Light Asymmetric Dark Matter on the Lattice: SU(2) Technicolor with Two Fundamental Flavors

0.29 MB·English

by  Randy Lewis

Light Asymmetric Dark Matter on the Lattice:
SU(2) Technicolor with Two Fundamental Flavors
♠ ∗ (cid:114) † (cid:114)‡
Randy Lewis , Claudio Pica , and Francesco Sannino
♠
Department of Physics and Astronomy,
York University, Toronto, M3J 1P3, Canada and
(cid:114) CP3-Origins & the Danish Institute for Advanced Study DIAS,
University of Southern Denmark, Campusvej 55, DK-5230 Odense M, Denmark
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n Abstract
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TheSU(2)gaugetheorywithtwomasslessDiracflavorsconstitutesthebuildingblockofseveral
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models of Technicolor. Furthermore it has also been used as a template for the construction of a

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h naturallightasymmetric,ormixedtype,darkmattercandidate. Weuseexplicitlatticesimulations
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to confirm the pattern of chiral symmetry breaking by determining the Goldstone spectrum and
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h thereforeshowthatthedarkmattercandidatecan,defacto,beconstitutedbyacomplexGoldstone
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boson. We also determine the phenomenologically relevant spin-one and spin-zero isovector
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spectrumanddemonstratethatitiswellseparatedfromtheGoldstonespectrum.
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9. Preprint: CP3-Origins-2011-28&DIAS-2011-15
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[email protected]
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[email protected]
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[email protected]
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I. INTRODUCTION
Thenatureofdarkmatter(DM)isanimportantprobleminmodernphysics. DMplays
a key role in the formation of large structures and the evolution of the Universe. It is
also widely expected to provide a link to physics beyond the Standard Model (SM). For
ff
these reasons much experimental, observational, and theoretical e ort has been devoted
to shedding light on DM. It is popular to identify DM with Weakly Interacting Massive
Particles(WIMPs).
ManypropertiesofaWIMParenotconstrainedbyourcurrentknowledgeofDM,for
example the WIMP may or may not be a stable particle [1] and it may or may not be
identifiedwithitsantiparticle[2]. AsymmetricDMreferstoascenariowheretheWIMP’s
antiparticle has been annihilated away, leaving the WIMP itself as the observed DM.
AsymmetricDMcandidateswereputforwardin[2]astechnibaryons,in[3]asGoldstone
bosons, and subsequently in many diverse forms [4–10]. There is also the possibility
of mixed DM [11], i.e. having both a thermally-produced symmetric component and an
asymmetricone.
/
Null results from several experiments, such as CDMS [12] and Xenon10 100 [13,
14], have placed stringent constraints on WIMP-nucleon cross sections. Interestingly
DAMA [15] and CoGeNT [16] have both produced evidence for an annual modulation
signature for DM, as expected due to the relative motion of the Earth with respect to the
ff
DMhalo. TheseresultssupportalightWIMPwithmassoforderafewGeV,whicho ers
the attractive possibility of a common mechanism for baryogenesis and DM production.
At first glance it seems that the WIMP-nucleon cross sections required by DAMA and
CoGeNT have been excluded by CDMS and Xenon upon assuming spin-independent
interactions between WIMPs and nuclei (with protons and neutrons coupling similarly
to WIMPs), but a number of resolutions for this puzzle have been proposed in the litera-
ture [17–22]. Interestingly, new results from the CRESST-II experiment report signals of
lightDM[23].
A composite origin of DM is an intriguing possibility given that the bright side of
the universe, constituted mostly by nucleons, is also composite. Thus a new strongly-
coupled theory could be at the heart of DM. In this work we investigate, for the first
time on the lattice, a technicolor-type extension of the SM expected to naturally yield a
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light DM candidate, as introduced in [8] and used in [22] to reconcile the experimental
observations.
II. ALIGHTDARKMATTERTEMPLATE:SU(2)TECHNICOLORWITHTWODIRACFLA-
VORS
AsymmetricDMrequiresatleastonecomplexfieldwitha(nearly)conservedAbelian
quantum number. It would be exciting if the theory predicting the existence of this state
is also directly involvedwith the breaking of electroweak symmetry. Therefore a natural
candidate is a technicolor model. Most of the states of the theory are much heavier than
afewGeV,butifthereareGoldstonebosonsnotabsorbedbythelongitudinaldegreesof
freedomoftheSMmassivegaugebosons,theybecomeprimarycandidatesforproducing
anaturalhierarchybetweentheGeVandtheTeVscale.
The minimal technicolor theory that breaks the electroweak symmetry and features
a light DM state was constructed first in [24, 25]. This theory is SU(2) technicolor with
two Dirac flavors, which has global symmetry SU(4) expected to break to Sp(4). Five
Goldstone bosons are generated of which three are eaten by the SM gauge bosons and a
complex one (a techni-diquark which is essentially a technibaryon of this SU(2) theory)
remainsinthespectrum. ThisisourcandidateforthelightasymmetricDMparticle.
The walking version of this model, known as Ultra Minimal Walking Technicolor
(UMT), has been constructed in [8] and it contains, besides the fermions in the funda-
mental representation, also a Dirac fermion in the adjoint. For the present exploratory
work, we begin with Section III of [8] but without the adjoint fermion. Moreover, when
implementing our lattice simulations we make two additional modifications: all elec-
troweak interactions are omitted from the lattice simulations because that physics is
well-understood by perturbative methods, and explicit technifermion masses are added
becauselatticesimulationswithexactlymasslessfermionsarenotpractical. Theresulting
Lagrangianisverysimple
1
L = − FaµνFaµν +U(iγµDµ −m)U+D(iγµDµ −m)D, (1)
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where U and D are the two techniquark fields having a common bare mass m, Fa is the
µν
field strength, and Dµ is the covariant derivative. The Dirac and technicolor indices of U
3
andDarenotshownexplicitly.
=
Latticesimulationswillbeusedtoextrapolatetom 0,andinthatlimittheLagrangian
hasaglobalSU(4)symmetrycorrespondingtothefourchiralfermionfields
1 1 1 1
U = (1−γ5)U , U = (1+γ5)U , D = (1−γ5)D, D = (1+γ5)D. (2)
L R L R
2 2 2 2
(cid:44)
For m 0, the SU(4) symmetry is explicitly broken to a remaining Sp(4) subgroup as
follows. TheLagrangianfrom(1)canberewrittenas
1 m m (cid:16) (cid:17)†
L = − FaµνFaµν +iUγµDµU+iDγµDµD+ QT(−iσ2)CEQ+ QT(−iσ2)CEQ , (3)
4 2 2
where
   
 U   0 0 1 0
 L   
   
   
   
Q = −iσD2CLUT  , E = −01 00 00 10 , (4)
 R   
   
 −iσ2CDT   0 −1 0 0
R
CisthechargeconjugationoperatoractingonDiracindices,andthePaulistructure−iσ2
isthestandardantisymmetrictensoractingoncolorindices. UnderaninfinitesimalSU(4)
transformationdefinedby
 
 (cid:88)15 
Q → 1+i αnTnQ, (5)
 
 
n=1
theLagrangian(3)becomes
im (cid:88)15 (cid:16) (cid:17)
L → L+ αnQT(−iσ2)C ETn +TnTE Q+h.c., (6)
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n=1
where Tn denotes the 15 generators of SU(4) and αn is a set of 15 constants. The only
generatorsthatleavetheLagrangianinvariantarethosethatobey
ETn +TnTE = 0 (7)
which is precisely the definition of an Sp(4) Lie algebra. From this, it is straightforward
toderivethetenSp(4)generatorsinaspecificrepresentation;seetheappendixof[8].
=
For m 0 the Lagrangian retains the full SU(4) symmetry but, by analogy with the
SU(3) theory of QCD, one might expect dynamical symmetry breaking associated with
theappearanceofanonzerovacuumexpectationvalue,
(cid:104) + (cid:105) (cid:44) .
UU DD 0 (8)
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Since this vacuum expectation value has the same structure as the terms containing m
→
in the Lagrangian, the dynamical breaking would also be SU(4) Sp(4). According to
Noether’stheorem,thefivebrokengeneratorswouldbeaccompaniedbyfiveGoldstone
bosons.
Of course this suggestion of dynamical symmetry breaking must be checked nonper-
turbatively using first-principles lattice simulations. As reported below, we have done
so and our lattice simulations provide direct verification of this dynamical symmetry
breaking.
There have been several previous lattice studies of SU(2) gauge theory with fermions
inthefundamentalrepresentation[26],mainlymotivatedbyinterestatnonzerochemical
potential, but all of these studies relied on the staggered action where the number of
=
fermions is a multiple of 4. Our minimal technicolor theory requires N 2 and thus
f
weusetheWilsonactioninsteadofstaggeredfermions. Whenstudyingchiralsymmetry
breaking scenarios with Wilson fermions attention must be paid to the presence, on a
lattice,oftheunphysicalAokiphase. Forfixedgaugecoupling,theAokiphaseisentered
as the quark mass is reduced. An analytic discussion of the Aoki phase symmetries for
thistheoryisprovidedin[27],andthreelatticestudiesarealsoavailable[28–30]. Forour
presentsimulations,weavoidtheAokiphaseandworkexclusivelyinthephysicalphase.
There have been very few previous lattice results reported for SU(2) gauge theory with
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2 fundamental fermions [30–32] and in each case the primary focus was on a di erent
>
action(eitherN 2oradjointrepresentationfermions). Ourworkisthefirstlatticestudy
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focused on the mass spectrum of the two-color two-flavor theory, which is the familiar
technicolortemplate.
III. LATTICEHADRONOPERATORSANDEFFECTIVEFIELDTHEORY
The creation operator for a meson is the Hermitian conjugate of its annihilation oper-
ator,andasetoflocalannihilationoperatorsformesonsis
O(Γ) ≡ U(x)ΓD(x),
UD
O(Γ) ≡ D(x)ΓU(x),
DU
(cid:18) (cid:19)
O(Γ) ≡ √1 U(x)ΓU(x)±D(x)ΓD(x) , (9)
UU±DD 2
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where Γ is a chosen Dirac structure. In this work we consider Γ = 1, γ5, γµ, or γµγ5. In
lattice simulations, meson masses are extracted from the time dependence of correlation
functions,forexample
(cid:88)(cid:88) (cid:16) (cid:17)†
C(Γ) (t −t ) = O(Γ) (y) O(Γ) (x)
x y
UD UD UD
(cid:126)x (cid:126)y
(cid:88)(cid:88) (cid:104) (cid:105)
= Tr ΓD(y)D(x)γ0Γ†γ0U(x)U(y) , (10)
(cid:126)x (cid:126)y
···
where Tr[ ] denotes a trace over Dirac and color indices and we have dropped the
vacuumexpectationvaluesforthepropagatorsasexplainedinAppendixA.
Perhaps surprisingly, local diquark (i.e. baryon) correlation functions provide no new
data in this theory. To understand why, notice that the available local diquark operators
are
O(Γ) ≡ UT(x)(−iσ2)CΓD(x),
UD
O(Γ) ≡ DT(x)(−iσ2)CΓU(x). (11)
DU
(Some operators containing UT···U or DT···D are identically zero.) The diquark corre-
lationfunctionistherefore
(cid:88)(cid:88) (cid:16) (cid:17)†
C(Γ) (t −t ) = O(Γ) (y) O(Γ) (x)
UD x y UD UD
(cid:126)x (cid:126)y
(cid:88)(cid:88) (cid:20) (cid:21)
= Tr ΓD(y)D(x)γ0Γ†C†(−iσ2)†γ0TUT(x)UT(y)(−iσ2)C (12)
(cid:126)x (cid:126)y
andthiscanberewrittenbyusingtwopropertiesofthechargeconjugationoperator: one
foraDiracmatrix,
γµT = −CγµC† , (13)
andtheotherfortheWilsonfermionmatrix,
(cid:18) (cid:19)T
C−1(−iσ2)−1 U(y)U(x) C(−iσ2) = U(x)U(y), (14)
toarriveat
(cid:88)(cid:88) (cid:104) (cid:105)
C(Γ) (t −t ) = Tr ΓD(y)D(x)γ0Γ†γ0U(x)U(y)
UD x y
(cid:126)x (cid:126)y
= C(Γ) (t −t ) (15)
x y
UD
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Γ
foranychoiceof . Thisconclusionmeansthatalatticesimulationwillfindnumerically-
identicalcorrelationfunctions,andthusidenticalmasses,forthemesonanddiquark. We
haveverifiedthisexplicitlyinthelatticesimulationsdescribedbelow.
Noticethatthedegeneratepairshaveequalangularmomentumbutoppositeparities,
forexample
(cid:16) (cid:17) (cid:16) (cid:17)
J O(Γ) = J O(Γ) , (16)
UD UD
(cid:16) (cid:17) (cid:16) (cid:17)
P O(Γ) = −P O(Γ) . (17)
UD UD
Thisrelationshipfor JP hasalsobeenmentioned,forexample,in[33]. Wemustconclude
thatthethreepseudoscalarmesonGoldstonebosonsareaccompaniedbytwoscalardiquark
Goldstone bosons. This is in contrast to the identification made in [8], where it was
ff
assumed that all five Goldstones would be pseudoscalars. Despite this, the e ective
Lagrangian in [8] remains unaltered because all scalar and pseudoscalar particles were
retainedinthatworkwiththecorrectassignmentwithrespecttothebrokenandunbroken
generatorsofthechiralsymmetrygroup1.
ff
Now that the five Goldstones have been identified, we can define an e ective theory
wherein those five are the only fields. Indeed, our lattice simulations (discussed below)
/
indicate that the non-Goldstone scalar pseudoscalar hadrons are even heavier than the
ff
vectormesons,makingitquitenaturaltointegrateallnon-Goldstonesoutofthee ective
theory. Theannihilationoperatorsareconvenientlycollectedintoacompactform,
(cid:88)(cid:18) (cid:19)
δL = QT(−iσ2)CΓTnQ Φ(Γ)n, (18)
n
whereΦ(Γ)nisagenericnameforthen’thparticlewithDiracstructureΓ. ThisδLrepresents
theeffectiveLagrangiancouplingsforexternalfieldsΦ(Γ)n. Forthespecialcaseofthefive
Goldstone bosons, we sum n only over the five broken generators and we must choose
Γ = γ5. Theresultis
δLG = QT(−iσ2)Cγ5GQ (19)
G Π+
wherethematrix containsthefiveGoldstonebosons(i.e. thepseudoscalarmesons ,
1InpracticetheonlychangeistheredefinitionofΠ(cid:101) andΠ(cid:101) attheunderlyingoperatorlevelwithΠ
UD UD UD
andΠ in[8].
UD
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Π−,Π0 andthescalardiquarksΠ ,Π ):
UD UD
√ √
 
 0 2Π Π0 2Π+ 
 √ UD √ 
 
G = i − 2ΠUD √0 2Π− √−Π0  . (20)
2  −Π0 − 2Π− 0 − 2Π 
 √ √ UD 
 
 
 − 2Π+ Π0 2Π 0 
UD
G
The matrix is closed under Sp(4) transformations: the Goldstone bosons form a five-
G
dimensional representation of Sp(4). The matrix is also identical to M as defined by
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equation(22)of[8]afterthenon-GoldstonefieldsareremovedfromM .
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ff
To summarize, an e ective field theory for the five Goldstone bosons is obtained by
using the matrix G in place of M in [8]. The three pseudoscalar Goldstones (Π±, Π0)
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Π
areresponsibleforelectroweaksymmetrybreaking,andthetwoscalarGoldstones( ,
UD
Π
)aretheDMcandidateanditsantiparticle.
UD
IV. METHODSFORNUMERICALSIMULATIONS
ThestandardWilsonaction,
β (cid:88)(cid:18) 1 (cid:19) (cid:88)
SW = 1− ReTrUµ(x)Uν(x+µˆ)Uµ†(x+νˆ)Uν†(x) +(4+m0) ψ¯(x)ψ(x)
2 2
x,µ,ν x
1 (cid:88)(cid:18) (cid:19)
− ψ¯(x)(1−γµ)Uµ(x)ψ(x+µˆ)+ψ¯(x+µˆ)(1+γµ)Uµ†(x)ψ(x) , (21)
2
x,µ
is used for this study of SU(2) gauge theory with two mass-degenerate fermions in the
fundamentalrepresentation. ConfigurationsweregeneratedusingtheHMCalgorithmas
implementedintheHiRepcode[34]. Atotalof12ensembleswerecreated,corresponding
ff ff β
tosixdi erentbarequarkmassesm foreachoftwodi erentgaugecouplings ;seeTable
0
β = .
I for details. We note that [30] also contains some simulations of this theory at 20
and our findings for the pseudoscalar meson mass and PCAC quark mass are consistent
with that paper. For other parameter choices, see [31] and [32]. In the present study,
all lattices are L3 × T = 163 × 32 with periodic boundary conditions in each direction.
Every ensemble contains 35 configurations separated by 20 unused configurations after
aninitialthermalizationof320configurations. Figures1and2displaythefluctuationsin
theaverageplaquettewithinselectedMarkovchains.
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TABLEI:Numericalvaluesofthegaugecouplingandthebarequarkmassusedto
generatethe12ensemblesofthisproject.
β m
0
2.0 -0.85,-0.90,-0.94,-0.945,-0.947,-0.949
2.2 -0.60,-0.65,-0.68,-0.70,-0.72,-0.75
0.575
m = -0.949
0.57 0
0.565
e m = -0.94
ett 0
u 0.56
q
a
pl
e
g
a 0.555
r
e
av m = -0.9
0
0.55
0.545
m = -0.85
0
0.54
0 200 400 600 800 1000
configuration number from an ordered start
β = .
FIG.1: Evolutionoftheaverageplaquetteforselectedbaremassesat 20. Our
ensembleof35configurationsiscomprisedofthosenumbered320,340,360,...1000.
Quark propagators are created from wall sources built of random U(1) phases at ev-
ery lattice site on one chosen time step, t. Inversions are performed with the standard
i
BiCGstabalgorithm. Toreducestatisticalfluctuations,correlationfunctionsareaveraged
≤ ≤
over all lattice times 0 t 31. Correlation functions then depend on the time t cor-
i f
responding to the annihilation operator. This annihilation operator is local and summed
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0.62
m = -0.75
0
0.615
m = -0.7
0
e
ett 0.61
u
q
a
pl m = -0.65
e 0
g
a
er 0.605
v
a
0.6
m = -0.6
0
0.595
0 200 400 600 800 1000
configuration number from an ordered start
β = .
FIG.2: Evolutionoftheaverageplaquetteforselectedbaremassesat 22. Our
ensembleof35configurationsiscomprisedofthosenumbered320,340,360,...1000.
overallspatialsitestoproduceazero-momentumhadron.
Ourmulti-statefits,
(cid:88)n (cid:18) (cid:18) T(cid:19)(cid:19)
= − ,
C(t) a cosh m t (22)
j j
2
j=1
= − >
useeverynonzerotimeseparationt t t 0,whichavoidsthesubjectivityofchoosing
f i
a fitting window. Pseudoscalar, vector and temporal-axial (A ) correlators are fitted to
4
= , , =
three states, n 3 in (22); scalar and spatial-axial (A A A ) correlators use n 2.
1 2 3
Statistical uncertainties are produced from each 35-configuration ensemble by creating
150bootstrapensembles(having35configurationseach).
ffi
Isovector hadrons are su cient for most topics addressed in this work, but isoscalars
are used to study a specific issue. Isoscalar operators require the computation of single-
site propagators, i.e. quark propagators that begin and end on a single lattice site. In
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