matrix element by 3-
point function methods or by invoking the Hellmann–Feynman relation through the study of the
quark-massdependenceofthenucleonmass.
Inthefirsttechnique,appropriateratiosof3-pointand2-pointcorrelationfunctionsareformed
toisolatethematrixelementsofinterest. Theevaluationofthecorrelationfunctionsinvolvestwo
distinct forms of Wick contraction, as depicted in Figure 2. In particular, there are contributions
4
Strangequarkcontentofthenucleonanddarkmattersearches R.D.Young
Figure 2: Schematic diagram displaying the two topologically distinct contractions in the evaluation of
the scalar 3-point function. The standard jargon is that the left is a connected insertion and the right a
disconnectedinsertion.
involving quark line connected or disconnected operator insertions. While standard methods typ-
ically yield strong signals for the connected correlation functions, it is well-known that the dis-
connected insertions are notoriously challenging, see eg. Refs. [23, 24, 25]. This of particular
significanceforthestrange-quarkmatrixelementsinthenucleon,whicharepurelydisconnected.
A common alternative to the three-point method, is to determine the qq matrix elements by
differentiationwithrespecttothequarkmasses,wheretheHellmann–Feynmanrelationgives[26,
27,28]
∂M
N
σ =m . (3.1)
q q
∂m
q
Here the Gell-Mann–Oakes–Renner (GOR) relation [29] is commonly imposed such that M is
N
expressed as a function of the squares of meson masses. One way to do this is to write M =
N
M (m2,m˜2), where m˜2 =m2 −m2/2 is the projection of the square of the kaon mass onto the
N π K K K π
SU(2)chirallimit(m →0). Withsuchaformulation,thesigmaterms(for2+1-flavoursimulations)
l
areeasilywrittenas
∂M ∂M
σ =m2 N , σ =m˜2 N . (3.2)
l π ∂m2 s K∂m˜2
π K
The main challenge of this approach is the difficulty in reliably parameterising the quark-mass
dependenceoverarangeoflightandstrangequarkmasses. Inparticular,itisonlyinrecentyears
that there have been large scale numerical simulations of baryons with 2+1-flavours of dynamical
quarks, eg. [30, 31, 32, 33, 34, 35, 36]. In constraining the two-dimensional parameter space, it
is also the case that typical lattice trajectories in the m –m plane approach the physical point for
l s
approximatelyconstantm ;thoughtheQCDSF-UKQCDarecurrentlypursuinganapproachwhich
s
keepsthesingletcombination(2m +m )aconstant[36].
l s
AsummaryoftheprogressionoflatticeresultsisdisplayedinFigures3and4. Withdiffering
degreesofanalysisintothevariousuncertaintiesofthecalculations,itwouldbedifficulttoformu-
late any rigourous aggregates. Nevertheless, two clear features are emergent. For σ , the values
l
revealed in lattice simulations are compatible with the range of phenomenologically determined
values. Secondly,themodernvaluesofσ areallatthelowerendofthepossiblevaluessuggested
s
byFigure1. Tohighlightthecurrentstatus,Figure5showsaclose-upoftherecentdeterminations
ofσ .
s
5
Strangequarkcontentofthenucleonanddarkmattersearches R.D.Young
0 20 40 60 80 100
Fukugitaet al.1995
(cid:242)(cid:242)
Donget al.1996
(cid:242)(cid:242)
SESAM1998
(cid:224)(cid:224)
Leinweberet al.2000
(cid:224)(cid:224)
Leinweberet al.2003
(cid:224)(cid:224)
Procuraet al.2003
(cid:224)(cid:224)
Procuraet al.2006
(cid:224)(cid:224)
ETM2008
(cid:224)(cid:224)
JLQCD2008
(cid:224)(cid:224)
QCDSF2011
(cid:224)(cid:224)
QCDSF2012
(cid:224)(cid:224)
Young&Thomas2009
(cid:230)(cid:230)
PACS(cid:45)CS2009
(cid:230)(cid:230)
Martin(cid:45)Camalichet al.2010
(cid:230)(cid:230)
Dürret al.2011
(cid:230)(cid:230)
QCDSF(cid:45)UKQCD2011
(cid:230)(cid:230)
Shanahanet al.2012
(cid:230)(cid:230)
Renet al.2012
(cid:230)(cid:230)
0 20 40 60 80 100
Σ (cid:72)MeV(cid:76)
l
Figure 3: Light-quark sigma term results based on lattice QCD. The colours denoted the number of dy-
namical flavours of quarks: green is N =0, blue N =2 and red N ≥2+1. References: Fukugita et
f f f
al.[37],Dongetal.[38],SESAM[39],Leinweberetal.(2000)[40],Leinweberetal.(2003)[41],Procura
etal.(2003)[42],Procuraetal.(2006)[43],ETM[44],JLQCD[45],QCDSF(2011)[46],QCDSF(2012)
[47],Young&Thomas[48],PACS-CS[49],Martin-Camalichetal.[50],Dürretal.[51],QCDSF-UKQCD
[52],Shanahanetal.[53],Renetal.[54].
4. DarkMatter
The smaller values of σ revealed in the recent lattice studies are particularly significant in
s
the context of the direct search for dark matter. The most precise limits on WIMP–nucleon cross
sectionsarebeingconstrainedbytheXENON100Collaboration,withthelatestupdateplacingan
upperboundonthecrosssectionoflessthan10−44cm2 overawiderangeofWIMPmasses[61].
Figure 3 of [61] suggests these limits are continuing to reduce the parameter space of potential
supersymmetriccandidatesfordarkmatter.
The XENON100 Collaboration results are plotted against predicted cross sections for some
favoured supersymmetric models [62, 63, 64]. The predicted cross section rates are based on a
determination of the strange quark sigma term, σ ,1 as outlined in Section 2. Hence σ in these
s s
studiesexhibitstheextremesensitivitytoσ displayedinFigure1.
l
AstheWIMP–nucleoninteractionsarelargelyHiggs-couplingdriven,thedifferencebetween
a small and large σ can have a dramatic influence on the predicted cross sections. This is high-
s
1Orinanalternativecommonnotation, fTs=σs/Mp,fortheprotonmassMp.
6
Strangequarkcontentofthenucleonanddarkmattersearches R.D.Young
0 100 200 300 400 500
Fukugitaet al.1995
(cid:242)(cid:242)
Donget al.1996
(cid:242)(cid:242)
Lewiset al.2002
(cid:242)(cid:242)
SESAM1998
(cid:224)(cid:224)
JLQCD2008
(cid:224)(cid:224)
JLQCD2010
(cid:224)(cid:224)
QCDSF2011
(cid:224)(cid:224)
Young&Thomas2009
(cid:230)(cid:230)
Toussaint&Freeman2009
(cid:230)(cid:230)
Martin(cid:45)Camalichet al.2010
(cid:230)(cid:230)
Dürret al.2011
(cid:230)(cid:230)
QCDSF(cid:45)UKQCD2011
(cid:230)(cid:230)
Freeman&Toussaint2012
(cid:230)(cid:230)
Shanahanet al.2012
(cid:230)(cid:230)
JLQCD2012
(cid:230)(cid:230)
Renet al.2012
(cid:230)(cid:230)
Engelhardt2012
(cid:230)(cid:230)
0 100 200 300 400 500
Σ (cid:72)MeV(cid:76)
s
Figure 4: Strange-quark sigma term results based on lattice QCD. Colours as in Figure 3. Fukugita et
al.[37],Dongetal.[38],Lewisetal.[55],SESAM[39],JLQCD(2008)[45],JLQCD2010[56],QCDSF
[46], Young & Thomas [48], Toussaint & Freeman [57], Martin-Camalich et al. [50], Dürr et al. [51],
QCDSF-UKQCD[52],Freeman&Toussaint[58],Shanahanetal.[53],JLQCD(2012)[59],Renetal.[54],
Engelhardt[60].
lightedinFigure6,whichshowshowthepredictedcrosssectionforaparticularconstrainedmin-
imal supersymmetric standard model (CMSSM) model2 depends strongly on Σ (with σ con-
πN s
strained by the phenomenological σ ) [65]. In contrast, the displayed ellipse shows the range of
0
predictedcrosssectionswithinthe95%confidencelevelintervalofthelatticeQCDdeterminations
ofσ andσ fromRefs.[48,57]. Itshouldbestressedthatthereducedvariationinthecrosssection
l s
isaconsequenceoftheincreasedprecisioninσ fromlatticeQCDinput—whichisnotrelianton
s
thepropogationofthephenomenologicaluncertaintyinσ .
0
Generic dark matter cross section packages, such as micrOMEGAs [68], have been designed
totakeasinputsσ andσ . WiththeimprovementinlatticeQCDresultsdiscussedabove,itwould
l 0
beadvantageoustoseethesepackagesreformulatedtotakeσ andσ asinputs3. Inthemeantime,
l s
with cross section predictions based on σ as an input, the reduction in uncertainty in σ may
0 s
be equivalently stated as a reduction in σ −σ , cf. Eq. (2.6). A crude, yet conservative view of
l 0
2Thefiguredisplaysthepredictedcross-sectionfor“modelC”,asoneofaclassofbenchmarkmodelsproposed
pre-LHC[66,67].
3Ofcoursethesearepreciselythesamethingwithanappropriatelyincludedcorrelationcoefficient.
7
Strangequarkcontentofthenucleonanddarkmattersearches R.D.Young
0 40 80 120 160 (cid:224)(cid:224)
JLQCD2008
(cid:224)(cid:224)
JLQCD2010
(cid:224)(cid:224)
QCDSF2011
(cid:224)(cid:224)
Young&Thomas2009
(cid:230)(cid:230)
Toussaint&Freeman2009
(cid:230)(cid:230)
Martin(cid:45)Camalichet al.2010
(cid:230)(cid:230)
Dürret al.2011
(cid:230)(cid:230)
QCDSF(cid:45)UKQCD2011
(cid:230)(cid:230)
Freeman&Toussaint2012
(cid:230)(cid:230)
Shanahanet al.2012
(cid:230)(cid:230)
JLQCD2012
(cid:230)(cid:230)
Renet al.2012
(cid:230)(cid:230)
Engelhardt2012
(cid:230)(cid:230)
0 40 80 120 160
Σ (cid:72)MeV(cid:76)
s
Figure5: ZoomedingraphicofFig.2showingmorerecentresultsonσ .
s
Figure6: Forapre-LHCCMSSMdarkmattermodel, thepredictedspin-independentcrosssection(σ )
SI
shows a strong dependence on Σ . This variation is a consequence of the large variation of σ , as con-
πN s
strained by the phenomenological σ , as seen in Figure 1. The ellipse show the range of σ using lattice
0 SI
inputsforΣ andσ [65].
πN s
8
Strangequarkcontentofthenucleonanddarkmattersearches R.D.Young
Figure5,maysuggestavalue4
2m
l
σ −σ = (40±30MeV)(cid:39)2.9±2.2MeV. (4.1)
l 0
m
s
Already at this scale of precision, there should be a substantial reduction in the uncertainties in
σ associated with the hadronic matrix elements. Importantly, with the reduction in the hadronic
SI
uncertainty,anydiscoveryofdarkmatterwillhavesignificantlymorediscriminationpoweramong
candidatemodels.
5. Summary
To summarise, an accurate determination of the strange quark sigma term is of principle im-
portanceinthereductionofhadronicuncertaintiesinthepredicteddarkmattercrosssectionsfora
widerange ofmodels. Inthe determinationof therelevant nucleonscalarmatrix elements, lattice
QCDsimulationshavemadesignificantprogressinrecentyears—particularlywiththeemergence
ofdynamicalsimulationswith2+1flavoursofdynamicalquarks. Asaconfirmationoflatticemeth-
ods,itisreassuringtoobservethatthepion–nucleonsigmatermappearstobecompatiblewiththe
reliably determined phenomenological extraction. Further, recent determinations of the strange
quark sigma term are significantly smaller than had previously been suggested — and lattice cal-
culations are now at a far greater precision. With the prospect of a discovery of dark matter in
the near future, it will be essential for lattice QCD simulations to further reduce these hadronic
uncertainties.
Acknowledgements
Ithankmycollaboratorsfortheircontributionstovariousaspectsoftheworkpresentedhere,
J. Giedt, P.E. Shanahan, A.W. Thomas and S.J. Underwood. This work was supported by the
Australian Research Council through the ARC Centre of Excellence for Particle Physics at the
TerascaleandgrantsDP110101265andFT120100821.
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