Comments on lattice chiral symmetry and minimal doubling
Michael Creutz
Brookhaven LabOutline• Chiral symmetry in mass parameter space• The anomaly and lattice complications• Topology and doublers• Minimal doubling• Point split operators• Counterterms

Michael Creutz

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Frame the discussion in the path integral formulation

Z=Z(dA)(dψ)(dψ) exp(−Sg (A) + ψDψ) =Z(dA)|D(A)|e−Sg (A)• Concentrate on the determinant of the Dirac matrix |D(A)|‘‘Continuum’’ picture

D =K+MK = γµ (∂µ + igAµ ) = −K † = −γ5 Kγ5M † = M = γ5 M γ5

• introduce Θ later
Michael Creutz

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2Chiral symmetry as a symmetry in parameter space• [K, γ5 ]+ = 0 implies
K = eiωi τi γ5 Keiωi τi γ5• since (Tr)τi = 0|eiωi τi γ5 | = exp(iωi Tr (τi γ5 )) = 1• therefore|K + M | = |K + eiωi τi γ5 M eiωi τi γ5 |Physics unchanged with a modified mass matrix

M −→ eiωi τi γ5 M eiωi τi γ5Michael Creutz

BNL

3Combined with the vector counterpart

M −→ e−iωi τi M eiωi τi ,SU (Nf ) ⊗ SU (Nf ) symmetry• symmetry of the massive theory in parameter space.

Specific example• for two degerate flavors the mass termsmψψ−mψψimψτ3 γ5 ψ• all give equivalent theories

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4Wine bottle analogyV ∼ (σ 2 + ~π 2 − v 2 )2Vπσ• all tilt directions are equivalent• basis for twisted masslattice artifacts improved

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The anomaly
M −→ eiθγ5 M

not a valid symmetry• mixes σ with η ′• η ′ gets extra mass from topology
details depend on specific cutoff
involve complications in defining γ5Fujikawa relates fermion measure and the index theorem• in gauge field with topology, K(A) has zero modes• eigenstates of γ5• winding number ν = n+ − n−Michael Creutz

BNL

6Use K to cutoff high modes of the Dirac operator• zero modes contribute to Tr γ5−K † K/Λ2Trγ5 → Tr(γ5 e)
X−K † K/Λ2=hψi |γ5 e|ψi ii=ν∼ZF F̃• expand in powers of K

at K 4 ultraviolet divergence cancels 1/Λ4In the path integral, configurations weighted by |eiθTrγ5 | = eiθNf νLeaving the usual CP violating parameter Θ of QCD

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m ↔ −m theories not equivalent for Nf odd• −1 is not in SU (2N + 1)
• −mψψ gives the Θ = π theory• spontaneous CP violation expected (Nf > 1)
purely non-perturbative

Two light flavors with non-degenerate mass
23dm2π0 ∼ mu +m−c(m−m)+O(mudq)2• c is a ‘‘low energy constant’’• associated with isospin breaking and π0 η η ′ mixing
lowers mπ0Michael Creutz

BNL

8Negative m2π0 can give pion condensation
md<π 0> != 0CP<σ> > 0mu<σ> < 0CP

Mass gap does not vanish when only one quark is massless• no singularity along mu = 0 axis

• scale dependence in defining a massless up quark

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9Isospin breaking
m2π0m2π+(md −mu )2= 1 − 2C md +mu• in continuum limit either
C diverges
quark mass ratios not constant

Mass independent regularization tricky• not natural with lattice regulator• perturbative matching requires caution

Michael Creutz

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Lattice complications
Lattice at finite volume has no infinities• how can an anomaly appear?
Consider any lattice Dirac operator D• assume gamma five hermiticity γ5 Dγ5 = D†
all operators in practice satisfy this (except twisted mass)

Divide D into hermitean and antihermitean parts

K = (D − D† )/2

M = (D + D† )/2Michael Creutz

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Then

[K, γ5 ]+ = 0[M, γ5 ]− = 0
M → eiθγ5 M an exact symmetry of the determinant• Where is the anomaly?

Naive fermions solve this with doublers• half use γ5 and half −γ5• the naive chiral symmetry is actually flavoredThis carries through to staggered and minimally doubled actions

Michael Creutz

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How about Wilson fermions?

• doublers given masses of order the cutoff• the rotation M → eiωi γ5 M also rotates their phases
Physical Θ is a relative angle• independently rotate the fermion mass and the Wilson term
Seiler and Stamatescu

The overlap operator• eigenvalues on a circle• each zero eigenmode has counterpart on the opposite side
• rotation of Hermitean part rotates heavy mode as well
• anomaly brings in γ̂5ν = Tr(γ5 + γ̂5 )/2Michael Creutz

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Message for continuum QCD:

• physical Θ can be moved around• placed on any one flavor at will

Θ can be entirely moved into the top quark phase• some aspects of the top quark are relevant to low energy physics!
• decoupling theorems don’t apply non-perturbatively

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Topology and doublers

Above effectively a restatement of Nielsen-Ninomiya

Consider the quark propagator at large momentum• D = ip/ + O(m, gAµ )a gauge choice needed to keep things smoothStudy the leading part D = ip/• maintain [ip,/ γ 5 ]+ = 0

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Convenient gamma matrix convention

γ5 = σ3 ⊗ 1 =


1 00 −10 −i
γ4 = σ2 ⊗ 1 =i 00 ~σ~γ = σ1 ⊗ ~σ =~σ 0

Then ip/ =0z(p)−z ∗ (p)0with z a quaternionz(p) = p4 + i~p · ~σ = |p| g(p)Michael Creutz

BNLg(p) ∈ SU (2) ∼ S316For the path integral we want

|D| = z ∗ z = |z|2 = p24 + p~2Consider a surface of constant |D|• a sphere in four dimensions of constant radius |p|p0pOver this sphere, z wraps non-trivially• around a similar sphere in quaternionic space
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On the lattice
Momentum bounded to the Brilloin zone

−π/a < pµ ≤ π/a• the propagator must be periodic in momentum space.
Problem• at the edge of the Brilloin zone
periodicity forbids wrapping
The propagator must unwrap somewhere• if z remains finite, there must be another zero.
Not a bad thing• anomaly says one flavor QCD has no chiral symmetry• with Nf flavors chiral symmetry is SU (Nf ) ⊗ SU (Nf ) ⊗ U (1)B
not U (Nf ) ⊗ U (Nf ).
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Note: zeros at non-zero momentum not a problem• redefining ψ(x) → eiαµ xµ ψ• moves a zero at pµ = 0 to pµ = αµcan always translate our zeros anywhere in momentum space.

Nielsen-Ninomiya conclusion:• local lattice action that anticommutes with γ5• must have an even number of fermion species
Note: 3 is also badMichael Creutz

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Wilson: unwrapping occurs at the doublers• doublers given mass by Wilson term
Overlap: unwrapping occurs at the opposite side of the overlap circle• each zero mode has a corresponding large real eigenvalue
Staggered: four tastes unwrap each other• rooting: unwrapping forces unphysical singularities
at mq = 0 even with non-degenerate quarks

SLAC fermions: unphysical singularities in momentum space
Minimal doubling• 2 species unwrap each other
Michael Creutz

BNL20Minimal doubling examples(units with a = 1)
Karsten (1981); Wilczek (1987)

K=iP3iγ4
γsin(p)+ii
i=1sin(α)

P4µ=1 cos(pµ ) − cos(α) − 3• propagator poles along p4 axis at p~ = 0, p4 = ±αp4+π0−π−π0+π pi• strictly antihermitean kinetic term, exactly anticommutes with γ5 .
• similar to a Wilson term for space momenta with an iγ4 factor

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Tatsu Misumi• ‘‘twisted ordering’’

K =iγ1 (sin(p1 ) + cos(p2 ) − 1)iγ2 (sin(p2 ) + cos(p3 ) − 1)iγ3 (sin(p3 ) + cos(p4 ) − 1)iγ4 (sin(p4 ) + cos(p1 ) − 1)

• propagator poles at p = (0, 0, 0, 0) and p = (π/2, π/2, π/2, π/2)

02π4π

• 1 → C shifts distance between poles

Michael Creutz

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Graphene motivated: generalize z = eip1 + eip2 + 1 toz = e+iσ1 p1 + e+iσ1 p2 + e−iσ1 p3 + e−iσ1 p4+ e+iσ2 p1 + e−iσ2 p2 + e−iσ2 p3 + e+iσ2 p4+ e+iσ3 p1 + e−iσ3 p2 + e+iσ3 p3 + e−iσ3 p4− 12C• minimal doubling for 12 < C < 1.
• poles along major diagonal pµ = pν = arccos(C)√C = 1/ 2 gives orthogonal lattice

C = cos(π/5) gives 4d analogue of graphene/diamondMichael Creutz

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Point splitting
One field gives two particles• separate them by combining fields at nearby points
For the Karsten Wilczek form above• start with free fields momentum
 space4 +α)u(q) = 21 1 + sin(qψ(q + αe4 )sin(α)
4 −α)
ψ(q − αe4 )d(q) = 21 Γ 1 − sin(qsin(α)• factor of Γ = iγ5 γ4 corrects for opposite chirality of one mode
Go to position space and insert gauge fields

ux =eiαx4dx =Γe−iαx4

Michael Creutz

22BNL

e−iα Ux,x−e4 ψx−e4 −eiα Ux,x+e4 ψx+e4
ψx + i2 sin(α)

e−iα Ux,x−e4 ψx−e4 −eiα Ux,x+e4 ψx+e4
ψx − i2 sin(α)
24Meson operators (α = π/2):iπ0 (x) =(4ψ x γ5 ψx + ψ x−e4 γ5 ψx−e4 + ψ x+e4 γ5 ψx+e416− ψ x+e4 U U γ5 ψx−e4 − ψ x−e4 U U γ5 ψx+e4 )η ′ (x) =1(ψ x−e4 U γ5 ψx − ψ x U γ5 ψx−e48+ ψ x+e4 U γ5 ψx − ψ x U γ5 ψx+e4 )• π0 connects sites of the same parity• η ′ connects even and odd sites; no on-site contribution
Tatsu and Taro: use uu and dd fields to split masses• variation on the Wilson term.

• use this form to study eigenvalue flow and the index theorem

Michael CreutzBNL25Counterterms
All actions pick a special direction• sµ points between the two zeros• hypercubic symmetry broken
Capitani, Weber, Wittig, MC• introduces new perturbative counterterms• the lattice gets distorted
the distance between the zeros
fermion speed of lightgluon speed of lightThree relevant operators

ψγµ sµ ψψγµ sµ sν ∂ν ψFµν Fµρ sν sρ
One operator of naive dimension 3, similar to Wilson• two of dimension 4
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Are the counterterms really necessary?
• Need to remove the special direction
Not yet realized
Two possible approaches:
Zeros half way around Brilloin zone
• pµ = 0 and pµ = pν = +π
• zeros in a body centered hyper-cubic arrangement

02π4π• adjusting C above gives extra zeros before reaching thisMichael Creutz

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Distort lattice to form a hyperdiamond• make 0, 0, 0, 0 to p, p, p, p distance equal p, p, p, p to 2π, 0, 0, 0
• each site has five equidistant neighbors• analog of graphene in 2-d• C = cos(π/5) reaches these points• but the action as we travel along different bonds not equivalent• brings in higher harmonics
Can non-nearest hoppings fix these examples?
• can we remain local over hypercubes?
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Summary
Chiral symmetry is a symmetry in mass parameter space
Breaking of continuum chiral symmetry gives rich physics• spontaneous• anomaly• masses
Understanding this on the lattice is challenging but instructive
Minimal doubling maintains some chiral symmetry• fast for simulations• avoids rooting• counterterm questions remain
Michael Creutz

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