Alex Kalloniatis

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CHIRALITY IN A DOMAIN MODEL FOR THE QCD VACUUM

• Alex Kalloniatis
• (CSSM, Adelaide)
• Sergei N. Nedelko
• (JINR, Dubna)

QCD_at_Work, Conversano, June 14-18, 2003.
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Singular Fields in the QCD Partition Function
In general, action will be infinite. exp(-action)
0. no contribution from such singular
fields. EXCEPTION when fields satisfy specific
conditions.
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Example Abelian singular field in YM theory
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4-dimensional Euclidean space
fields in a given domain interact with
fluctuations in rest of system
replace with a mean field in each domain and
treat effective domains as noninteracting
On average, each domain has

• hyperspherical geometry
• same radius
• constant field strength
• self-dual (SD) or anti-self-dual (ASD)
• colour direction Abelian

2 Model parameters average radius, R average
field strength, B

• domain positions
• EB field orientations
• colour orientations in Abelian subgroup
• random assignment of SD and ASD

Collective coordinates (to be summed/integrated
over in path-integral)
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Topological charge
absolute value of topological charge per domain
Topological susceptibility
Area law in Wilson loop string tension
C
2L
for large L
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large (c.f. 0.01 GeV4)
non-integer
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Dirac operator in a single domainKalloniatis
Nedelko, PRD 66 074020 (2002).
In domain model
BC explicitly breaks chiral symmetry
UA(1)? Z2
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Solution
angular momentum quantum numbers k,m1,m2
colour projection
spin-field projection
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• discrete spectrum for each k
• principal quantum , n
• spectrum asymmetric
• similar behaviour for all k
• NO ZERO MODES
• spectrum in a single domain
• NOT the entire ensemble!

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Comparison with chirality measurements on the
lattice Dirac eigenmodes as filters for gluonic
structures in the vacuum.

• To compare lattice results with domain model
• Mimic lattice spacing smear over centre.
• Do for each a.
• Distribute a randomly through ensemble.

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Spectral Asymmetry and Zeta Function
Regularisation of Fermion determinants
Free energy and quark determinant for single
domain
Trick for computing spectral sums
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Representation of square Dirac operator
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QUARK CONDENSATE
Zeta function with mass term
h(1) breaks symmetry between minima of free
energy
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Domain Boundary Condition
Infinitesimal mass term
Thermodynamic Limit
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Domain model confines static quarks and
provides for spontaneously broken chiral symmetry
Kalloniatis, Nedelko PRD 64, 114025
(2001), Kalloniatis, Nedelko PRD 66, 074020
(2002).