SLIDE 1
Baryon in the sextet gauge model
Zoltan Fodor, Kieran Holland, Julius Kuti, Santanu Mondal, Daniel Nogradi, Chik Him Wong Lattice Higgs Collaboration (LatHC)
SLIDE 2 Composite Higgs Model
- Strongly coupled gauge theory.
- Electroweak symmetry breaking is realized by spontaneous
chiral symmetry breaking; and three Goldstone bosons are eaten up to give three massive gauge bosons (W ±,Z 0).
- Higgs is a composite particle of vacuum quantum numbers
0++
SLIDE 3
Possible dark matter candidates
Electroweak singlet Goldstone boson → Possible for the models with more than three PNGBs. Weak singlet ⇒ can be light.
Nf = 2 pseudo-real SU(2) gauge model: Flavour symmetry group is SU(2Nf ) Most attractive channel breaks SU(4) − → Sp(4) ⇒ 5 Goldstone bosons. One of these can be a DM candidate (Lewis,Pica,Sannino in PRD 85, 014504 (2012)).
SLIDE 4
Dark Baryon
Stable, electrically neutral, neutron like state, but has weak hypercharge. ⇒ Needed to be heavy enough from experimental constraint Nf = 2 SU(3) gauge theory with fermions in sextet representation: Symmetry breaking: SU(2)×SU(2) − → SU(2) : 3 PNGBs → three massive electroweak gauge bosons.
SLIDE 5
Why Nf = 2, SU(3) Sextet Gauge model?
Minimal realization of composite Higgs mechanism. Walking behaviour
LatHC: Phys. Lett. B 718, 657, PoS of LATTICE 2014; Kogut and Sinclair, Phys. Rev. D 84, 074504 (2011), Phys. Rev. D 81, 114507 (2010)
Expectedly low S parameter S ∼ N(N +1) 2 .Nf 2 Estimate from resonance spectrum shows it is not QCD like
(T. Appelquist and F. Sannino, Phys. Rev. D 59, 067702 (1999) [hep-ph/9806409] ).
SLIDE 6
Symmetry braking: SU(2)R ×SU(2)L − → SU(2)V Exactly three Goldstone modes − → eaten up to give the three massive gauge bosons. Can give a light composite scalar state with Higgs quantum numbers (0++)
LatHC: PoS LATTICE 2013, 062 (2014); Z. Fodor, PoS of LATTICE 2014
SLIDE 7
Is neutral baryon stable?
Charge assignments: u − → +2 3 d − → −1 3 Sextet neutron udd : electrically neutral Lightest because of elctromagnetic correction Stable .....unlike QCD
SLIDE 8
Constructing nucleon operator in continuum
Color singlet: 6⊗6⊗6 = 1⊕2×8⊕10⊕10⊕3×27⊕28⊕2×35 (1) → Only one singlet possible. Ψ ≡ (Ψ0,Ψ1,...,Ψ5) a vector in the sextet representation of SU(3) TABC ΨA ΨB ΨC ≡ T ′
aa′bb′cc′ ψaa′ ψbb′ ψcc′
= εabc εa′b′c′ ψaa′ ψbb′ ψcc′ (2) ψaa′ − → ψ′
aa′ = Uab Ua′b′ ψbb′
(3) Then εabc εa′b′c′ ψaa′ ψbb′ ψcc′ − → εabc εa′b′c′ Uad Ua′d′ Ube Ub′e′ Ucf Uc′f ′ ψdd′ ψee′ ψff ′ = εdef εd′e′f ′ detU detU ψdd′ ψee′ ψff ′ = εdef εd′e′f ′ ψdd′ ψee′ ψff ′ (4) since detU = 1.
SLIDE 9
→ Singlet → TABC symmetric. Correct JPC ⇒ nucleon operator antisymmetric under exchange of spin indices. ⇒ Symmetric in flavour (Spin Statistics Theorem).
SLIDE 10 Flavour SU(2) irrep: 2×2×2 = 1A +2×2M +4s (5) Thus sextet nucleon belongs to 2M irrep. An example: Tritium isotope H3 with pnn or the Helium isotope He3 with ppn as baryon ground states. Color singlet contituents ⇒ spin-flavour structure will be similar as
This comes from a Slater determinant combining mixed representations of permutations.
SLIDE 11 sextet baryon in quark language
In quark language our two fermions have SU(2) flavor symmetry and eight states can be formed: uuu, uud, udu, udd, duu, dud, ddu, ddd They are groupped into an isospin quadruplet and two isospin doublets. The quadruplet belongs to the symmetric rep.
2, 3 2
2, 1 2
√ 3
2,−1 2
√ 3
2,−3 2
SLIDE 12 We also have two doublets which have mixed symmetries:
2, 1 2
- = −(2ddu −udd −dud)/sqrt6
- 1
2,−1 2
where the mixed symmetry means symmetry under 1 → 2 and 2→1 and no definite symmetry under 1→3. The other doublet:
2, 1 2
2,−1 2
anti-symmetric under 1→2 and no symmetry under 1→3. From the combination of the two mixed reps it is possible to construct an anty-symmetric spin-flavor wave function.
SLIDE 13
Nucleon operator in lattice with staggered fermion
Bαi(x) = TABC uαi
A (x) [uβj B (x) (Cγ5)βγ (C ∗γ∗ 5)ij dγj C (x)]
Looking for a operator as local as possible. Staggered fields: uαi = 1 8 ∑
η
Γαi
η χu(η)
where η ≡ (η1,η2,η3,η4), Γ(η) = γη1
1 γη2 2 γη3 3 γη4 4 .
Diquark ≡ [...] = − 1 82 ∑
ηη′
Tr(Cγ5Γ′
ηCγ5ΓT η )χB u (η′)χC d (η)
= − 1 82 ∑
ηη′
δηη′S(η)χB
u (η′)χC d (η)
= − 1 82 ∑
η
S(η)χB
u (η)χC d (η),
S(η) is a sign factor
SLIDE 14 Diquark populates 16 corner of the hypercube. Writing the third quark in staggered basis: Bαi(x) = −TABC 1 83 ∑
η′
Γαi
η′ χA u (η′) ∑ η
S(η)χB
u (η)χC d (η)
To make the operator confined in a single time-slice an extra term has to be added to or subtracted from the diquark part. → Similar to the construction of single time-slice staggered meson
→ corresponds to the parity partner.
Bαi(x) = −TABC 1 83 ∑
η′
Γαi
η′ χA u (η′) ∑ η
S(η)χB
u (η)χC d (η)
η ≡ (η1,η2,η3), η′ ≡ (η′
1,η′ 2η′ 3)
SLIDE 15
Hence Bαi(x) is sum of 64 terms with proper sign. Local terms vanish individually when contracted with TABC in color-space → Different from normal QCD where the color contraction tensor is εabc, antisymmetric. The next simple type of terms is diquark sitting one of the 8 corners and the third quark in any other corner. We use operators of this type for our pilot calculation.
SLIDE 16 Operators used
IV VIII
x y z
Table: Operator set a
Label Operators IVxy χu(1,1,0,0) χu(0,0,0,0) χd(0,0,0,0) IVyz χu(0,1,1,0) χu(0,0,0,0) χd(0,0,0,0) IVzx χu(1,0,1,0) χu(0,0,0,0) χd(0,0,0,0) VIII χu(1,1,1,0) χu(0,0,0,0) χd(0,0,0,0)
SLIDE 17 IV VIII
x y z
Table: Operator set b
Label Operators IVxy χu(0,0,0,0) χu(1,1,0,0) χd(1,1,0,0) IVyz χu(0,0,0,0) χu(0,1,1,0) χd(0,1,1,0) IVzx χu(0,0,0,0) χu(1,0,1,0) χd(1,0,1,0) VIII χu(0,0,0,0) χu(1,1,1,0) χd(1,1,1,0)
SLIDE 18 Small ensemble test with different operators
11 12 13 14 15
0.45 0.5 0.55 0.6 0.65 MN VIII IV_xy IV_yz IV_zx
3x64, β = 3.20, m = 0.007, tmax = 20
11 12 13 14 15
tmin
0.45 0.5 0.55 0.6 0.65 MN
Set a Set b
Figure: Comparing nucleon mass obtained by different operators.
SLIDE 19 Chiral extrapolation
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008
m
0.2 0.3 0.4 0.5 0.6
MN
MN = a + b m, a=0.33(2), b=38(2) χ
2/dof = 0.5
3X64, β = 3.200
Preliminary
Figure: Chiral extrapolation
SLIDE 20 Hadron-Spectrum so far
0.002 0.004 0.006 0.008
m
0.1 0.2 0.3 0.4 0.5 0.6
M
MN Ma1 Mρ Ma0 Mπ
M = M0 + C m
Figure: Hadron masses versus quark mass
SLIDE 21 Direct Detection
PHYSICAL REVIEW D 88, 014502 (2013)
10−2 10−1 100 101 102 Mχ = MB [TeV] 10−16 10−14 10−12 10−10 10−8 10−6 10−4 10−2 100 102 104 Rate, event / (kg·day)
Nf = 2 Nf = 6 XENON100 [1207.5988], 95% CL exclusion
SLIDE 22 Conclusion and outlook
- The value of nucleon mass in sextet gauge model, from our
preliminary calculation is 0.33(2) in lattice unit, which is 3193±167 GeV when converted to physical unit.
- We also have ensembles on 403 ×80 and 483 ×96, and also at
a finer lattice spacing 3.25, thus more systematic studies can be done.
- Construction of operators with no mixing in taste space is
needed for more precise calculation.
- Calculations of magnetic moment and electric charge radius
are needed to compare with direct detection experiments.
