Comments on non-degenerate
staggered fermions, staggeredWilson and Overlap fermions,and the application of Chiral
Perturbation theory to lattice
fermions
Stephen R. Sharpe
University of Washington

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 20121

Outlineunrooted staggered fermions to simulate 2+2,
• Using
2+1+1 and 1+1+1+1 flavors: Is it practical?
• Staggered-Wilson fermions---Adams version• Staggered-Wilson fermions---Hoelbling-like versions
low energy coefficients in ChPT using• Constraining
Weingarten mass inequalities (Flash talk?)

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 20122

Caveatunrooted staggered fermions to simulate 2+2,
• Using
2+1+1 and 1+1+1+1 flavors: Is it practical?
• Staggered-Wilson fermions---Adams version• Staggered-Wilson fermions---Hoelbling-like versions
low energy coefficients in ChPT using• Constraining
Weingarten mass inequalities

Work very much in progress!

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 20123

Outlineunrooted staggered fermions to simulate 2+2,
• Using
2+1+1 and 1+1+1+1 flavors: Is it practical?
• Staggered-Wilson fermions• Staggered-Wilson fermions---Hoelbling-like versions
low energy coefficients in ChPT using• Constraining
Weingarten mass inequalities

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 2012

4

Staggered fermions• Very OLD type of fermion! [Susskind, 1976]
component on each site, describing 4• Single
continuum fermions (tastes)• Computationally efficient
practice, each continuum flavor described by a
• Inrooted
staggered fermion• Why not instead use 4 tastes to describe u,d,s & c?
non-degenerate staggered fermions discussed• Suchlong ago by [Golterman & Smit (1984)]• Take a NEW look at this possibility
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 2012

5

Action & Bilinears
Sunimproved =�χ̄(Dst + m)χ =

�
n

χ̄n

�

�

�

ηµ (n)∇µ + m χnµ

• Covariant bilinears transform covariantly under lattice symmetries (unlike
hypercube bilinears)

� 1�χ̄n (γS ⊗ ξF )n,n+S−F Un,n+∆ χn+∆N∆n|∆|=|S−F |�∼ a4 d4 x Q̄(γS ⊗ ξF )Q

cov
OS⊗F
=

S & F are “hypercube vectors”: Sμ={0,1}
Δ includes forward & backward differences (gives “symmetric shifts”),
e.g. S-F=(1100) Δ=(1100),(1,-1,00),(-1,1,00),(-1,-1,00)

γ12 = γ1 γ2 , . . .
e.g normal mass term in this notation

cov
OI⊗I
=

�

χ̄n χnn

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 20126

Breaking degeneracy
Snon−degen =

•
•

�

cov
cov
cov
cov
χ̄Dst χ + m̄OI⊗I+ mA OI⊗ξ
+m
iO
+
m
iO
12
34
I⊗ξ12
I⊗ξ34
5
� �� �

0-link2-linkΓ55 Γ52-link4-linkMany choices: use even # links so determinant is real, positive [de Forcrand]
Use diagonal matrices (Weyl basis)

M = m̄

�

1 0
0 1

�

+mA

�

−1 0
0 1

�

+m12

�

σ3
0

0
σ3

�

+m34

�

−σ3
0

0
σ3

�

• m12=m34=0 gives 2+2 (Adams-type)• m12=±m34 gives 2+1+1 (modified Hoelbling)• Here we take all lattice masses to be of O(a), i.e. physical• Each mass is independently multiplicatively renormalized, with no mixing [G+S]✴ Follows because in different irreps of lattice symm. group✴ No fine tuning, only usual tuning

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 20127

Lattice symmetries [G+S]S=

Symmetries:�covχ̄Dst χ + m̄OI⊗I{C0 , Ξµ , Rµν , Is } ×

�
U1

����

if m̄=0
Lattice Charge Conjugation
Acts on spin & taste
Shifts (with eip removed)
Q → (1 ⊗ ξµ )Q
Form Γ4 subgroup

Spatial inversion
Acts on spin & taste
Hypercubic rotations
Act on spin & taste

Only small discrete subgroup of SU(4) taste ispreserved (and mixed up with spin)
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 20128

Lattice symmetries (2+2 ﬂavors)
SA =

�

cov
cov
χ̄Dst χ + m̄OI⊗I + mA OI⊗ξ5• Useful properties:

Ξµ : (γS ⊗ ξF ) → (γS ⊗ ξµ ξF ξµ ) = (γS ⊗ ξF )(−)

✴ Adams mass flips sign under all shifts�ν�=µ FνIµ : (γS ⊗ ξF ) → (γµ5 γS γ5µ ⊗ ξµ5 ξF ξ5µ ) = (γS ⊗ ξF )(−)Sµ +Fµ✴ Adams mass flips sign under all axis inversions (and Is=I1I2I3)⇒

�Ξ µ = Ξµ I µ

are unbroken

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 20129

Lattice symmetries (2+2 ﬂavors)
SA =

�

cov
cov
χ̄Dst χ + m̄OI⊗I + mA OI⊗ξ5

Symmetries:{C0 , Ξ�µ , Rµν } ×

U1�
����

if m̄=mA =0

Discrete group is halved in size, with alltransformations acting both on spin and taste.
Rotation symmetry unchanged.

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201210

Lattice symmetries (2+1+1 ﬂavors)
S2+1+1 =

�

cov
cov
χ̄Dst χ + m̄OI⊗I+ mA OI⊗ξ+ m1234 i
5

Symmetries:{CT , Ξ�µ , R12 , R34 , R13 R24 } ×

�

�

cov
cov
OI⊗ξ
+
O
I⊗ξ34
12



��



0
0 
0 2σ3

�

�

U1�
����

if m̄=mA =m1234 =0
Modified C [Misumi]2
CT = R21 R13
C0

Compared to SA, rotational symmetry broken
from 192 element SW4 to 64 element subgroup

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201211

Lattice symmetries (2+1+1 ﬂavors a la Hoelbling)
�
cov
cov
SH =
χ̄Dst χ + m̄OI⊗I+ mA OI⊗ξ
+
m
O
H
H
5
OH = i

�

cov
cov
OI⊗ξ
+
O
I⊗ξ34
12

�




��



0
0 
0 2σ3

�

�

+i

�

cov
cov
OI⊗ξ
+
O
I⊗ξ42
13

�




��



0
0 
0 −2σ2

�

�

+i

�

cov
cov
OI⊗ξ
+
O
I⊗ξ23
14

�




��



0
0 
0 2σ1

�

�

Symmetries:�{CT , Ξµ , R12 R43 , R14 R32 } ×

�
U1

����

if m̄=mA =mH =0

Compared to S2+1+1, rotational symmetry groupreduced from 64 to 8 elements

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201212

Lattice symmetries (1+1+1+1 ﬂavors)
S1+1+1+1 =

�

cov
cov
cov
cov
χ̄Dst χ + m̄OI⊗I+ mA OI⊗ξ
+
m
iO
+
m
iO
12
34
I⊗ξ12
I⊗ξ34
5

Symmetries:�{CT , Ξµ , R12 , R34 } ×

�
U1

����

if m̄=mA =m12 =m34 =0Compared to S2+1+1, rotational symmetry groupreduced from 64 to 16 elements.
Compared to SH, rotation group here is larger
but is Abelian.

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201213

Pseudoscalar operators for 2+2 theory
SA =

�

cov
cov
χ̄Dst χ + m̄OI⊗I + mA OI⊗ξ5• Consider pseudoscalars at rest
✴ In continuum, have light-light, heavy-light and heavy-heavy states✴ On lattice, classify operators by timeslice group✴ For standard staggered this is: {C0 , Ξj , Rjk , Is }✴ For 2+2 theory it reduces to: {C0 , Ξ�j , Rjk }• Use timeslice operators with spin-taste: (γ5 ⊗ ξF )• Use methods of “toolkit” [Kilcup & SS]

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201214

Pseudoscalar operators for 2+2 theory• Pion irreps from original staggered theory mix in pairs¯ and h̄h(γ5 ⊗ I) and (γ5 ⊗ ξ5 ) mix Create ��¯ and h̄�(γ5 ⊗ ξ4 ) and (γ5 ⊗ ξ45 ) mix Create �h(γ5 ⊗ ξj4 ) and (γ5 ⊗ ξj45 ) mix¯ j � and h̄σj h States in 3-d irrep
Create �σ(γ5 ⊗ ξj ) and (γ5 ⊗ ξj5 ) mix¯ j h and h̄σj �Create �σStates in 3-d irrep

Good news: discrete symmetries enoughto have 3-d irreps as in continuum

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201215

Problems for 2+2 theory¯ and h̄h(γ5 ⊗ I) and (γ5 ⊗ ξ5 ) mix Create ��• Both correlators have disconnected contractions• Cannot separate heavy from light states¯ and h̄�(γ5 ⊗ ξ4 ) and (γ5 ⊗ ξ45 ) mix Create �h• Cannot separate l-bar h and h-bar l (lattice induces FCNC!)(γ5 ⊗ ξj4 ) and (γ5 ⊗ ξj45 ) mix¯ j � and h̄σj h States in 3-d irrep
Create �σ• Cannot separate heavy and light states(γ5 ⊗ ξj ) and (γ5 ⊗ ξj5 ) mix¯ j h and h̄σj �Create �σStates in 3-d irrep• Cannot separate l-bar h and h-bar l

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201216

Generalize to 1+1+1+1 theory
S1+1+1+1 =

�

cov
cov
cov
cov
χ̄Dst χ + m̄OI⊗I+ mA OI⊗ξ
+
m
iO
+
m
iO
12
34
I⊗ξ12
I⊗ξ34
5

�
{C
,
Ξ
T
• Timeslice group reduces to
j , R12 }(γ5 ⊗ I), (γ5 ⊗ ξ5 ), (γ5 ⊗ ξ34 ) and (γ5 ⊗ ξ12 ) mix (and have disconnected contractions)create ūu, d̄d, s̄s and c̄c(γ5 ⊗ ξ4 ), (γ5 ⊗ ξ45 ), (γ5 ⊗ ξ3 ) and (γ5 ⊗ ξ35 ) mix

create ūs, d̄c, s̄u and c̄d(γ5 ⊗ ξ1 ), (γ5 ⊗ ξ2 ), (γ5 ⊗ ξ15 ) and (γ5 ⊗ ξ25 ) mix

create ūc, d̄s, s̄d and c̄u(γ5 ⊗ ξ14 ), (γ5 ⊗ ξ24 ), (γ5 ⊗ ξ13 ) and (γ5 ⊗ ξ23 ) mix

create ūd, d̄u, s̄c and c̄s

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201217

Outlook for non-degenerate
staggered fermions• Non-degenerate staggered fermions seem impractical• Only advantage over Wilson is lack of fine tuning

in simplest case (pseudoscalars at rest) one• Evencannot separately create states with different tastes
problem with 2+1+1 twisted mass “solved” in practice using partial• Similar
quenchingin motion, baryons, and weak operators would• States
likely be a nightmare• Bottom line: too few symmetries
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201218

Outlineunrooted staggered fermions to simulate 2+2,
• Using
2+1+1 and 1+1+1+1 flavors: Is it practical?
• Staggered-Wilson fermions---Adams version• Staggered-Wilson fermions---Hoelbling-like versions
low energy coefficients in ChPT using• Constraining
Weingarten mass inequalities

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201219

General set-up
same actions as described above, but with lattice• Usemasses now of O(1) [physical masses ~1/a]• Requires fine-tuning to obtain light states just as with Wilson fermions• 2 light-flavor example [Adams]
SA =�χ̄Dst χ + r

�

[de Forcrand et al, 2012]

cov
cov
OI⊗I
− OI⊗ξ
5

�

cov+ mOI⊗I

Shift spectrum so one branch is lightas described above, but now• Symmetries
cannot treat masses as small, so canhave any number of mass insertions• Only interested in light sector: ξ =+1
5

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201220

Questions and issues• What additional terms are allowed in the action?
[Adams] Schematically : χ̄Dst (1 ⊗ ξ5 )χ

In physical sector, this simply renormalizes kinetic term

• What is the symmetry group in the light sector?
• What is the spectrum of light states?
are the impact of discretization errors in pion
• Whatand vacuum sectors?
• Computational efficiency?
• Utility as kernel for overlap?
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201221

Some preliminary answers• What is the symmetry group in the light sector?
Full group is:{C0 , Ξ�µ , Rµν }(m~1/a implies loss of approximate U1ε)

Proposed method: keep only transformations whichtake ξ5=1 subspace into itself, and drop “heavy part”
Results:2Ξ�j Ξ�4 Rj4= Ξj Ξ4 ∼ (1 ⊗ ξj4 ) → (1 ⊗ σj )2�jkl Ξ�k Ξ�l Rkl∼ (1 ⊗ σj )22Ξ�4 R34
R12= Ξ4 Is ∼ (γ4 ⊗ I) = P

2
C0 Ξ�2 Ξ�4 R24∼ Ccont

Rotations Rμν: Rij and Rk4 act simultaneously in spin, spaceand flavor (as an SU(2) rotation about iso-axis k by π/2)

• Same result holds for overlap version
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201222

Some preliminary answers• What is the spectrum of pions at rest?
Can use earlier analysis of 2+2 flavor staggered theory, keeping only light-light statesη:¯ and h̄h Has disconnected contractions(γ5 ⊗ I) and (γ5 ⊗ ξ5 ) mix Create ��π:(γ5 ⊗ ξj4 ) and (γ5 ⊗ ξj45 ) mix¯ j � and h̄σj h States in 3-d irrep
Create �σ• Symmetries sufficient to have degenerate pion triplet• Same holds for overlap version• Expect symmetry breaking for pions in motion, other mesons and for baryons

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201223

Some preliminary answers

are the impact of discretization errors in pion
• Whatand vacuum sectors?
Method adapted from those for Wilson and staggered ChPT [SS & Singleton, Lee & SS]
down all dimension 5 & 6 terms allowed by lattice symmetries; these would• Writebe needed to improve the action, and so, without improvement, tell us the form of
discretization errors in Symanzik’s continuum effective action• Project these terms into the physical subspace (new step)
the projected terms into the continuum Symanzik effective action (here for 2• Map
flavors)• Match the operators into the chiral effective theory
their effects (particularly those of SU(2) breaking operators) on the• Analyze
vacuum (e.g. is there an Aoki phase?) and pion spectrum
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201224

Very preliminary analysis
��Symmetries : {C0 , Ξµ , Rµν } (no U1 )Allowed operators of dimension 3 & 4
Mass and kinetic terms collapse to standard 2 flavor continuum formson ξ5=1 subspacee.g. Q̄(1 ⊗ ξ5 )Q −→ ψ̄ψAllowed operators of dimension 5e.g. for Q̄(iσµν Gµν ⊗ ξF )Q

Only ξF = I and ξ5 allowed

only standard flavor singlet clover term in physical subspaceψ̄iσµν Gµν ψ
no flavor breaking at O(a)S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201225

Very preliminary analysis
Allowed operators of dimension 6• Compared to normal staggered analysis, loss of U1ε increases operator count (24 to 35)• Loss of Is implies doubling of operators (so 70 in all)• Examples of new operators:
Q̄(I ⊗ I)QQ̄(I ⊗ ξ5 )Q,
In physicalsubspace:•

Also get

(ψ̄ψ)

2

�

�

µ�=µ

j�=k�=��=j�jQ̄(γµν ⊗ ξµν )QQ̄(γµν ⊗ ξµν )Q(ψ̄γjk ⊗ σ� ψ)ψ̄σj ψ ψ̄σj ψ ∝ (ψ̄ψ)22�j(ψ̄γj4 ⊗ σj ψ)

2(by Fierz)• Only flavor-breaking occurs in operators with correlated spin and flavor indices• These do not contribute to potential in ChPT, since require derivatives• Conclusion: vacuum structure analysis identical to that for Wilson fermions at O(a2)S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201226

Many open questions• Can one understand from ChPT analysis (to all orders) why there is no isospin
breaking in the pion states at rest, as deduced from the lattice symmetries?

• Do the symmetries of the resulting chiral Lagrangian match those determined above?
(A cross check on the methodology)• What are the symmetries of the chiral Lagrangian describing the overlap version?
Chiral symmetry will add additional constraints, but flavor-breaking should still enter.

• Can one understand how the pion spectrum evolves as mA and m vary from 0 (usual
staggered) to ~1/a (Adams)? Must have level crossings since Goldstone pion becomesthe η in Adams theory. Can analyze using ChPT for mA < ΛQCD

• ....

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201227

Outlineunrooted staggered fermions to simulate 2+2,
• Using
2+1+1 and 1+1+1+1 flavors: Is it practical?
• Staggered-Wilson fermions---Adams version• Staggered-Wilson fermions---Hoelbling-like versions
low energy coefficients in ChPT using• Constraining
Weingarten mass inequalities

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201228

Perils of fewer symmetries
S12 =

�

χ̄Dst χ + r

�

�

cov
cov
cov
OI⊗I
− iOI⊗ξ
+mO
I⊗I
12
�
��
�

 

1 0   σ3 0 

−
0 1
0 σ3

• Gives 2+2 theory, but with different physical subspacefrom Adams theory• Has advantage of using 2-link mass term (instead of 4-link)• Disadvantage is breaking of rotation symmetries��{C0 , Ξµ , Rµν } −→ {CT , Ξµ , I3 , R12 , R34 }S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201229

Perils of fewer symmetries��{C0 , Ξµ , Rµν } −→ {CT , Ξµ , I3 , R12 , R34 }• Smaller group allows new kinetic terms (with independent coefficients)(p1 γ1 + p2 γ2 ⊗ I), (p1 γ1 + p2 γ2 ⊗ iξ12 ),(p3 γ3 + p4 γ4 ⊗ I), and (p4 γ3 + p4 γ4 ⊗ iξ12 )• Projecting onto physical 2 flavor subspace (i ξ12=+1) gives 2 independent terms:

ψ̄(p1 γ1 + p2 γ2 )ψ and ψ̄(p3 γ3 + p4 γ4 )ψ
➡ O(1) breaking of rotation invariance in the continuum limit!

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201230

Nf=2 �Hoelbling-like theoryS=χ̄Dst χ + r i
�

�

cov
cov
OI⊗ξ
+
O
I⊗ξ34
12



0
0

��



0 
2σ3

�

�

• Gives 1+2+1 flavor theory, with 2 flavor subspace at origin• Variant of proposal of [de Forcrand, Kurkela and Panero]• Symmetries include new charge-conjugation C’T [Misumi]• C’T forbids mixing with other mass terms ➡No tuning!
• Spectrum is left-right symmetric if average {U, U*}
�
�
{C
,
Ξ
• Symmetry group:Tµ , R12 , R34 , R24 R31 }• Only kinetic terms consistent with symmetries are: (pµ γµ ⊗ I) and (pµ γµ ⊗ ξ5 )• Both give standard kinetic term when projected onto 2 flavor subspace➡ Rotational invariance recovered in the continuum limit?
• Same holds for theory with Hoelbling mass, despite smaller symmetry group
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201231

Examples from analysis
Symmetry group:{CT� , Ξ�µ , R12 , R34 , R24 R31 }Ξ�µ and rotations allow : ([p1 γ1 + p2 γ2 ] ⊗ ξ12 ) + ([p3 γ3 + p4 γ4 ] ⊗ ξ34 )Forbidden by C’TΞ�µ and CT� allow : (p1 γ15 ⊗ ξ15 )Forbidden by (R12)2

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201232

Gluonic counterterms?
“12+34” Symmetry group:{CT� , Ξ�µ , R12 , R34 , R24 R31 }G212 + G234 and G213 + G223 + G214 + G224 can appear with different coefficients➡ Need one gluonic counterterm to restore rotational invariance“Hoelbling” Symmetry group:{CT� , Ξ�µ , R12 R43 , R14 R32 }G212 + G234 and G213 + G224 and G214 + G223 can appear with different coefficients➡ Need two gluonic counterterms to restore rotational invarianceDavid was right!
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201233

Nf=1 Hoelbling-like theoryS=�χ̄Dst χ + r

�

cov
cov
cov
iOI⊗ξ
+
iO
−2O
I⊗ξ34
I⊗I
12

�

��

√
or OH / 3

�

• Shift so that left-hand branch is near origin
•

�

cov
+ mOI⊗I

fine tuning

cov
Symmetries allow mixing with OI⊗ξ5

➡ spectrum not symmetric, but this is not important[de Forcrand, Kurkela and Panero]• Symmetries allow mixing with rotationally non-invariant kinetic terms, e.g.
([p1 γ1 + p2 γ2 ] ⊗ ξ12 ) + ([p3 γ3 + p4 γ4 ] ⊗ ξ34 )• All such terms reduce to standard kinetic term when projected onto 1-flavor subspace• Holds for both “12+34” and Hoelbling mass terms (even though latter has smaller
symmetry group so more intermediate terms arise)• Gluonic counterms will be needed as for Nf=2 branches• Will hold also for overlap versions, since inherit symmetries and subspace
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201234

Outlook for new fermions• Adams N =2 theory passes some basic testsf• Flavor-symmetry breaking enters beyond NNLO in ChPT

N =1 and N =2 Hoelbling-like fermions• requiregluonic tuning due to breaking offfrotation symmetry• Counterbalances attractive features (e.g. no mass tuning for N =2)f

use in practice, need to understand• To
complications of constructing operators (e.g.
for weak matrix elements)?

cases where have mixing with lower-dimension operators,• Inreduced
symmetry group may lead to problems• Is the computational gain sufficient?
S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201235

Outlineunrooted staggered fermions to simulate 2+2,
• Using
2+1+1 and 1+1+1+1 flavors: Is it practical?
• Staggered-Wilson fermions---Adams version• Staggered-Wilson fermions---Hoelbling-like versions
low energy coefficients in ChPT using• Constraining
Weingarten mass inequalities (flash talk?)

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201236

Partially Quenched Wilson ChPT

• SU(2) ×SU(2) → SU(2+N |N ) ×SU(2+N |N )• Construct L including a effects
L

R

V

2

χ

V L

V

V R

[SS & Singleton; Bar, Rupak & Shoresh;
Aoki]

2
f2
f
L0 =
�∂µ Σ∂µ Σ† � −2B0 �M † Σ + Σ† M �4
4− â2 W6� �Σ + Σ† �2 − â2 W7� �(Σ − Σ† )2 � − â2 W8� �Σ2 + (Σ† )2 �Σ ∈ SU (2 + NV |NV )Supertrace•
one constrain the signs of the low-energy coefficients• Can(LECs)?
Phase structure (Aoki vs. first-order) determined by
c2 = −8â2 (2W6� + W8� )S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201237

Can signs of LECs be predicted?

• General issue in effective field theories• Sometimes can use causality•
argument from ε-regime study in• Hermiticity
WChPT implies W ’<0[Pham & Truong, A. Adams et al.]

Doesn’t apply here

8

[Akemann, Damgaard, Splittorff & Verbaarschot]• Important question: Is this argument correct?

recent method is to use QCD mass
• Anotherinequalities to constrain LECs
we derived W ’<0, by• Incalculating a PQ pion massin ChPT and comparing[Bar, Golterman & Shamir][Hansen & SS, arxiv:1111.2404]8

to constraint from Weingarten-like mass inequalities

S. Sharpe, “Comments on new fermions” 2/17/12 @ Kyoto workshop “New types of fermions on the lattice”
Friday, February 17, 201238