Staggered Heavy Baryon Chiral Perturbation Theory
Staggered Heavy Baryon Chiral Perturbation Theory
Jon A. Bailey∗
Washington University in St. Louis
(Dated: August 7, 2021)
Abstract
Although taste violations significantly affect the results of staggered calculations of pseudoscalar
and heavy-light mesonic quantities, those entering staggered calculations of baryonic quantities
have not been quantified. Here I develop staggered chiral perturbation theory in the light-quark
baryon sector by mapping the Symanzik action into heavy baryon chiral perturbation theory. For
2+1 dynamical quark flavors, the masses of flavor-symmetric nucleons are calculated to third order
in partially quenched and fully dynamical staggered chiral perturbation theory. To this order the
expansion includes the leading chiral logarithms, which come from loops with virtual decuplet-
like states, as well as terms of O(m3π), which come from loops with virtual octet-like states. Taste
violations enter through the meson propagators in loops and tree-level terms of O(a2). The pattern
of taste symmetry breaking and the resulting degeneracies and mixings are discussed in detail. The
resulting chiral forms are appropriate to lattice results obtained with operators already in use and
could be used to study the restoration of taste symmetry in the continuum limit. I assume that
the fourth root of the fermion determinant can be incorporated in staggered chiral perturbation
theory using the replica method.
∗jabailey@wustl.edu
http://arxiv.org/abs/0704.1490v2
mailto:jabailey@wustl.edu
I. INTRODUCTION
The staggered formulation [1] of lattice QCD incorporates the rooting conjecture of [2]
to eliminate remnant sea quark doublers. The past year has seen good progress in our
understanding of this conjecture [3, 4] and the corresponding replica or quark flow method
[5, 6, 7, 8] of staggered chiral perturbation theory [9]. However, the results of [9] have yet
to be extended to the baryon sector, and many of the assumptions of [4, 9] have not yet
been tested. Lacking proof of these assumptions, successful calculations of experimentally
well-known hadronic quantities serve as essential tests of the rooting conjecture and the
replica method.
The advantages [10] of staggered QCD have allowed successful calculations of many
hadron masses, decay constants, form factors, and other quantities of phenomenological
importance [11, 12, 13, 14, 15, 16]. However, to date this success has been most pronounced
in the pseudoscalar and heavy-light meson sectors, where the development of staggered chi-
ral perturbation theory (SχPT) [7, 8, 17, 18] facilitated calculations of quark masses, meson
masses, and meson decay constants [13, 14, 16, 19]. Progress calculating baryonic quantities
has been impeded by comparatively large statistical errors and two sources of systematic
errors: questions of interpretation specific to the staggered valence sector, and a lack of
control over the chiral and continuum extrapolations.
In contemporary studies employing staggered fermions, three physical quark flavors are
explicitly included in the action [15]; the four pseudoflavors per physical quark flavor are
called “tastes.” The presence of taste quantum numbers implies that the valence sector of
staggered QCD contains baryons that have no counterpart in nature. However, assuming
taste symmetry is restored in the continuum limit, the staggered baryons become degenerate
within irreps of the taste symmetry group, SU(4)T , and baryons composed of valence quarks
of the same taste are seen to correspond to physical states [20]. At nonzero lattice spacing,
taste violations lift the SU(4)T degeneracies and mix baryons with different SU(4)T quantum
numbers. The resulting spectrum contains many sets of nearly degenerate, mixed states.
Since states with the same conserved (lattice) quantum numbers mix, interpolating fields
typically overlap multiple nearly degenerate, mixed members of these sets.
This situation complicates the task of extracting the masses of physical baryons. To
date, staggered calculations of the masses of light-quark baryons have been performed by
first taking the continuum limit of lattice data and then using continuum baryon chiral
perturbation theory to take the chiral limit [15]. In principle this approach is perfectly
rigorous, but using it to quantify the effects of taste violations, which are O(α2sa2), is difficult
and has not been done. However, using SχPT to study the splittings and mixings due to
taste violations is comparatively straightforward.
Reference [20] considered the questions of interpretation specific to the mass spectrum
of light-quark staggered baryons. It was found that, in calculations of the masses of the
nucleon, the lightest decuplet, and certain excited states, one can avoid the complications
due to the splittings and mixings introduced by taste violations provided one appropriately
chooses the interpolating operators and quark masses. The work reported here enhances our
control over the chiral and continuum limits by developing SχPT in the baryon sector. As an
example, the masses of certain staggered nucleons are calculated in staggered heavy baryon
chiral perturbation theory (SHBχPT) through leading chiral logarithms; the resulting chiral
forms could be used to study the restoration of taste symmetry in the continuum limit and
to improve calculations of the nucleon mass. Calculations of the chiral forms for the masses
of other states highlighted in [20] are in progress and will be reported elsewhere [21].
In SHBχPT, the calculation of baryon masses through leading chiral logarithms requires
the inclusion of leading loops, which enter at third order in the staggered chiral expansion.
After discussing the power counting in Sec. II, I write down the Symanzik action and use
its symmetries to map the quark-gluon operators into the corresponding interactions of
SHBχPT. As anticipated in [17], the O(a2) Lagrangian breaks the continuum spin-taste
symmetry down to the hypercubic group of the lattice. However, in the rest frame of the
heavy baryons, a new remnant spin-taste symmetry eliminates O(a2) mixing between spin-1
and spin-3
baryons. For calculations to third order, this symmetry allows one to consider
the baryon mass matrix in a smaller spin-flavor-taste subspace.
The nucleon operators currently in use interpolate to baryons that are completely sym-
metric in flavor [15]. In Sec. IIIA I recall the symmetry considerations used in [20, 22]
to identify staggered baryons that are degenerate with the nucleon in the continuum limit
and to identify the irreducible representations (irreps) of interpolating fields for the flavor-
symmetric nucleons. Sec. III B contains a discussion of the symmetries of the various terms
in the staggered chiral expansion of the masses of the flavor-symmetric nucleons and the im-
plied form of the mass matrix. Sec. IIIC contains the staggered chiral forms for the masses
of the flavor-symmetric nucleons in partially quenched and fully dynamical SHBχPT. In
Sec. IIID, I show that the pattern of symmetry breaking furnishes the connection between
the staggered chiral forms and the irreps of the corresponding interpolating fields; the rea-
soning is the same as that used to establish the connection between continuum states and
lattice irreps [20, 23]. Sec. IV summarizes and discusses future work.
II. THE STAGGERED HEAVY BARYON LAGRANGIAN
Counting a2 as O(mq), where mq is any of the light quark masses, the staggered chiral
power counting is a straightforward generalization of the power counting of the continuum
heavy baryon chiral perturbation theory [24, pp. 360-362, 476-477]. The staggered heavy
baryon Lagrangian is expanded in increasing powers of the quark masses, the squared lattice
spacing, and derivatives of the heavy baryons and pseudoscalar mesons. The baryon masses
are calculated in increasing powers of the square roots of the quark masses, the residual
off-shell momentum of the heavy baryon, the momenta of the pseudoscalar mesons, the
squared lattice spacing, and the average mass splitting between the lightest spin-1
spin-3
baryons. For convenience
O(ε) ≡ O(m1/2q ) = O(k) = O(a) = O(∆),
where k is the residual heavy baryon momentum or pseudoscalar meson momentum and ∆
is the octet-decuplet mass splitting in the continuum and chiral limits.
In the continuum theory, the leading order (LO) heavy baryon Lagrangian is of O(ε),
while the LO meson Lagrangian is of O(ε2); the next-to-leading order (NLO) heavy baryon
Lagrangian is of O(ε2), while the NLO meson Lagrangian is of O(ε4), and so on. First order
contributions to the baryon self-energies do not arise, while tree-level contributions of O(ε3)
vanish. One-loop diagrams first enter the expansion at O(ε3) and can only include vertices
from the LO Lagrangian; two-loop diagrams first enter at O(ε5). Tree-level contributions
at O(ε3) vanish. Contributions at O(ε2) can contain at most one vertex from the NLO
heavy baryon Lagrangian, together with vertices from the LO heavy baryon and meson
Lagrangians; in fact, no diagrams with vertices from the LO Lagrangians arise.
Nontrivially, the situation in SHBχPT is the same. The staggered symmetries forbid
corrections to the Symanzik action ofO(a) and O(a3) [25, 26], so the power counting remains
the same through NLO. Calculating the baryon masses to O(ε3) requires at most the LO
heavy baryon and meson Lagrangians for one-loop diagrams, which are O(ε3), and the NLO
heavy baryon Lagrangian for O(ε2) tree-level diagrams. Tree-level contributions to the
baryon masses do not arise at O(ε) and O(ε3). To derive the LO heavy baryon and meson
Lagrangians and the NLO heavy baryon Lagrangian needed for O(ε2) tree-level corrections,
we require the Symanzik action through O(a2).
In partially quenched staggered QCD [7, 17], the Symanzik action to O(a2) is
Seff = S4 + a
d4x(L4 + a2L6),
where L4 = 12Tr(FµνFµν) +Q( /D +m)Q
and L6 = Lglue6 + Lbilin6 + L
FF (A)
6 + L
FF (B)
Lglue6 contains terms constructed from gluon fields only, Lbilin6 contains quark bilinears, and
LFF (A,B)6 contain four-fermion operators constructed from products of quark bilinears. The
terms in LFF (A)6 respect the remnant taste symmetry Γ4 ⋊ SO(4)T [17], while the terms in
LFF (B)6 break the continuum rotation-taste symmetry down to the hypercubic group of the
staggered lattice.
Because the staggered formulation introduces four quark tastes for each physical quark
flavor, the chiral symmetry of L4 is SU(24|12)L × SU(24|12)R. There are twelve valence
quarks, twelve sea quarks, and twelve ghost quarks; Q is a 36-element column vector in
flavor-taste space. The quark mass matrix is
m = diag(mx, my, mz;mu, md, ms;mx, my, mz)⊗ I4,
where mx,y,z are the masses of the valence quarks, mu,d,s, the masses of the sea quarks, and
I4 is the identity in taste space. The masses of the ghost quarks are set equal to the masses
of the valence quarks so that valence quarks do not contribute to the fermion determinant.
In the fully dynamical limit, one can drop the graded formalism and consider a Symanzik
theory with the chiral symmetry SU(12)L × SU(12)R. For the fully dynamical case with
N physical flavors, the symmetries of the various terms in S4 and S6 are given in detail in
the Appendix of [7]. The fourth root of the fermion determinant is implemented in SχPT
calculations by the method of quark flows [7, 27] or, equivalently, the replica method [5, 9].
In the continuum limit, the resulting chiral forms are equivalent to the forms that result by
considering a partially quenched theory with only three flavors of sea quarks.
The development of SχPT in the baryon sector proceeds as in the meson sector [7, 8].
In addition to the pseudoscalar mesons, the Lagrangian now includes the lightest spin-1
and spin-3
baryons. In the fully dynamical continuum chiral theory [28], the lightest spin-1
baryons transform in the adjoint of SU(3)F , so they can be collected in a 3 × 3 matrix in
flavor space:
Σ0 + 1√
Λ Σ+ p
Σ− − 1√
Σ0 + 1√
Ξ− Ξ0 − 2√
In the quenched and partially quenched continuum chiral theories [27, 29], the lightest spin-
baryons transform within mixed-symmetry irreps of the appropriate graded symmetry
groups; the presence of more than three light quark flavors implies that the baryons no
longer transform in the adjoint. The above matrix parameterization remains valid only
in restricted subsectors of the flavor space. The generalization to an arbitrary number of
flavors involves embedding the independent fields in a rank-3 mixed-symmetry tensor of
the graded group, Bijk. The embedding respects the defining indicial symmetries of this
tensor but is otherwise arbitrary; any convenient basis of the graded irrep suffices. In the
fully dynamical case (alternatively, restricting the indices to fermionic quarks), the standard
defining relations of B are
Bikj = Bijk and Bijk +Bjki +Bkij = 0. (1)
Under the chiral symmetry group, the spin-1
baryon tensor transforms as
Bijk → U li U
k Blmn,
where U is a local, nonlinear realization of the chiral symmetry defined by the transformation
law of the meson fields [18, 24, 28, 30]
Σ → σ′ = LσU † = UσR†.
The tastes of SχPT are effectively additional quark flavors, so the situation in SχPT is
similar to that in the quenched and partially quenched continuum theories: Even in the
fully dynamical case, one makes use of the rank-3 mixed-symmetry tensor in Eq. (1). If
the indices range from 1 to N , then the number of independent components is 1
N(N2 − 1).
For three quark flavors, N = 12, so the lightest spin-1
baryons transform within a 572M of
SU(12)f , the diagonal (vector) subgroup of SU(12)L × SU(12)R.
The lightest spin-3
baryons are incorporated in SχPT similarly. In the fully dynamical
case, the independent fields are embedded in a completely symmetric rank-3 tensor, Tijk:
Tijk = Tjki = Tkij = Tjik = Tikj = Tkji,
Tijk → U li U
k Tlmn.
The dimension of the symmetric irrep is 1
N(N +1)(N +2), so for three quarks, the lightest
spin-3
baryons transform within a 364S of SU(12)f .
So far the spacetime indices of the heavy baryon fields have been suppressed. B is
a Dirac spinor, while T is a spin-3
Rarita-Schwinger field and therefore carries a spinor
index and a vector index. The anti-particle components of both fields are projected out in
an arbitrary but specific Lorentz frame, and the relativistic chiral Lagrangian is expanded
in increasing powers of 1/mB, where mB is the average octet baryon mass in the chiral
limit [28, 31]. The resulting heavy baryon Lagrangian contains no explicit reference to the
Dirac matrices; they are replaced by the classical baryon four-velocity vµ and the covariant
spin vector Sµ ≡ i2γ5σµνv
ν [24, 31]. For convenience in making contact with the literature on
the continuum theory, I use Minkowski space notation throughout; accordingly, the Lorentz
indices here and below take values from {0, 1, 2, 3}. The Green’s functions of SHBχPT
are defined in Euclidean space; a Wick rotation is implicit throughout.
A. O(a0) Lagrangian
In the continuum chiral theory [29, 31], the heavy baryon Lagrangian to O(ε2) is con-
structed of all Hermitian operators containing one derivative or quark mass matrix that
respect chiral symmetry, Lorentz invariance, parity, and, before non-relativistic reduction,
charge conjugation. The building blocks are the heavy baryon fields, the covariant deriva-
tives Dµ of the heavy baryon fields, the mesonic axial vector current Aµ ≡ i2(σ∂µσ
†−σ†∂µσ),
the mass matrices M± ≡ 12(σ
†mσ† ± σmσ), the four-vectors vµ and Sµ, and the Lorentz-
invariant tensors, gµν and ǫ
µναβ . In the heavy baryon Lagrangian, the vector current
Vµ ≡ 12(σ∂µσ
† + σ†∂µσ) and derivatives of the heavy baryon fields only appear together
in the covariant derivative Dµ; the explicit definition of Dµ is given in [29] but will not be
required here. For calculating the baryon masses to O(ε3), the covariant derivative may be
replaced with the partial derivative ∂µ, M− does not contribute, and M+ may be replaced
with the quark mass matrix, m. The meson fields therefore enter the required operators in
the heavy baryon Lagrangian only through Aµ.
In SχPT, operators mapped from S4 obey all the symmetries of the continuum theory
(after Wick rotation) except for the chiral symmetries, which are simply enlarged to the
appropriate flavor-taste group:
SU(6|3)L × SU(6|3)R → SU(24|12)L × SU(24|12)R
SU(3)L × SU(3)R → SU(12)L × SU(12)R
in the partially quenched or fully dynamical cases, respectively. It follows that the stag-
gered heavy baryon Lagrangian through O(ε2) mapped from L4 has the same form as the
continuum Lagrangian through O(ε2) [29], where the flavor indices in the latter are simply
replaced by flavor-taste indices:
L4 → L(1)φB + L
φB + . . .
L(1)φB ≡ Biv
µDµB + 2αBS
µBAµ + 2βBS
− T ν(ivµDµ −∆)Tν +
AµB +BAµT
+ 2HT
SµAµTν
L(2)φB ≡ 2αMBBM+ + 2βMBM+B + 2σ
MBBstr(M+)
+ 2γMT
M+Tµ − 2σ′MT
Tµstr(M+) + . . .
where the notation is closely based on that of [17, 27, 29]; α, β, αM , βM , C, H , γM , σ
and σ′M are low-energy couplings (LECs), ∆ is the average 572M-364S mass splitting in the
continuum and chiral limits, and “str” denotes the supertrace. The couplings σ′M and σ
M are
normalized so that the supertrace terms reproduce the continuum tree-level contributions to
the baryon masses in the continuum limit; σ′M =
σM and σ
σM . The neglected terms
in L(2)φB are higher order in 1/mB, contain two derivatives, or are O(a2); through O(ε3), only
the terms of O(a2) contribute to the masses of the lightest spin-1
baryons. Of the operators
listed above, those that contribute are
L(1)′φB = Biv
µ∂µB +
BSµB∂µφ+
BSµ(∂µφ)B (2)
− T ν(ivµ∂µ −∆)Tν +
(∂µφ)B +B(∂µφ)T
L(2)′φB,m = 2αMBBm+ 2βMBmB + 2σ
MBBstr(m)
where the definitions of M+, Aµ, and σ = e
iφ/(2f) = 1 + iφ
+ O(φ2) [7] were used; f is the
pion decay constant in the continuum and chiral limits. Terms with explicit factors of φ
contribute vertices for the leading loops, and terms with explicit factors of m contribute
analytically at O(ε2).
B. O(a2) baryon operators for analytic corrections
In addition to the chiral operators in L(1)φB and L
φB mapped from L4, one must find
the chiral operators in L(2)φB corresponding to L6. Here I consider only the subset of these
operators needed to calculate irrep-dependent corrections to the masses of the lightest spin-
baryons through O(ε3); through this order, such operators contribute analytic tree-level
corrections of O(a2). I neglect operators contributing only generic O(a2) corrections to the
LECs in the Lagrangian mapped from L4 and operators that contribute only to the masses
of spin-3
baryons.
Operators containing the quark mass matrix, the axial vector current, and derivatives of
the heavy baryon fields can be ignored because such operators first contribute at O(a2mq) =
O(ε4). Tree-level corrections come only from operators that are bilinear in the heavy baryon
fields and have terms that are free of meson fields. The previous set of building blocks is thus
reduced to the heavy baryon fields, the four-vectors vµ and Sµ, and the Lorentz-invariant
tensors, gµν and ǫ
µναβ . In addition to these building blocks, one may promote the taste
matrices to spurion fields and use the meson field σ to construct objects that transform in the
adjoint under vector (chiral diagonal) flavor-taste transformations. Expanding σ = 1+O(φ)
at the end of the enumeration, one recovers the Lagrangian that contributes at tree-level.
Finally, the fact that the heavy baryon Lagrangian may be obtained as the non-relativistic
limit of the relativistic chiral theory implies that Sµ occurs at most once in each baryon
bilinear, while it suffices for the present purposes to consider operators in which vµ occurs
at most once for each heavy baryon field.
Three cases arise. Chiral operators corresponding to Lglue6 and Lbilin6 respect all the
symmetries of the LO Lagrangian except for spacetime rotation invariance, which is broken
to the hypercubic rotations [7, 17]. Chiral operators corresponding to LFF (A)6 break chiral
flavor-taste to the remnant flavor-taste group; in the fully dynamical case, we have
U(3)l × U(3)r × [Γ4 ⋊ SO(4)T ],
where U(3)l ×U(3)r is the analog of the U(1)vec ×U(1)A of the single-flavor case [7]. These
operators are invariant under all other symmetries of the LO Lagrangian, including spacetime
rotations. Allowing the taste matrices to transform as spurions, these operators also respect
chiral flavor-taste transformations. Finally, chiral operators corresponding to LFF (B)6 break
chiral flavor-taste and spacetime rotations to the symmetry group of the lattice theory [17,
26]. Allowing the taste matrices to transform appropriately, one can work with all the
symmetries except spacetime rotations, which are broken to the hypercubic rotations. We
will consider each of these cases in turn.
1. Lglue6 and Lbilin6
The chiral operators corresponding to Lglue6 and Lbilin6 contribute nothing but a generic
O(a2) correction to the average spin-1
baryon mass in the chiral limit. The Symanzik
operators contain no taste matrices, so there are no spurions in the corresponding chiral
operators. Displaying flavor-taste indices explicitly, the only possibilities that respect parity
are, up to additional factors of vµ,
Bijkv
µvµgµµ
Tµ,ijk + T
µ Bijk)v
µ Tµ,ijkv
ijkgµµ
where the repeated index µ is summed from 0 to 3. Chiral operators containing Sµ do not
arise because it is an axial vector; bilinears containing it are odd under parity. The second
operator is constructed so that it is invariant under charge conjugation. One Lorentz index
is raised in each operator because I have Wick rotated to Minkowski space.
The first term is a velocity-dependent correction to the average spin-1
baryon mass in
the chiral and continuum limits. The underlying lattice theory is not Lorentz invariant, so
there is no prohibition against such corrections. The second term mixes spin-1
and spin-3
baryons; the continuum spin and taste symmetries are broken by L6. The last two terms are
generic corrections to the average difference ∆ between the masses of the spin-1
and spin-3
baryons. Specializing to the rest frame of the heavy baryon, v = (1, 0), and only the first
and last terms survive. The other two vanish because [31]
µ = 0, ∀ v =⇒ T 0 = 0 for v = (1, 0).
For v = 0, the surviving terms respect all the symmetries of the LO heavy baryon La-
grangian, including invariance under spatial rotations. The spacetime symmetry is restored
for this special case. Operators with additional factors of vµ yield nothing new; such opera-
tors either vanish or reduce to the first type.
The fact that spin-1
and spin-3
baryons do not mix in the rest frame considerably simpli-
fies, in this frame, the analysis of Sec. III and calculations of the baryon masses. Moreover,
simulations performed with zero momentum sources must be analyzed in this frame. Ac-
cordingly, the baryon masses given in Sec. IIIC are specific to this case.
2. LFF (A)6
There are twelve types of operators in LFF (A)6 ; following [17, 26], I label these with the
spacetime-taste irreps in which the quark bilinears transform:
LFF (A)6 ∼
[V × S] + [V × P ] + [V × T ] + (V → A)
[S × V ] + [S ×A] + (S → P )
+ [T × V ] + [T × A]. (3c)
In these operators, the indices of the spin and taste matrices are contracted separately; e.g.,
[A× T ] ≡
Q(γρ5 ⊗ ξµν)QQ(γ5ρ ⊗ ξνµ)Q.
Here ξµν is shorthand for I9 ⊗ ξµν , where I9 is the identity matrix in flavor space. We recall
the definitions of the taste matrices:
ξµν ≡ iξµξν and ξρ5 ≡ iξρξ5, where ξ5 ≡ ξ1ξ2ξ3ξ4,
and similarly for the spin matrices. In Sec. III B we will specialize to the Weyl basis for the
taste matrices, in which
 0 −iσ
 and ξ4 =
 0 I2
. (4)
For now, the taste basis is arbitrary; different choices correspond to different interpreta-
tions of the underlying staggered degrees of freedom in terms of continuum taste degrees of
freedom.
Promoting the taste matrices to spurion fields and displaying the chiral structures of
these operators explicitly, one can write each of the operators in LFF (A)6 in one of three
distinct forms [17, 26]. The six operators with vector and axial spin structure (3a) can each
be written in the form
OF = ±
QR(γµ ⊗ FR)QR ±QL(γµ ⊗ FL)QL
, (5)
where FL,R are spurions transforming so that OF is invariant under chiral transformations,
Euclidean rotations, parity, and charge conjugation:
SU(N)L × SU(N)R : FL → LFLL† (6)
FR → RFRR†
SO(4)E : FL → FL
FR → FR
P : FL ↔ FR
C : FL ↔ F TR .
The four operators with scalar and pseudoscalar spin structure (3b) can be written in the
O′F =
QL(I4 ⊗ F̃L)QR ±QR(I4 ⊗ F̃R)QL
, (7)
where the spurions F̃L,R transform as follows:
SU(N)L × SU(N)R : F̃L → LF̃LR† (8)
F̃R → RF̃RL†
SO(4)E : F̃L → F̃L
F̃R → F̃R
P : F̃L ↔ F̃R
C : F̃L → F̃ TL
F̃R → F̃ TR .
The two tensor operators (3c) can be written
O′′F =
QL(γµν ⊗ F̃L)QR
QR(γνµ ⊗ F̃R)QL
, (9)
where the spurions are the same as in O′F .
For convenience in mapping to the heavy baryon theory, I use the meson fields σ to
construct objects with the spurions that transform in the adjoint of the vector subgroup of
the chiral group. Let
l ≡ σ†FLσ and l̃ ≡ σ†F̃Lσ†
r ≡ σFRσ† r̃ ≡ σF̃Rσ.
SU(N)L × SU(N)R : l → UlU † l̃ → Ul̃U † (10)
r → UrU † r̃ → Ur̃U †
SO(4)E : l → l l̃ → l̃
r → r r̃ → r̃
P : l ↔ r l̃ ↔ r̃
C : l ↔ rT l̃ → l̃T
r̃ → r̃T .
The spurion-meson objects l, r, l̃, and r̃ can be further combined to construct definite-parity
objects that are Hermitian:
l ⊗ l ± r ⊗ r and l ⊗ r ± r ⊗ l (11)
l̃ ⊗ l̃ + r̃ ⊗ r̃ l̃ ⊗ r̃ + r̃ ⊗ l̃
−i(l̃ ⊗ l̃ − r̃ ⊗ r̃) −i(l̃ ⊗ r̃ − r̃ ⊗ l̃)
The objects involving l̃ and r̃ are effectively Hermitian because one sets the left- and right-
handed spurion fields equal to one another at the end of the enumeration, which implies
l̃† = r̃. This relation also implies that the above list is complete. Moreover, in constructing
the chiral operators, the only axial vector that is allowed is Sµ, which can occur once.
However, LFF (A)6 is Lorentz invariant, and v·S = 0 is the only Lorentz invariant containing Sµ
that can be constructed. Therefore, Sµ does not appear in the chiral operators corresponding
to LFF (A)6 , and only the parity-even spurion-meson combinations in (11) are allowed. Lorentz
invariance also implies that operators mixing the spin-3
fields Tµ and the spin-
fields do
not appear; to be Lorentz invariant, such operators at this order must contain either v · T
or S · T , but both vanish.
A priori, chirally invariant baryon bilinears may contain zero, one, two, or three pairs of
contractions between the flavor-taste indices of the baryon fields and the flavor-taste indices
of the meson fields. Explicitly,
WBijk B
Xni Bnjk B
ij Bnqk B
ijkBnql, (12)
where W , X , Y , and Z are constructed out of the non-baryonic building blocks and are
quadratic in the spurions. However, it turns out that only operators with two pairs of con-
tractions are relevant here. The W - and X-type operators contribute only irrep-independent
corrections to the masses, while Z-type operators only modify the LECs of the twice-
contracted (Y -type) operators.
First consider the W -type operators. Since there are no contractions between the flavor-
taste indices of the meson fields inW and the baryon fields, W can be any operator appearing
in the remnant-taste symmetric potential V of the meson sector [7, 17]. Setting Σ = 1+O(φ)
in these terms, V collapses to a single constant. Therefore, W -type operators contribute an
O(a2), irrep-independent (generic) correction to the masses of the baryons.
The X- and Z-type operators reduce similarly. For example, consider
Xni ≡ (lr + rl) ni and Z
ijk ≡ (l
j + r
j )(Σ + Σ
†) lk .
The taste matrices all square to the identity, so setting the spurions to their final values and
σ = 1 gives Xni ∝ δni , and the X-type operators reduce to W -type operators. As for the
Z-type operators, the fact that all operators must be quadratic in the spurions means that
only Σ can be used to provide a third set of indices in Z. But Σ is set to unity at the end
of the enumeration, so these operators reduce to the twice-contracted form. Therefore, all
chiral operators leading to distinct spin-taste breaking (irrep-dependent) corrections to the
baryon masses have two pairs of contractions between the flavor-taste indices on the baryon
fields and those on the meson fields.
Assuming the indicial symmetries in Eqs. (1), there are only two linearly independent,
Y -type contraction structures:
ij Bnqk and B
ij Bknq. (13)
Although the symmetries in Eqs. (1) are not generally valid for the partially quenched case,
for which various minus signs must be inserted, only required are those O(a2) operators
needed for the calculation of tree-level corrections to masses of baryons composed of valence
quarks. In this case, the graded indicial symmetries reduce to those given in Eqs. (1), so
the latter are correct for the operators considered here, and the contraction structures (13)
exhaust the possibilities for the relevant operators.
For operators with vector or axial spin structure (3a, 5), setting
ij = l
j + r
j , l
j + r
inserting the scalar, pseudoscalar, and tensor taste matrices in the spurions, and introducing
constants of proportionality gives
[V × S] and [A× S] → (−(±b1 + b2) + 2(±c1 + c2))B
[V × P ] and [A× P ] → ±b1B
(σ†ξ5σ)
†ξ5σ)
j + (σξ5σ
†) ni (σξ5σ
+ b2B
(σ†ξ5σ)
i (σξ5σ
j + (σξ5σ
†) ni (σ
†ξ5σ)
± c1B
(σ†ξ5σ)
†ξ5σ)
j + (σξ5σ
†) ni (σξ5σ
+ c2B
(σ†ξ5σ)
i (σξ5σ
j + (σξ5σ
†) ni (σ
†ξ5σ)
[V × T ] and [A× T ] → ±b1B
(σ†ξµνσ)
†ξνµσ)
j + (σξµνσ
†) ni (σξνµσ
+ b2B
(σ†ξµνσ)
i (σξνµσ
j + (σξµνσ
†) ni (σ
†ξνµσ)
± c1B
(σ†ξµνσ)
†ξνµσ)
j + (σξµνσ
†) ni (σξνµσ
+ c2B
(σ†ξµνσ)
i (σξνµσ
j + (σξµνσ
†) ni (σ
†ξνµσ)
Here the taste matrix ξτ is being used as shorthand for I3 ⊗ ξτ , where I3 is the identity
matrix in valence quark flavor space.
For operators with scalar or pseudoscalar spin structure (3b, 7), setting
ij = l̃
j + r̃
j , l̃
j + r̃
inserting the vector and axial taste matrices in the spurions, and introducing constants of
proportionality gives
[S × V ] and [P × V ] → e1B
(σ†ξνσ
†) ni (σ
j + (σξνσ)
i (σξνσ)
± e2B
(σ†ξνσ
†) ni (σξνσ)
j + (σξνσ)
+ f1B
(σ†ξνσ
†) ni (σ
j + (σξνσ)
i (σξνσ)
± f2B
(σ†ξνσ
†) ni (σξνσ)
j + (σξνσ)
[S × A] and [P ×A] → e1B
(σ†ξν5σ
†) ni (σ
†ξ5νσ
j + (σξν5σ)
i (σξ5νσ)
± e2B
(σ†ξν5σ
†) ni (σξ5νσ)
j + (σξν5σ)
†ξ5νσ
+ f1B
(σ†ξν5σ
†) ni (σ
†ξ5νσ
j + (σξν5σ)
i (σξ5νσ)
± f2B
(σ†ξν5σ
†) ni (σξ5νσ)
j + (σξν5σ)
†ξ5νσ
Finally, for operators with tensor spin structure (3c, 9), one proceeds as before except
that, noting the form of O′′F (9), one omits operators corresponding to cross-terms of left-
and right-handed spurions. Setting
ij = l̃
j + r̃
inserting vector and axial taste matrices, and introducing constants of proportionality gives
[T × V ] → e′1B
(σ†ξνσ
†) ni (σ
j + (σξνσ)
i (σξνσ)
+ f ′1B
(σ†ξνσ
†) ni (σ
j + (σξνσ)
i (σξνσ)
[T × A] → e′1B
(σ†ξν5σ
†) ni (σ
†ξ5νσ
j + (σξν5σ)
i (σξ5νσ)
+ f ′1B
(σ†ξν5σ
†) ni (σ
†ξ5νσ
j + (σξν5σ)
i (σξ5νσ)
Although these operators have been deduced within the heavy baryon framework, all the
operators listed here do in fact arise when deriving the heavy baryon Lagrangian from the
relativistic baryon chiral Lagrangian. To confirm this assertion, one may map LFF (A)6 to
the relativistic chiral theory in Euclidean space, Wick rotate, and then execute the non-
relativistic reduction. For the operators listed above, this analysis has been carried out.
3. LFF (B)6
The development parallels that for LFF (A)6 . There are four types of operators in
LFF (B)6 [17, 26]:
LFF (B)6 ∼ [Vµ × Tµ] + [Aµ × Tµ] (14a)
+ [Tµ × Vµ] + [Tµ × Aµ]. (14b)
In these operators, the indices of the spin and taste matrices are contracted together; e.g.,
[Vµ × Tµ] ≡
ν 6=µ
Q(γµ ⊗ ξµν)QQ(γµ ⊗ ξνµ)Q−Q(γµ ⊗ ξµν5)QQ(γµ ⊗ ξ5νµ)Q
Up to a taste singlet component that has no influence on the map to the chiral theory [26],
these four operators may each be written in one of two forms. The two operators with vector
and axial spin structure (14a) can be written in the form
OFF (B)F = ±
QR(γµ ⊗ FRµ)QR ±QL(γµ ⊗ FLµ)QL
, (15)
where (FL,R)µ are spurions transforming so that OFF (B)F is invariant under chiral transfor-
mations, Euclidean hypercubic rotations, parity, and charge conjugation. Except for the
Euclidean spacetime index on the spurions, OFF (B)F is the same as OF (5). Therefore, ex-
cepting Euclidean rotations, the spurions transform as in (6); under Euclidean rotations,
the spurions transform in the fundamental representation (rep):
SO(4)E : FLµ → ΛµνFLν
FRµ → ΛµνFRν .
The two operators with tensor spin structure (14b) can be written in the form
O′′FF (B)F =
ν 6=µ
QL(γµν ⊗ F̃Lµ)QRQR(γνµ ⊗ F̃Rµ)QL, (16)
where, once again excepting Euclidean rotations, the spurions (F̃L,R)µ transform as in (8).
Under Euclidean rotations, the spurions again transform as vectors:
SO(4)E : F̃Lµ → ΛµνF̃Lν
F̃Rµ → ΛµνF̃Rν .
For purposes of mapping to the heavy baryon theory, I again use the meson fields σ to
construct spurion-meson objects that transform in the adjoint of the vector subgroup of the
chiral group. Let
lµ ≡ σ†FLµσ and l̃µ ≡ σ†F̃Lµσ†
rµ ≡ σFRµσ† r̃µ ≡ σF̃Rµσ.
Then these objects transform as in (10) but are to be treated as vectors under Euclidean
rotations. The Hermitian, definite-parity objects are constructed as before (cf. (11)). How-
ever, unlike the operators in LFF (A)6 , the operators in L
FF (B)
6 break Lorentz invariance. The
argument given in the paragraph following (11) no longer applies, and the operators can
contain both Sµ and parity-odd spurion-meson objects.
A priori, the chiral operators can have any of the contraction structures given in (12).
However, the previous arguments against X- and Z-type operators go through unchanged,
while W -type operators cannot contribute because they require derivatives to break Lorentz
invariance and are therefore necessarily O(a2k2)=O(ε4) or higher, in accord with the central
result of [17]. Therefore only the two contraction structures given in (13) enter.
For operators with vector or axial spin structure (14a, 15), setting
ij = vµv
µ((lµ)
i (lµ)
j + (rµ)
i (rµ)
j ), vµv
µ((lµ)
i (rµ)
j + (rµ)
i (lµ)
vµSµ((lµ)
i (rµ)
j − (rµ) ni (lµ)
inserting the tensor taste matrices in the spurions, introducing constants of proportionality,
and summing the remaining tensor index (cf. [26]) gives
[Vµ × Tµ] and [Aµ × Tµ] →
ν 6=µ
(σ†ξµνσ)
†ξνµσ)
j + (σξµνσ
†) ni (σξνµσ
ν 6=µ
(σ†ξµνσ)
i (σξνµσ
j + (σξµνσ
†) ni (σ
†ξνµσ)
+ h′2
ν 6=µ
SµBknq
(σ†ξµνσ)
i (σξνµσ
j − (σξµνσ†) ni (σ†ξνµσ)
ν 6=µ
(σ†ξµνσ)
†ξνµσ)
j + (σξµνσ
†) ni (σξνµσ
ν 6=µ
(σ†ξµνσ)
i (σξνµσ
j + (σξµνσ
†) ni (σ
†ξνµσ)
+ p′2
ν 6=µ
SµBnqk
(σ†ξµνσ)
i (σξνµσ
j − (σξµνσ†) ni (σ†ξνµσ)
Two points deserve additional comment. First, the operator multiplying h′2 is not Hermitian.
The culprit is the coincidence of the second contraction structure in (13) with a spurion-
meson object that is anti-symmetric under simultaneous interchange of the upper and lower
flavor-taste indices. This consideration did not arise in mapping LFF (A)6 because parity and
Lorentz invariance conspired to allow only symmetric spurion-meson objects. The remedy
is to find a Hermitian linear combination of the contraction structures given in (13) and
replace the operator multiplying h′2 with its Hermitian counterpart. For anti-symmetric
spurion-meson objects, the sum of the operators multiplying h′2 and p
2 is anti-Hermitian.
Using the cyclic property in (1), one may take
ij Bqkn
as the second linearly independent contraction structure for anti-symmetric spurion-meson
objects. Then the operator multiplying h′2 above is replaced with the Hermitian operator
ν 6=µ
SµBqkn
(σ†ξµνσ)
i (σξνµσ
j − (σξµνσ†) ni (σ†ξνµσ)
Second, at the level of the relativistic baryon chiral Lagrangian, charge conjugation invari-
ance and Hermiticity conspire to forbid operators containing the spurion-meson object
ij = v
µSµ((lµ)
i (lµ)
j − (rµ) ni (rµ)
and such operators have been eliminated from the above list (h′1 = p
1 ≡ 0). For tree-level
corrections (σ = σ† = 1) or quantities calculated in the rest frame (v = 0), this detail does
not affect the results.
For operators with tensor spin structure (14b, 16), one proceeds as before except that,
noting the form of O′′FF (B)F (16), one omits operators corresponding to direct-terms of left-
and right-handed spurions. Setting
ij = vµv
µ((l̃µ)
i (r̃µ)
j + (r̃µ)
i (l̃µ)
j ), −ivµSµ((l̃µ) ni (r̃µ)
j − (r̃µ) ni (l̃µ)
inserting vector and axial taste matrices, and introducing constants of proportionality gives
[Tµ × Vµ] → s
(σ†ξµσ
†) ni (σξµσ)
j + (σξµσ)
SµBqkn
(σ†ξµσ
†) ni (σξµσ)
j − (σξµσ) ni (σ†ξµσ†)
(σ†ξµσ
†) ni (σξµσ)
j + (σξµσ)
−ivµBijkSµBnqk
(σ†ξµσ
†) ni (σξµσ)
j − (σξµσ) ni (σ†ξµσ†)
[Tµ ×Aµ] → s
(σ†ξµ5σ
†) ni (σξ5µσ)
j + (σξµ5σ)
†ξ5µσ
SµBqkn
(σ†ξµ5σ
†) ni (σξ5µσ)
j − (σξµ5σ) ni (σ†ξ5µσ†)
(σ†ξµ5σ
†) ni (σξ5µσ)
j + (σξµ5σ)
†ξ5µσ
−ivµBijkSµBnqk
(σ†ξµ5σ
†) ni (σξ5µσ)
j − (σξµ5σ) ni (σ†ξ5µσ†)
where the procedure followed for the h′2 operator above has already been performed; opera-
tors multiplying s′ have been replaced with their Hermitian counterparts.
As before, all the operators given here do arise when deriving the heavy baryon Lagrangian
from the relativistic baryon chiral Lagrangian; I have mapped LFF (B)6 to the relativistic chiral
theory in Euclidean space, performed the Wick rotation, and performed the non-relativistic
reduction. In addition to the operators given here, there are operators that mix spin-1
spin-3
fields and operators that contain more powers of the baryon four-velocity. In the rest
frame, the former vanish, while the latter contribute nothing new. I therefore ignore such
operators.
C. Staggered heavy baryon Lagrangian for O(ε3) baryon octet masses
In the rest frame of the heavy baryon, v = 0, so the only non-vanishing component of
the four-velocity is v0 = 1. But v = 0 implies that S0 = 0, and it follows from the form of
the enumerated operators that O(a2) operators containing Sµ do not contribute to the rest
frame baryon masses. Setting σ = σ† = 1 in the remaining operators and introducing an
LEC for each taste-violating term in the Lagrangian that is distinct at tree level,
L(2)A′
φB,a2
= A1B
Bknq(ξ5)
i (ξ5)
j + A2B
Bnqk(ξ5)
i (ξ5)
j (17)
+ A3B
(ξµν)
i (ξµν)
j + A4B
(ξµν)
i (ξµν)
+ A5B
Bknq(ξν)
i (ξν)
j + A6B
Bnqk(ξν)
i (ξν)
+ A7B
Bknq(ξν5)
i (ξν5)
j + A8B
Bnqk(ξν5)
i (ξν5)
L(2)B′
φB,a2
= B1B
ν 6=4
(ξ4ν)
i (ξ4ν)
j +B2B
ν 6=4
(ξ4ν)
i (ξ4ν)
j (18)
+ B3B
Bknq(ξ4)
i (ξ4)
j +B4B
Bnqk(ξ4)
i (ξ4)
+ B5B
Bknq(ξ45)
i (ξ45)
j +B6B
Bnqk(ξ45)
i (ξ45)
In the fully dynamical case, the flavor-taste indices ijknq take the values 1 to 12. Recalling
that ξτ for τ ∈ {I, µ, µν(µ < ν), µ5, 5} is shorthand for I3⊗ξτ , we see that the O(a2) heavy
baryon Lagrangian is invariant under arbitrary flavor transformations, SU(3)F ⊂ SU(12)f .
However, the taste matrices in L(2)A′
φB,a2
break taste down to the remnant taste group of [17],
Γ4 ⋊ SO(4)T ⊂ SU(4)T ,
while in an arbitrary reference frame, the chiral operators arising from LFF (B)6 (Wick rotated
to Euclidean space) break taste to the lattice symmetry group,
Γ4 ⋊ SW4,diag ⊂ [Γ4 ⋊ SO(4)T ]× SO(4)E.
SW4,diag is the hypercubic group in the diagonal of the taste and spacetime SO(4)’s; Γ4 is
the Clifford group generated by shifts by one lattice site [17].
In contrast, the operators appearing in L(2)B′
φB,a2
are manifestly invariant under a larger
group: Under the taste subgroup that corresponds to the spatial rotations, which excludes
(taste) boosts, the taste matrices ξν for ν = 1, 2, 3 transform as components of a three-
vector, and the taste matrices ξ4 and ξ5 are invariant. Moreover, in the rest frame we
Term in Lχ Case Flavor-taste-spacetime symmetry
L(1)′φB + L
φB,m + L
φ isospin limit, m 6= 0, any v, a SU(8)x,y × SU(4)z × SO(4)E
V any m, v, a U(3)l × U(3)r × [Γ4 ⋊ SO(4)T ]× SO(4)E
L(2)A′
φB,a2
any m, v = 0, any a U(3)l × U(3)r × [Γ4 ⋊ SO(4)T ]× SU(2)E
L(2)B′
φB,a2
any m, v = 0, any a U(3)l × U(3)r × [Γ4 ⋊ SU(2)T ]× SU(2)E
TABLE I: The valence quark symmetries of terms in the Euclidean staggered heavy baryon La-
grangian that are needed for computing the octet baryon masses to O(ε3). The taste symmetry in
the rest frame of the heavy baryons is not completely broken to the lattice symmetry group.
are free to make arbitrary spatial rotations. Recalling the taste subgroup corresponding to
spatial rotations of spinors, SU(2)T ⊂ SO(4)T , one concludes that the O(a2) Lagrangian is
invariant under
U(3)l × U(3)r × [Γ4 ⋊ SU(2)T ]× SU(2)E , (19)
where SU(2)E is the group of spatial rotations. Table I lists the symmetries of the various
terms of the Lagrangian used in the calculation,
Lχ = L(1)′φB + L
φB,m + L
φ − a
2V + a2L(2)A′
φB,a2
+ a2L(2)B′
φB,a2
where L(1)′φ denotes the leading staggered chiral Lagrangian in the meson sector at zero
lattice spacing, and V is the generalized Lee-Sharpe potential [7, 17].
III. THE FLAVOR-SYMMETRIC NUCLEONS
A. Identifying staggered nucleons and irreps of interpolating fields
As shown in [20], the 572M and 364S contain not only baryons that are degenerate in
the continuum limit with the octet and decuplet states of nature, but also baryons that have
unphysical masses; the latter states are degenerate with certain partially quenched octet and
decuplet baryons. To calculate the masses of baryons degenerate with a given member of
the octet or decuplet, one must choose a basis in the flavor-taste space in which degeneracies
with the desired states are evident.
Consider the valence sector of the partially quenched theory. Decomposing the 572M
and 364S of SU(12)f into irreps of the flavor-taste subgroup gives
SU(12)f ⊃ SU(3)F × SU(4)T
572M → (10S, 20M)⊕ (8M, 20S)⊕ (8M, 20M)⊕ (8M, 4̄A)⊕ (1A, 20M) (20a)
364S → (10S, 20S)⊕ (8M, 20M)⊕ (1A, 4̄A) (20b)
Assume that taste is restored in the continuum limit. Taking the valence quark masses equal
so that SU(3)F is exact in the valence sector, SU(12)f is a good symmetry as well, and all
the baryons of the 572M are degenerate, as are all the baryons of the 364S. The symmetry
of the spin-1
baryons is the same as the symmetry of the octet, and the symmetry of the
spin-3
baryons, the same as that of the decuplet. Therefore, setting the masses of all three
valence quarks and the masses of two sea quarks equal to the average up-down quark mass
while setting the mass of the remaining sea quark equal to that of the strange quark mass,
the baryons of the 572M are degenerate with the nucleon (in the limit of exact isospin), and
those of the 364S, with the ∆.
Increasing the mass of the strange valence quark to its physical value does not change
the masses of baryons that do not contain a strange valence quark; such baryons remain de-
generate with the nucleon or the ∆. In particular, the isospin-3
members of the (10S, 20M)
remain degenerate with the nucleon. These states are flavor-symmetric; interpolating fields
for the single-flavor members of the (10S, 20M) were constructed in [23]. In the contin-
uum limit, the SU(8)x,y ×SU(4)z symmetry of the valence sector allows one to rotate these
flavor-symmetric nucleons into those with physical flavor and simple taste structure, i.e.,
baryons that are manifestly degenerate with the nucleon [20].
The interpolating fields for the flavor-symmetric nucleons fall into irreps of the geometrical
time-slice group (GTS) [23]. Decomposing the continuum irrep of the flavor-symmetric
nucleons into irreps of GTS gives [23]
, 20M) → 3(8)⊕ 16,
where a direct product with continuum parity is suppressed on both sides of the decom-
position. At nonzero lattice spacing, discretization effects lift the degeneracy of the GTS
irreps and introduce mixing among corresponding members of the 8’s. Interpolating fields
transforming in the 8 of GTS overlap corresponding members of each of the three 8’s, while
interpolating fields transforming in the 16 overlap corresponding members of the 16.
B. Taste symmetry and the mass matrix
Consider one of the single-flavor, isospin-3
members of the 10S of the (10S, 20M). At
nonzero lattice spacing, taste violations partially lift the degeneracy of the members of the
20M. The baryons are degenerate within irreps of the remnant taste symmetries respected by
the relevant propagators and vertices at a given order in the staggered chiral expansion. To
O(ε3) in the staggered chiral expansion, the masses receive analytic contributions of O(mq)
and O(a2) at tree level, non-analytic contributions of O(m3/2q ) from loops with virtual spin-
baryons, and non-analytic contributions of O(∆mq lnmq) from loops with virtual spin-32
baryons.
The analytic contributions proportional to mq come from L(2)′φB,m; referring to Table I,
we see that they respect SU(8)x,y × SU(4)z × SO(4)E. The vertices and heavy baryon
propagators in the loops come from L(1)′φB , while the meson propagators come from L
φ and
V. Therefore the loops respect SU(2)x,y ×U(1)z × [Γ4⋊SO(4)T ]×SO(4)E; the loops break
taste from SU(4)T to the remnant taste symmetry Γ4 ⋊ SO(4)T . Under Γ4 ⋊ SO(4)T , the
20M of SU(4)T decomposes into one 12-dimensional irrep and two 4-dimensional irreps:
SU(4)T ⊃ Γ4 ⋊ SO(4)T
20M → 12⊕ 2(4).
Finally, analytic contributions proportional to a2 come from L(2)A′
φB,a2
and L(2)B′
φB,a2
. The former
do not break taste further than the loops; the latter break taste to Γ4 ⋊ SU(2)T . Under
Γ4 ⋊ SU(2)T , the 12 of Γ4 ⋊ SO(4)T decomposes further into an 8 and a 4, while the two
4’s of Γ4 ⋊ SO(4)T are irreducible under Γ4 ⋊ SU(2)T :
Γ4 ⋊ SO(4)T ⊃ Γ4 ⋊ SU(2)T
12 → 8⊕ 4
4 → 4.
Taste violations in the loops and in the O(a2) terms from L(2)A′
φB,a2
lift the continuum degener-
acy between baryons in the 12 and those in the two 4’s of Γ4⋊SO(4)T and introduce mixing
between corresponding states in the two 4’s; such states have the same Γ4 ⋊ SO(4)T quan-
tum numbers. Taste violations in the O(a2) terms from L(2)B′
φB,a2
lift the remaining degeneracy
between baryons in the 8 and those in the 4 of Γ4⋊SU(2)T and mix corresponding states in
the three 4’s of Γ4 ⋊ SU(2)T ; in this case, corresponding states have the same Γ4 ⋊ SU(2)T
quantum numbers.
The quantum numbers of these remnant taste symmetries are eigenvalues of maximal
sets of commuting observables (MSCO) corresponding to each symmetry. To construct such
sets, first consider the decomposition of the Γ4⋊SO(4)T irreps under the SO(4)T subgroup:
Γ4 ⋊ SO(4)T ⊃ SO(4)T
(21a)
(21b)
where the isomorphism SO(4) ≃ SU(2) × SU(2)/Z2 has been used to label the SO(4)T
irreps; the conserved observables are simply the spins corresponding to the SU(2)’s. The
decomposition (21b) is explicit in the Weyl representation of the taste matrices, so it is
convenient to work in this representation. In the Weyl representation, the only diagonal
member of Γ4 that commutes with but is not redundant with the spins is ξ5. Therefore, one
MSCO of Γ4 ⋊ SO(4)T is {ξ5, J2L, J2R, JLz, JRz}, where JL and JR denote the spins.
Tree-level corrections of O(a2) from L(2)B′
φB,a2
break Γ4⋊SO(4)T to Γ4⋊SU(2)T by breaking
SO(4)T to SU(2)T , where SU(2)T is the subgroup of SO(4)T obtained by rotating left-
handed and right-handed Weyl spinors together; decomposing the SO(4)T irreps under
SU(2)T is an elementary exercise in addition of angular momentum:
SO(4)T ⊃ SU(2)T
(22a)
(22b)
The sum of left- and right-handed spins is respected by Γ4⋊SU(2)T , and the corresponding
MSCO is {ξ5, (JL + JR)2, JLz + JRz}. Table II lists the MSCO’s of the remnant taste
symmetries and notation for the corresponding quantum numbers.
It is not difficult to see that Γ4 contains a taste parity operation: ξ4 interchanges the
left- and right-handed spin quantum numbers in the product irreps of (21); moreover, under
Remnant taste symmetry MSCO Quantum numbers
Γ4 ⋊ SO(4)T ξ5, J
R, JLz, JRz ξ5, jL, jR, mL, mR
Γ4 ⋊ SU(2)T ξ5, (JL + JR)
2, JLz + JRz ξ5, j, m
TABLE II: Maximal sets of commuting observables (MSCO’s) for each remnant taste symmetry.
The quantum numbers of each MSCO are used to distinguish generically nonzero mixing elements
in the mass matrix from off-diagonal elements that must vanish.
SU(2)T in Γ4 ⋊ SU(2)T ,
Γ4 ⋊ SU(2)T ⊃ SU(2)T
8 → 3
(23a)
4 → 1
(23b)
where taste parity interchanges the two SU(2)T irreps in each of the decompositions (23).
Taste parity allows one to identify irreps of the remnant taste symmetries from the spin
irreps appearing in the decompositions (21) and (23).
The explicit form of the mass matrix depends on the basis chosen for the 20M. Taking
tensor products of fundamental representations of SU(4)T , projecting onto the 20M, and
demanding that the states also be eigenvectors of the MSCO of Γ4 ⋊ SO(4)T leads to
|Naab〉 ≡ 1√6 (|aab〉 + |aba〉 − 2|baa〉) , a 6= b, a, b ∈ {1, 2, 3, 4} (24a)
|Σabc〉 ≡ 1√12 (|abc〉 + |bac〉 − 2|cab〉+ |acb〉+ |bca〉 − 2|cba〉) (24b)
|Λabc〉 ≡ 12 (|abc〉 + |acb〉 − |bac〉 − |bca〉) (24c)
where in (24b) and (24c), abc ∈ {123, 124, 341, 342}. The names for the states are motivated
by analogy with the states of the mixed irrep of SU(3)F ; of the 20 vectors spanning the 20M,
12 have a taste structure analogous to the flavor structure of the nucleon, 4 have a taste
structure analogous to the flavor structure of the neutral Σ, and 4, taste structure analogous
to the flavor structure of the Λ.
The convenience of this basis is evident when one considers taste violations from loops
and tree-level O(a2) contributions respecting Γ4 ⋊ SO(4)T . Although they are both linear
combinations of states with the same taste labels, Σabc and Λabc do not mix because they are
State of 20M ξ5
jL, jR
mL mR
N112 +1
N221 +1
N334 −1
N443 −1
Λ341 +1
Λ342 +1
Λ123 −1
Λ124 −1
N331 +1
+1 +1
N332 +1
+1 −1
N441 +1
−1 +1
N442 +1
−1 −1
Σ341 +1
Σ342 +1
N113 −1
N114 −1
N223 −1
N224 −1
Σ123 −1
Σ124 −1
TABLE III: Quantum numbers of Γ4⋊SO(4)T for each of the states in the 20M. States having the
same set of Γ4⋊SO(4)T eigenvalues are mixed by taste violations. States with distinct eigenvalues
are not mixed by interactions that respect Γ4 ⋊ SO(4)T .
eigenstates of J2L and J
R with different eigenvalues; the Σ’s are symmetric under interchange
of the first two indices, and so are triplets, jL(R) = 1, while the Λ’s are antisymmetric under
interchange of the first two indices, and so are singlets, jL(R) = 0. The Γ4 ⋊ SO(4)T
quantum numbers of each of the states in the basis (24) are listed in Table III. As a set, the
Γ4⋊SO(4)T quantum numbers of members of the 12 differ from the Γ4⋊SO(4)T quantum
numbers of the two 4’s; interactions respecting Γ4 ⋊ SO(4)T do not mix members of the 12
with members of the two 4’s. In the same way, the Γ4 ⋊SO(4)T quantum numbers serve to
completely distinguish members within each of the Γ4 ⋊ SO(4)T irreps; within each irrep,
the mass submatrix of contributions that respect Γ4⋊SO(4)T is diagonal. This observation
is consistent with the fact that members of a given irrep are not mixed by interactions
respecting the corresponding symmetry group. However, each member of each 4 has the
same quantum numbers as one member of the other 4. The taste violations mix N112 with
Λ341, N221 with Λ342, and so on. An 8-dimensional submatrix corresponding to the two 4’s
contains generically nonzero off-diagonal elements due to contributions from the loops and
tree-level O(a2) corrections that respect Γ4 ⋊ SO(4)T . Note that states having the same
Γ4 ⋊ SO(4)T quantum numbers are nonetheless distinct states; they have different SU(4)T
quantum numbers.
The form of the contributions to the mass matrix that respect Γ4 ⋊ SO(4)T follows from
the symmetry of the quark flows and the Γ4 ⋊ SO(4)T symmetry; in the basis given in
Table III, the 8-dimensional submatrix corresponding to the two 4’s (for any given member
of the 10S) has the form

c1 0 0 0 c3 0 0 0
0 c1 0 0 0 −c3 0 0
0 0 c1 0 0 0 c3 0
0 0 0 c1 0 0 0 −c3
c3 0 0 0 c2 0 0 0
0 −c3 0 0 0 c2 0 0
0 0 c3 0 0 0 c2 0
0 0 0 −c3 0 0 0 c2

To O(ε3) in the staggered chiral expansion, the contributions parameterized here are given
in Sec. IIIC. The results have been checked to verify that they conform to the pattern of
degeneracies and mixings given here. The minus signs in the off-diagonal elements arise
because some of the states in Table III are unconventionally normalized with respect to the
ladder operators. Specifically,
JR−|N112〉 = −|N221〉
JL−|N334〉 = −|N443〉
while
JR−|Λ341〉 = |Λ342〉
JL−|Λ123〉 = |Λ124〉.
In the continuum limit, the restoration of SU(4)T implies that c3 must vanish and c1 must
equal c2. The tree-level taste violations vanish trivially; a straightforward exercise with the
loop contributions completes the consistency check (cf. Appendix A).
Tree-level O(a2) contributions from L(2)B′
φB,a2
may be parameterized in much the same way.
The taste violations in such contributions split the baryons in the Γ4 ⋊ SO(4)T 12 between
the 8 and a 4 of Γ4 ⋊ SU(2)T and introduce mixing between corresponding states in the
resulting three 4’s of Γ4 ⋊ SU(2)T . The two 4’s of Table III are also irreducible under
Γ4 ⋊ SU(2)T ; however, because the MSCO of Γ4 ⋊ SU(2)T includes the total angular mo-
mentum (JL + JR)
2, the components JLz and JRz are no longer separately conserved, and
contributions from L(2)B′
φB,a2
are not diagonal in the basis given in Table III for the 12. Con-
structing eigenstates of the new MSCO gives the bases for the 8 and 4 of Γ4 ⋊ SU(2)T
shown in Table IV. These states are related to those in Table III by the Clebsch-Gordan
coefficients for adding spin 1 and spin 1
From the eigenvalues listed in Table IV, we see that the submatrix containing generically
nonzero off-diagonal elements is 12-dimensional and corresponds to mixing among corre-
sponding members of the three 4’s of Γ4⋊SU(2)T . In the basis of Table IV, the form of the
mixing matrix implied by the symmetry of the quark flows and Γ4 ⋊ SU(2)T is

c4 0 0 0 c7 0 0 0 c8 0 0 0
0 c4 0 0 0 −c7 0 0 0 −c8 0 0
0 0 c4 0 0 0 c7 0 0 0 c8 0
0 0 0 c4 0 0 0 −c7 0 0 0 −c8
c7 0 0 0 c5 0 0 0 c9 0 0 0
0 −c7 0 0 0 c5 0 0 0 c9 0 0
0 0 c7 0 0 0 c5 0 0 0 c9 0
0 0 0 −c7 0 0 0 c5 0 0 0 c9
c8 0 0 0 c9 0 0 0 c6 0 0 0
0 −c8 0 0 0 c9 0 0 0 c6 0 0
0 0 c8 0 0 0 c9 0 0 0 c6 0
0 0 0 −c8 0 0 0 c9 0 0 0 c6

The contributions parameterized here are presented in Sec. IIIC; a straightforward exercise
shows that they are consistent with this parameterization (cf. A). The minus signs again
arise from the phase convention for |N221〉 and |N443〉.
C. Masses of the flavor-symmetric nucleons
We now consider the staggered chiral forms for the masses of the flavor-symmetric nu-
cleons having degenerate valence quarks of mass mx. The forms are given through O(ε3)
in fully dynamical and partially quenched SHBχPT. Let the tree-level contributions pro-
portional to mq be denoted by Tree(mq), the tree-level contributions proportional to a
by Tree(a2), the loops with virtual spin-1
baryons, by Loop(1
), and the loops with vir-
tual spin-3
baryons, by Loop(3
). These contributions are of O(mq), O(a2), O(m3/2q ), and
O(∆mq lnmq), respectively. The mass matrix is given by
M = M0 + Σ(0), (27)
Σ(0) = Tree(mq) + Tree(a
2) + Loop(1
) + Loop(3
) (28)
where M0 is the average mass of the 572M in the continuum and chiral limits, and Σ(0) is
the heavy baryon self-energy evaluated at v · r = 0; here r is the residual 4-momentum of
the heavy baryon.
State of 20M ξ5 j m
N112 +1
N221 +1
N334 −1 12 +
N443 −1 12 −
Λ341 +1
Λ342 +1
Λ123 −1 12 +
Λ124 −1 12 −
N332 − 1√
Σ341 +1
N441 +
Σ342 +1
N114 − 1√
Σ123 −1 12 +
N223 +
Σ124 −1 12 −
N331 +1
N332 +
Σ341 +1
N441 +
Σ342 +1
N442 +1
N113 −1 32 +
N114 +
Σ123 −1 32 +
N223 +
Σ124 −1 32 −
N224 −1 32 −
TABLE IV: Eigenvalues of the MSCO of Γ4 ⋊ SU(2)T for each of the eigenstates in the 20M.
States with the same eigenvalues are mixed by contributions respecting Γ4 ⋊ SU(2)T .
In accord with the discussion of Sec. III B, M0 and Tree(mq) are diagonal matrices in the
baryon-taste subspace corresponding to the 20M of SU(4)T ; explicitly,
M0 + Tree(mq) = [M0 − 2(αM + βM)mx − 2σM(mu +md +ms)] I20,
where I20 is the identity matrix. Loop(
), Loop(3
), and the contributions of L(2)A′
φB,a2
Tree(a2) are diagonal in the 12 of Γ4 ⋊ SO(4)T but have the form given in (25) in the
subspace corresponding to the two 4’s of Γ4 ⋊ SO(4)T . The contributions of L(2)B′φB,a2 to
Tree(a2) are diagonal in the 8 of Γ4 ⋊ SU(2)T but have the form given in (26) in the three
4’s of Γ4 ⋊ SU(2)T .
For convenience in writing the results, let
Loop(1
σconn1/2 + σ
Loop(3
(2π)2
σconn3/2 + σ
Tree(a2) = −a2
σFF (A) + σFF (B)
where σconn is the sum of loops with connected meson propagators, σdisc is the sum of
loops with disconnected (hairpin) propagators, and σFF (A,B) are tree-level contributions
from L(2)A,B′
φB,a2
. At tree-level, the pseudoscalar meson masses are degenerate within irreps of
Γ4 ⋊ SO(4)T , so we label them with an index t ∈ {I, V, T, A, P} denoting the SO(4)T
irrep:
(mtij)
2 = λ(mi +mj) + a
where i, j ∈ {u, d, s, x, y, z}, λ is proportional to the chiral condensate, and ∆t is the
meson taste-splitting [7]. Let τ ∈ {I, µ, µν(µ < ν), µ5, 5} be the meson taste index, and
st, the set of meson tastes in irrep t: sI = {I}, sV = {µ}, sT = {µν(µ < ν)}, sA = {µ5},
and sP = {5}. Finally, let nt denote the number of meson tastes corresponding to irrep t.
1, and
f(τ) =
for any function f(τ). If f is a function of t only, then
f(τ) =
f(t) =
ntf(t).
For the contributions to Loop(1
), we have
σconn1/2 =
csea1/2,t
(mtxq)
3 + cval1/2,t(m
σdisc1/2 = 6(2(α + β))
2iDIXX + (4c
1/2,V + c
1/2,V )iD
XX + (V → A)
where q ∈ {u, d, s}, the DtXX are hairpin loop integrals, and csea1/2,t and cval1/2,t are symmetric
State(s) 〈cval
1/2,I
〉 〈cval
1/2,V
〉 〈cval
1/2,T
〉 〈cval
1/2,A
〉 〈cval
1/2,P
N112 (
, 2, 10) (0, 0, 0) (1, −20, −28) (0, 0, 0) (7
, 2, 10)
N331 (
, 2, 10) (1
, −20
, −28
) (11
, −20
, −28
, −10
, −14
Λ123 (
, 2, 10) (5, −4, 4) (−3, −12, −24) (5, −4, 4) (−5
, 2, −2)
N112|Λ341 (0, 0, 0) (
6) (0, 0, 0) (−
6, −4
6, −8
6) (0, 0, 0)
TABLE V: Coefficients of (α2, β2, αβ) for the distinct, nontrivial matrix elements of cval
1/2,t
matrices depending on the irrep t and containing the LECs. In dimensional regularization,
DIXX ≡ iµ4−n
(2π)n
(S · k)2
v · k + iǫ
(k2 − (mIxx)2 + iǫ)2
XX ≡ a
2δ′V,Aµ
(2π)n
(S · k)2
v · k + iǫ
(k2 − (mV,Axx )2 + iǫ)2 (k2 −m2η′
+ iǫ)
where
(k2 − (mtuu)2 + iǫ) (k2 − (mtdd)2 + iǫ) (k2 − (mtss)2 + iǫ)
(k2 −m2πt + iǫ) (k2 −m2ηt + iǫ)
Contributions from quark flows that contain a sea quark circulating in the loop are rep-
resented by csea
1/2,t
; an overall factor of 1
included in csea
1/2,t
accounts for taking the fourth
root of the fermion determinant in simulations. Contributions from quark flows containing
only valence quarks in the meson propagator are represented by cval
1/2,t
. In the basis (24) of
Table III, the diagonal elements of csea
1/2,t
〈csea1/2,t〉 = nt(58α
2 + β2 + 1
and the off-diagonal elements of csea
1/2,t
all vanish. Unlike csea
1/2,t
, cval
1/2,t
depends on the rep-
resentation used for the taste matrices. Employing the Weyl representation, the distinct,
nontrivial matrix elements of cval
1/2,t
are listed in Table V. The results for c
sea,val
1/2,t
are consis-
tent with the degeneracies and mixings parameterized in (25). Appendix A gives the general
forms of the matrix elements of cval1/2,t in terms of the elements of the taste matrices. These
forms are independent of the representation used for the taste matrices and can therefore be
used to show that the chiral forms of continuum HBχPT are recovered in any representation.
For the contributions to Loop(3
), we have
σconn3/2 =
csea3/2,t
F(mtxq) + cval3/2,tF(mtxx)
σdisc3/2 = (4c
3/2,V + c
3/2,V )iE
XX + (V → A)
where
F(m) ≡ ∆
(3m2 − 2∆2) ln(m2/µ2)− 4m2 + 10
m3g(∆/m),
g(x) =
(1− x2)3/2 arccosx 0 ≤ x ≤ 1
(x2 − 1)3/2 ln(x+
x2 − 1) x > 1
XX ≡ a
2δ′V,Aµ
(2π)n
kνkλP
v · k −∆+ iǫ
(k2 − (mV,Axx )2 + iǫ)2 (k2 −m2η′
+ iǫ)
the projection operator P νλ, in n spacetime dimensions, is
P νλ = vνvλ − gνλ − 4
SνSλ.
As before, the csea
3/2,t
and cval
3/2,t
are symmetric matrices in the baryon taste space. However,
the coefficients for spin-3
contributions are simpler than for the spin-1
contributions. The
diagonal elements of csea3/2,t are
〈csea3/2,t〉 = 12nt,
while the off-diagonal elements vanish. In the Weyl representation, the distinct, nontrivial
matrix elements of cval3/2,t are listed in Table VI. As for the spin-
contributions, the results for
sea,val
3/2,t
are consistent with the degeneracies and mixings parameterized in (25). The general
forms of the matrix elements of cval
3/2,t
, in terms of the elements of the taste matrices, are
given in Appendix A.
Like the loops, σFF (A) is a symmetric matrix. In the Weyl representation, the distinct,
nontrivial matrix elements of σFF (A) are listed in Table VII; they are consistent with the
degeneracies and mixings parameterized in (25). The LECs Ai, i = 1, . . . , 8 are defined in
Eq. (17). Appendix B contains the general forms of the matrix elements of σFF (A) in terms
of the taste matrices appearing in (17).
For the contributions to the symmetric matrix σFF (B), we consider the basis of Table IV.
In the Weyl representation, the distinct, nontrivial matrix elements are listed in Table VIII;
State(s) 〈cval
3/2,I
〉 〈cval
3/2,V
〉 〈cval
3/2,T
〉 〈cval
3/2,A
〉 〈cval
3/2,P
N112 −2 0 20 0 −2
N331 −2 203
Λ123 −2 4 12 4 −2
N112|Λ341 0 −4
6 0 4
TABLE VI: The distinct, nontrivial matrix elements of cval
3/2,t
State(s) A1 A3 A5 A7 A2 A4 A6 A8
N112 −12 −1 0 0 1 −4 0 0
N331 −16 −
0 −1 −1 0 −3 0 0
N112|Λ341 0 0 0 0 0 0
TABLE VII: Coefficients of Ai, i = 1, . . . , 8, for the matrix elements of σ
FF (A).
they are consistent with the degeneracies and mixings parameterized in (26). The LECs Bi,
i = 1, . . . , 6 are defined in Eq. (18). Appendix B contains the general forms of the matrix
elements of σFF (B) in terms of the taste matrices appearing in (18).
Taking the continuum and isospin limits, the staggered chiral forms reduce to that of
State(s) B1 B3 B5 B2 B4 B6
N112 −12 0 0 −2 0 0
N331 −16 −
Λ123 0 −14 −
N332 − 1√
Σ341 −23
N112|Λ341 0 0 0 0
N112|
N332 − 1√3 Σ341 0 −
Λ341|
N332 − 1√
TABLE VIII: Coefficients of Bi, i = 1, . . . , 6, for the matrix elements of σ
FF (B).
continuum HBχPT [29, 32] for the partially quenched nucleon. For the fully dynamical 2+1
flavor case, the hairpin loop integrals simplify significantly; we have
DIUU = iµ
(2π)n
(S · k)2
v · k + iǫ
UU = a
2δ′V,Aµ
(2π)n
(S · k)2
v · k + iǫ
X ′V,A
(k2 −m2
+ iǫ)
UU = a
2δ′V,Aµ
(2π)n
kνkλP
v · k −∆+ iǫ
X ′V,A
(k2 −m2
+ iǫ)
where
X ′t ≡
(k2 − (mtuu)2 + iǫ)2
(k2 − (mtss)2 + iǫ)
(k2 −m2πt + iǫ)(k2 −m2ηt + iǫ)
Recalling the residue relations of [8], evaluating the integrals, and renormalizing the tree-
level LECs gives
DIUU →
(m3ηI − 3m
UU → −
ia2δ′V,A
R[3,1]πV, Am
πV, A
+R[3,1]ηV, Am
ηV, A
[3,1]
ia2δ′V,A
(2π)2
R[3,1]πV, AF(mπV,A) +R
[3,1]
ηV, A
F(mηV, A) +R
[3,1]
F(mη′
where R
[3,1]
lV, A
is shorthand for the residue R
[3,1]
lV, A
({mπV, A, mηV, A , mη′V, A}; {m
ss }).
Combining these results for the hairpin integrals with the coefficients csea and those listed
in Tables V, VI, VII, and VIII, the parameters of (25) and (26) are, in the fully dynamical
2+1 flavor case,
c1 = M0 − 2(αM + βM + 2σM )mu − 2σMms (31)
192πf 2
α2 + β2 + 1
t nt(2m
+m3Kt)
α2 + 2β2 + 10αβ)(m3πI +m
) + (α2 − 20β2 − 28αβ)m3πT
+ 4(α + β)2(m3ηI − 3m
)− 16a2(5
α2 + β2 + 1
δ′V (R
[3,1]
m3πV +R
[3,1]
m3ηV +R
[3,1]
) + (V → A)
t nt[2F(mπt) + F(mKt)]
− 2[F(mπI) + F(mπP )] + 20F(mπT )
− 8a2
δ′V (R
[3,1]
F(mπV ) +R[3,1]ηV F(mηV ) +R
[3,1]
F(mη′
)) + (V → A)
A1 −A3 + A2 − 4A4
c2 = M0 − 2(αM + βM + 2σM)mu − 2σMms (32)
192πf 2
α2 + β2 + 1
nt(2m
+m3Kt)
α2 + 2β2 + 10αβ)m3πI + (5α
2 − 4β2 + 4αβ)(m3πV +m
+ (−3α2 − 12β2 − 24αβ)m3πT + (−
α2 + 2β2 − 2αβ)m3πP
+ 4(α+ β)2(m3ηI − 3m
α2 + β2 + 1
αβ) + (5α2 − 4β2 + 4αβ)
δ′V (R
[3,1]
m3πV +R
[3,1]
m3ηV +R
[3,1]
) + (V → A)
t nt[2F(mπt) + F(mKt)]
− 2[F(mπI ) + F(mπP )] + 4[F(mπV ) + F(mπA)] + 12F(mπT )
− 12a2
δ′V (R
[3,1]
F(mπV ) +R[3,1]ηV F(mηV ) +R
[3,1]
F(mη′
)) + (V → A)
A1 − A5 − A7 − 3A4
c3 = −
192πf 2
6(α2 + 4β2 + 8αβ)(m3πV −m
)− a2
6(α2 + 4β2 + 8αβ)× (33)
δ′V (R
[3,1]
m3πV +R
[3,1]
m3ηV +R
[3,1]
)− (V → A)
6[F(mπV )− F(mπA)]
+ 4a2
δ′V (R
[3,1]
F(mπV ) +R[3,1]ηV F(mηV ) +R
[3,1]
F(mη′
))− (V → A)
6 [A6 −A8] ,
c4 = −a2
B1 − 2B2
, (34)
c5 = −a2
B3 − 14B5 −
, (35)
c6 = −a2
, (36)
c7 = −a2
, (37)
c8 = −a2
B4 − 12√6B6
, (38)
c9 = −a2
B1 − 16B3 −
. (39)
In addition to the parameters ci, i = 1, . . . , 9, which correspond to members of Γ4⋊SU(2)T
4’s, define the parameters d1 and d2 to respectively correspond to the Γ4 ⋊ SO(4)T 12 and
Γ4 ⋊ SU(2)T 8. Then
d1 = M0 − 2(αM + βM + 2σM)mu − 2σMms (40)
192πf 2
α2 + β2 + 1
t nt(2m
+m3Kt)
α2 + 2β2 + 10αβ)m3πI +
(α2 − 20β2 − 28αβ)(m3πV +m
(11α2 − 4β2 + 16αβ)m3πT +
α2 − 10β2 − 14αβ)m3πP
+ 4(α+ β)2(m3ηI − 3m
α2 + β2 + 1
αβ) + 1
(α2 − 20β2 − 28αβ)
δ′V (R
[3,1]
m3πV +R
[3,1]
m3ηV +R
[3,1]
) + (V → A)
nt[2F(mπt) + F(mKt)]
− 2F(mπI) + 203 [F(mπV ) + F(mπA)] +
F(mπT ) + 103 F(mπP )
δ′V (R
[3,1]
πV F(mπV ) +R
[3,1]
ηV F(mηV ) +R
[3,1]
F(mη′
)) + (V → A)
A1 + 2A3 + A5 + A7 + 2A2 −A4 + 4A6 + 4A8],
d2 = a
B5 + 2B2 + 2B4 + 2B6
. (41)
The mass submatrices equal to d1 I12 and d2 I8 were suppressed in (25) and (26), respectively.
In the 2+1 flavor partially quenched case, double poles in the integrands of the hairpin
loop integrals complicate the results for c1, c2, c3, and d1. Evaluating the integrals and
renormalizing the tree-level LECs gives
DIXX → −
[2,2]
· (mIxx) +D
[2,2]
XI ,XI
(mIxx)
[2,2]
ηI , XI
(mηI )
ia2δ′V,A
[3,2]
XV, A
· (mV,Axx ) +D
[3,2]
XV, A, XV, A
(mV,Axx )
[3,2]
ηV, A,XV, A
(mηV, A)
[3,2]
,XV, A
XX → −
ia2δ′V,A
(2π)2
[3,2]
XV, A
F ′(mV,Axx )
[3,2]
XV, A, XV, A
F(mV,Axx )
[3,2]
ηV, A, XV, A
F(mηV, A) +D
[3,2]
, XV, A
F(mη′
where XI, V, A denote the states with masses m
I, V, A
xx , and, adapting the notation of [8], R
[3,2]
lV, A
is shorthand for R
[3,2]
lV, A
({mV,Axx , mηV, A, mη′V, A}; {m
uu , m
ss }),
[3,2]
lV, A, XV, A
xx )2]
[3,2]
lV, A
[2,2]
is shorthand for R
[2,2]
({mIxx, mηI}; {mIuu, mIss})|m2→−m2 , and
[2,2]
lI , XI
≡ + d
d[(mIxx)
[2,2]
Working in Minkowski space leads to the plus signs before the derivatives and the necessity
of evaluating R
[2,2]
with all masses m replaced, relative to the conventions of [8], by im,
where i2 = −1; R[3,2]lV, A is invariant under this substitution. The prime on F
′(m) represents
the derivative with respect to m.
Combining the results for the hairpin integrals with the coefficients csea, the coefficients
listed in Tables V and VI, and the results (29) and (30), the parameters c1, c2, c3, and d1
are, in the 2+1 flavor partially quenched case,
c1 = M0 − 2(αM + βM)mx − 2σM(2mu +ms) (42)
192πf 2
α2 + β2 + 1
t nt (2(m
3 + (mtxs)
α2 + 2β2 + 10αβ)
(mIxx)
3 + (mPxx)
+ (α2 − 20β2 − 28αβ)(mTxx)3
− 8(α+ β)2
[2,2]
· (mIxx) +D
[2,2]
XI , XI
(mIxx)
[2,2]
ηI , XI
(mηI )
+ 16a2(5
α2 + β2 + 1
[3,2]
· (mVxx) +D
[3,2]
XV , XV
(mVxx)
[3,2]
ηV , XV
(mηV )
[3,2]
+ (V → A)
t nt[2F(mtxu) + F(mtxs)]
− 2[F(mIxx) + F(mPxx)] + 20F(mTxx)
+ 8a2
[3,2]
F ′(mVxx)
2(mVxx)
[3,2]
XV ,XV
F(mVxx)
[3,2]
ηV ,XV
F(mηV ) +D
[3,2]
F(mη′
+ (V → A)
A1 −A3 + A2 − 4A4
c2 = M0 − 2(αM + βM)mx − 2σM(2mu +ms) (43)
192πf 2
α2 + β2 + 1
t nt (2(m
3 + (mtxs)
α2 + 2β2 + 10αβ)(mIxx)
3 + (5α2 − 4β2 + 4αβ)((mVxx)3 + (mAxx)3)
+ (−3α2 − 12β2 − 24αβ)(mTxx)3 + (−52α
2 + 2β2 − 2αβ)(mPxx)3
− 8(α+ β)2
[2,2]
· (mIxx) +D
[2,2]
XI , XI
(mIxx)
[2,2]
ηI ,XI
(mηI )
α2 + β2 + 1
αβ) + (5α2 − 4β2 + 4αβ)
[3,2]
· (mVxx) +D
[3,2]
XV , XV
(mVxx)
[3,2]
ηV ,XV
(mηV )
[3,2]
+ (V → A)
t nt[2F(mtxu) + F(mtxs)]
− 2[F(mIxx) + F(mPxx)] + 4[F(mVxx) + F(mAxx)] + 12F(mTxx)
+ 12a2
[3,2]
F ′(mVxx)
2(mVxx)
[3,2]
XV ,XV
F(mVxx)
[3,2]
ηV , XV
F(mηV ) +D
[3,2]
F(mη′
+ (V → A)
A1 − A5 − A7 − 3A4
c3 = −
192πf 2
6(α2 + 4β2 + 8αβ)((mVxx)
3 − (mAxx)3) + a2
6(α2 + 4β2 + 8αβ)× (44)
[3,2]
· (mVxx) +D
[3,2]
XV ,XV
(mVxx)
[3,2]
ηV , XV
(mηV )
[3,2]
+ (V → A)
6[F(mVxx)− F(mAxx)]
− 4a2
[3,2]
F ′(mVxx)
2(mVxx)
[3,2]
XV ,XV
F(mVxx)
[3,2]
ηV ,XV
F(mηV ) +D
[3,2]
F(mη′
+ (V → A)
6 [A6 −A8] ,
d1 = M0 − 2(αM + βM)mx − 2σM (2mu +ms) (45)
192πf 2
α2 + β2 + 1
t nt (2(m
3 + (mtxs)
α2 + 2β2 + 10αβ)(mIxx)
3 + 1
(α2 − 20β2 − 28αβ)((mVxx)3 + (mAxx)3)
(11α2 − 4β2 + 16αβ)(mTxx)3 + 13(
α2 − 10β2 − 14αβ)(mPxx)3
− 8(α + β)2
[2,2]
· (mIxx) +D
[2,2]
XI ,XI
(mIxx)
[2,2]
ηI ,XI
(mηI )
α2 + β2 + 1
αβ) + 1
(α2 − 20β2 − 28αβ)
[3,2]
· (mVxx) +D
[3,2]
XV ,XV
(mVxx)
[3,2]
ηV , XV
(mηV )
[3,2]
+ (V → A)
t nt[2F(mtxu) + F(mtxs)]
− 2F(mIxx) + 203 [F(m
xx) + F(mAxx)] + 43F(m
xx) +
F(mPxx)
[3,2]
F ′(mVxx)
2(mVxx)
[3,2]
XV , XV
F(mVxx)
[3,2]
ηV ,XV
F(mηV ) +D
[3,2]
F(mη′
+ (V → A)
A1 + 2A3 + A5 + A7 + 2A2 − A4 + 4A6 + 4A8].
The distinct, nontrivial elements of the mass matrix of the single-flavor nucleons are
simply related to the parameters ci and dj , j = 1, 2:
〈N112|M |N112〉 = c1 + c4 (46a)
〈Λ341|M |Λ341〉 = c2 + c5 (46b)
N332 − 1√3 Σ341|M |
N332 − 1√3 Σ341〉 = d1 + c6 (46c)
〈N331|M |N331〉 = d1 + d2, (46d)
〈N112|M |Λ341〉 = c3 + c7 (46e)
〈N112|M |
N332 − 1√3 Σ341〉 = c8 (46f)
〈Λ341|M |
N332 − 1√3 Σ341〉 = c9. (46g)
The O(a2) tree-level terms can be roughly estimated by noting that Tree(mq) − a2σFF (A)
absorbs the renormalization-scale dependence of the loop diagrams with virtual spin-3
baryons [33]. At a given scale, one expects the counterterms to be at least as large as
the change in the loop diagrams under a change in the renormalization scale by an amount
of order unity. If at some scale the counterterms were much smaller, then changing the scale
would make the magnitude of the counterterms comparable to the change in the loops. For
quark masses and lattice spacings for which the staggered chiral power counting is meaning-
ful, the tree-level corrections of O(mq) should be roughly equal to the tree-level corrections
of O(a2); one expects the expansion in the tree-level masses of the staggered mesons to be
meaningful at tree-level. Varying the renormalization scale in the loops and associating the
resulting O(a2) terms with valence and sea contributions of O(mq) gives
Tree(a2) & −2(αM + βM)
nt + c
3/2,t)− 2σM
where the term proportional to (αM + βM) comes from the valence sector, and the term
proportional to σM , from the sea. Given an estimate of the LECs αM , βM , σM , and λ from
the continuum and the measured values of the meson mass splittings ∆t [14], one estimates
the splittings to be roughly 10-40 MeV for lattice spacings of current interest. This splitting
in the flavor-symmetric nucleon masses may have been observed already with unimproved
staggered quarks [34]. Different nucleon operators transforming in the 8 of GTS would
generically have differing overlaps with the nondegenerate states in the 20M, which could
generate the observed operator dependence of the central mass values.
D. Correspondence between chiral forms and interpolating fields
The connection between the interpolating operators and the staggered chiral forms may
be obtained by decomposing the irreps of the forms into irreps of the operators. To O(ε3),
the staggered chiral forms respect the parity–spin–remnant-taste symmetry P × SU(2)E ×
[Γ4⋊SU(2)T ], while the operators transform within irreps of GTS. For the flavor-symmetric
nucleons, the relevant decompositions are
SU(2)E × [Γ4 ⋊ SU(2)T ] ⊃ GTS
, 8) → 16
, 4) → 8,
where the direct product with continuum parity has been suppressed on both sides of the
decompositions. Operators transforming in the 8 of GTS overlap members of SU(2)E ×
[Γ4 ⋊ SU(2)T ] (
, 4)’s; operators transforming in the 16 overlap members of (1
, 8)’s. To
O(ε3) in SχPT, members of a given SU(2)E × [Γ4 ⋊ SU(2)T ] irrep are degenerate, so the
exact linear combination of states within the SU(2)E × [Γ4 ⋊ SU(2)T ] multiplet created by
a given operator is unimportant; the staggered chiral forms are degenerate.
To O(ε3), the flavor-symmetric nucleons transform in one (1
, 8) and three (1
, 4)’s.
Operators transforming in the GTS 8 overlap states in the three 4’s of Γ4 ⋊ SU(2)T : Rep-
resentative members of these irreps are the N112, Λ341, and
N332 − 1√3 Σ341. Operators
transforming in the 16 overlap states in the 8 of Γ4⋊SU(2)T : for example, the N331. For the
case of 2+1 fully dynamical flavors, the staggered chiral forms for the masses of these states
are given in Eq. (46) with Eqs. (31) through (41); for the 2+1 flavor partially quenched case,
the corresponding results for c1,2,3 and d1 are given in Eqs. (42), (43), (44), and (45).
IV. SUMMARY AND FURTHER DIRECTIONS
Staggered, partially quenched chiral perturbation theory has been formulated in the
light-quark baryon sector by introducing taste degrees of freedom in heavy baryon chi-
ral perturbation theory and breaking the taste symmetry by mapping the operators of the
O(a2) Symanzik action to the heavy baryon Lagrangian. Including operators of O(a2) in the
Symanzik action allows one to calculate octet and decuplet baryon masses toO(ε3)=O(m3/2q )
in the staggered chiral expansion. As an example, the masses of the single-flavor nucleons
have been calculated; the result for the mass matrix is consistent with the pattern of degen-
eracies and mixings implied by the remnant taste symmetries, Γ4⋊SO(4)T and Γ4⋊SU(2)T .
In the rest frame of the heavy baryon, the symmetry Γ4⋊SU(2)T emerges as the taste sym-
metry of the chiral Lagrangian mapped from type B four-fermion operators in the Symanzik
action [17]; the resulting spin-taste SU(2)E× [Γ4⋊SU(2)T ] protects against mixing between
states of different spin. In the continuum limit, taste restoration forces all off-diagonal ele-
ments of the mass matrix to vanish, while the diagonal elements reduce to the result obtained
for the partially quenched nucleon in continuum HBχPT [29, 32].
The splittings in the nucleon mass must vanish as taste is restored, so they could be used
to test taste restoration. The staggered chiral forms given here are those needed to quantify
taste violations in simulation results obtained from the local corner-wall operators of [15]. At
small quark masses, the lattice spacing and quark mass dependences of the data qualitatively
agree with the staggered chiral forms given here. At larger quark masses, the O(ε3) chiral
forms decrease as −m3φ, while the data continues to increase. Fits to continuum χPT that
include analytic terms of O(m2q) seem to accurately describe the general trend of the data
at large quark mass. At sufficiently large quark mass and fixed lattice spacing, continuum
O(ε4)=O(m2q) terms will dominate over terms that are formally O(ε4)=O(mqa2)=O(a4), so
it might be possible to describe the data by supplementing the results given here with con-
tinuum terms of O(m2q). Alternatively, some authors argue that dimensional regularization
in BχPT incorporates spurious high-energy physics [35]. If true, then it is possible that the
data at larger quark masses would be better described by using a cut-off regulator. Such an
approach amounts to resumming the perturbative expansion and might not require terms
of O(ε4). Minimally, departing from dimensional regularization would require recalculating
the loop integrals at O(ε3).
Operators transforming in the 8 of GTS interpolate to three nucleon states and two
∆ states. Operators transforming in the 16 of GTS interpolate to only one nucleon, but
to three ∆’s [20]. For extracting the mass spectrum, one would prefer operators that do
not interpolate to states in SU(4)T -degenerate multiplets, which are split and mixed by
discretization effects. By these criteria, operators that would be ideal for extracting the
masses of the nucleon and the lightest decuplet were identified and constructed in [20]. The
associated chiral forms for the nucleon, the ∆, and the Ω− have been calculated to O(ε3);
these and the chiral forms for the Σ∗ and Ξ∗ will be reported in a future publication. There
are plans to use these operators and chiral forms in the near future [36].
Acknowledgments
The guidance and assistance of C. Bernard were essential at nearly every stage of this
project. Funding was supplied in part by the U.S. Department of Energy under grant DE-
FG02-91ER40628.
APPENDIX A: QUARK-FLOW COEFFICIENTS AND TASTE MATRICES
Here I write down the matrix elements of the quark-flow coefficients c
sea,val
1/2,t
and c
sea,val
3/2,t
in terms of the elements of the taste matrices ξτ , τ ∈ {I, µ, µν(µ < ν), µ5, 5}. These
explicit forms can be used to verify that, in the Weyl representation of the taste matrices,
the loop contributions are consistent with the parameterization (25). Because they are
independent of the representation used for the taste matrices, they can also be used to verify
that the continuum limits of the O(ε3) staggered chiral forms equal, in any representation,
the corresponding O(ε3) chiral forms of continuum HBχPT. The result is immediate if one
uses the completeness relation of the taste matrices:
ξτabξ
cd = 4δadδbc,
where ξτ ≡ ξτ and ξτab ≡ 〈a|ξτ |b〉, the (a, b)-element of ξτ in the fundamental representation
of SU(4)T .
In the basis (24) of Table III, the diagonal elements of csea1/2,t and c
1/2,t are
〈Naab|csea1/2,t|Naab〉 = nt(58α
2 + β2 + 1
〈Naab|cval1/2,t|Naab〉 =
(ξτabξ
ba + 10ξ
bb + 11(ξ
(−(ξτaa)2 − 5ξτabξτba + 4ξτaaξτbb)
+ 2αβ
(−7ξτabξτba + 11ξτaaξτbb + 4(ξτaa)2)
〈Σabc|csea1/2,t|Σabc〉 = nt(58α
2 + β2 + 1
〈Σabc|cval1/2,t|Σabc〉 =
(11(ξτaaξ
bb + ξ
ba) +
(ξτacξ
ca + ξ
cb) + 5(ξ
cc + ξ
(2(ξτaaξ
bb + ξ
ba) + 5(ξ
ca + ξ
cb)− 4(ξτaaξτcc + ξτbbξτcc))
+ 2αβ
(4(ξτaaξ
bb + ξ
ba)− 72(ξ
ca + ξ
cb) +
(ξτaaξ
cc + ξ
〈Λabc|csea1/2,t|Λabc〉 = nt(58α
2 + β2 + 1
〈Λabc|cval1/2,t|Λabc〉 =
(ξτaaξ
bb − ξτabξτba + 52(ξ
ca + ξ
cb) + 3(ξ
cc + ξ
+ β2 [2(ξτaaξ
bb − ξτabξτba)− (ξτacξτca + ξτbcξτcb)]
+ 2αβ
(4(ξτaaξ
bb − ξτabξτba) + (ξτacξτca + ξτbcξτcb) + 3(ξτaaξτcc + ξτbbξτcc))
The remnant taste symmetry Γ4⋊SO(4)T forces many of the off-diagonal elements to vanish.
The distinct off-diagonal elements that do not vanish by this symmetry are
〈Naab|csea1/2,t|Λdfg〉 = 0
〈Naab|cval1/2,t|Λdfg〉 =
(7ξτag(δadξ
bf − δafξτbd) + 4ξτbg(δadξτaf − δafξτad)
+ 3δag(ξ
bf − ξτafξτbd) + 11ξτag(δbfξτad − δbdξτaf))
(3δag(ξ
bd − ξτadξτbf) + δbfξτag(ξτad − ξτaf )
+ ξτag(δafξ
bd − δadξτbf) + 2ξτbg(δadξτaf − δafξτad))
+ 2αβ
(ξτbg(δafξ
ad − δadξτaf ) + 5ξτag(δadξτbf − δafξτbd)
+ 6δag(ξ
bf − ξτafξτbd) + 4ξτag(δbfξτad − δbdξτaf))
The matrix element 〈Naab|csea1/2,t|Λdfg〉 vanishes accidentally; calculation shows it is propor-
tional to δag(δbfδad − δbdδaf ), and no two of the indices dfg are ever equal for the states
Λdfg. In the Weyl representation, these matrix elements of c
sea,val
1/2,t
are consistent with the
degeneracies and mixings parameterized in (25). In the Weyl representation, the distinct,
nontrivial matrix elements of cval1/2,t are listed in Table V.
The diagonal elements of csea
3/2,t
and cval
3/2,t
〈Naab|csea3/2,t|Naab〉 = 12nt
〈Naab|cval3/2,t|Naab〉 =
[(ξτaa)
2 + 5ξτabξ
ba − 4ξτaaξτbb]
〈Σabc|csea3/2,t|Σabc〉 = 12nt
〈Σabc|cval3/2,t|Σabc〉 =
[2(ξτaaξ
bb + ξ
ba) + 5(ξ
ca + ξ
cb)− 4(ξτaaξτcc + ξτbbξτcc)]
〈Λabc|csea3/2,t|Λabc〉 = 12nt
〈Λabc|cval3/2,t|Λabc〉 =
[−2(ξτaaξτbb − ξτabξτba) + ξτacξτca + ξτbcξτcb] ,
while the off-diagonal elements that are not required to vanish by symmetry are
〈Naab|csea3/2,t|Λdfg〉 = 0
〈Naab|cval3/2,t|Λdfg〉 =
(3δag(ξ
bd − ξτadξτbf) + δbfξτag(ξτad − ξτaf )
+ ξτag(δafξ
bd − δadξτbf) + 2ξτbg(δadξτaf − δafξτad))
In the Weyl representation, these matrix elements of c
sea,val
3/2,t
are consistent with the degen-
eracies and mixings parameterized in (25). In this representation, the distinct, nontrivial
matrix elements of cval
3/2,t
are listed in Table VI.
APPENDIX B: FOUR-FERMION CONTRIBUTIONS AND TASTE MATRICES
Here I write down the matrix elements of σFF (A) and σFF (B) in terms of the elements
of the taste matrices appearing in (17) and (18). These results can be used to show that,
in the Weyl representation, σFF (A) and σFF (B) are consistent with the forms (25) and (26),
respectively.
For the diagonal σFF (A) contributions, we have
〈Naab|σFF (A)|Naab〉 = A1
(2(ξ5aa)
2 + ξ5abξ
ba + ξ
(2ξµνaa ξ
aa + ξ
ba + ξ
(2ξνaaξ
aa + ξ
ba + ξ
(2ξν5aaξ
aa + ξ
ba + ξ
((ξ5aa)
2 − 4ξ5abξ5ba + 5ξ5aaξ5bb)
(ξµνaa ξ
aa − 4ξ
ba + 5ξ
(ξνaaξ
aa − 4ξνabξνba + 5ξνaaξνbb)
(ξν5aaξ
aa − 4ξν5ab ξν5ba + 5ξν5aaξν5bb )
〈Σabc|σFF (A)|Σabc〉 = A1
(4(ξ5abξ
ba + ξ
bb) + ξ
ca + ξ
cc + ξ
cb + ξ
ba + ξ
bb ) + ξ
ca + ξ
cc + ξ
cb + ξ
(4(ξνabξ
ba + ξ
bb) + ξ
ca + ξ
cc + ξ
cb + ξ
(4(ξν5ab ξ
ba + ξ
bb ) + ξ
ca + ξ
cc + ξ
cb + ξ
(2(ξ5abξ
ba + ξ
bb)− 4ξ5acξ5ca + 5ξ5aaξ5cc − 4ξ5bcξ5cb + 5ξ5bbξ5cc)
ba + ξ
bb )− 4ξµνac ξµνca + 5ξµνaa ξµνcc − 4ξ
cb + 5ξ
(2(ξνabξ
ba + ξ
bb)− 4ξνacξνca + 5ξνaaξνcc − 4ξνbcξνcb + 5ξνbbξνcc)
(2(ξν5ab ξ
ba + ξ
bb )− 4ξν5ac ξν5ca + 5ξν5aaξν5cc − 4ξν5bc ξν5cb + 5ξν5bb ξν5cc )
〈Λabc|σFF (A)|Λabc〉 = A1
(ξ5acξ
ca + ξ
cc + ξ
cb + ξ
(ξµνac ξ
ca + ξ
cc + ξ
cb + ξ
(ξνacξ
ca + ξ
cc + ξ
cb + ξ
(ξν5ac ξ
ca + ξ
cc + ξ
cb + ξ
(2(ξ5aaξ
bb − ξ5abξ5ba) + ξ5aaξ5cc + ξ5bbξ5cc)
(2(ξµνaa ξ
bb − ξ
ba ) + ξ
cc + ξ
(2(ξνaaξ
bb − ξνabξνba) + ξνaaξνcc + ξνbbξνcc)
(2(ξν5aaξ
bb − ξν5ab ξν5ba ) + ξν5aaξν5cc + ξν5bb ξν5cc )
while the nontrivial off-diagonal terms are
〈Naab|σFF (A)|Λdfg〉 = A1
(δaf (ξ
bg + ξ
bd) + 2δbdξ
− δad(ξ5afξ5bg + ξ5agξ5bf )− 2δbfξ5adξ5ag)
(δaf (ξ
bg + ξ
bd ) + 2δbdξ
− δad(ξµνaf ξ
bg + ξ
bf )− 2δbfξ
(δaf (ξ
bg + ξ
bd) + 2δbdξ
− δad(ξνafξνbg + ξνagξνbf )− 2δbfξνadξνag)
(δaf (ξ
bg + ξ
bd ) + 2δbdξ
− δad(ξν5afξν5bg + ξν5agξν5bf )− 2δbfξν5adξν5ag )
(δaf (ξ
bg − 2ξ5agξ5bd) + δad(2ξ5bfξ5ag − ξ5afξ5bg)
+ δbfξ
ag − δbdξ5afξ5ag + 3δag(ξ5adξ5bf − ξ5afξ5bd))
(δaf (ξ
bg − 2ξµνag ξ
bd ) + δad(2ξ
ag − ξ
+ δbfξ
ag − δbdξ
ag + 3δag(ξ
bf − ξ
bd ))
(δaf (ξ
bg − 2ξνagξνbd) + δad(2ξνbfξνag − ξνafξνbg)
+ δbfξ
ag − δbdξνafξνag + 3δag(ξνadξνbf − ξνafξνbd))
(δaf (ξ
bg − 2ξν5agξν5bd ) + δad(2ξν5bf ξν5ag − ξν5afξν5bg )
+ δbfξ
ag − δbdξν5afξν5ag + 3δag(ξν5adξν5bf − ξν5afξν5bd ))
Repeated meson taste indices µν, ν, and ν5 are summed as in (17); as for the quark-
flow coefficients, the matrix is symmetric: 〈Λdfg|σFF (A)|Naab〉 = 〈Naab|σFF (A)|Λdfg〉. In the
Weyl representation, the matrix elements of σFF (A) are consistent with the degeneracies
and mixings parameterized in (25). For this representation the distinct, nontrivial matrix
elements are listed in Table VII.
For the diagonal σFF (B) contributions, first consider
〈Naab|σFF (B)|Naab〉 = B1
(2ξ4νaaξ
aa + ξ
ba + ξ
(2ξ4aaξ
aa + ξ
ba + ξ
(2ξ45aaξ
aa + ξ
ba + ξ
(ξ4νaaξ
aa − 4ξ4νab ξ4νba + 5ξ4νaaξ4νbb )
(ξ4aaξ
aa − 4ξ4abξ4ba + 5ξ4aaξ4bb)
(ξ45aaξ
aa − 4ξ45abξ45ba + 5ξ45aaξ45bb )
〈Σabc|σFF (B)|Σabc〉 = B1
(4(ξ4νab ξ
ba + ξ
bb ) + ξ
ca + ξ
cc + ξ
cb + ξ
(4(ξ4abξ
ba + ξ
bb) + ξ
ca + ξ
cc + ξ
cb + ξ
(4(ξ45abξ
ba + ξ
bb ) + ξ
ca + ξ
cc + ξ
cb + ξ
(2(ξ4νab ξ
ba + ξ
bb )− 4ξ4νac ξ4νca + 5ξ4νaaξ4νcc − 4ξ4νbc ξ4νcb + 5ξ4νbb ξ4νcc )
(2(ξ4abξ
ba + ξ
bb)− 4ξ4acξ4ca + 5ξ4aaξ4cc − 4ξ4bcξ4cb + 5ξ4bbξ4cc)
(2(ξ45abξ
ba + ξ
bb )− 4ξ45acξ45ca + 5ξ45aaξ45cc − 4ξ45bc ξ45cb + 5ξ45bb ξ45cc )
〈Λabc|σFF (B)|Λabc〉 = B1
(ξ4νac ξ
ca + ξ
cc + ξ
cb + ξ
(ξ4acξ
ca + ξ
cc + ξ
cb + ξ
(ξ45acξ
ca + ξ
cc + ξ
cb + ξ
(2(ξ4νaaξ
bb − ξ4νab ξ4νba ) + ξ4νaaξ4νcc + ξ4νbb ξ4νcc )
(2(ξ4aaξ
bb − ξ4abξ4ba) + ξ4aaξ4cc + ξ4bbξ4cc)
(2(ξ45aaξ
bb − ξ45abξ45ba ) + ξ45aaξ45cc + ξ45bb ξ45cc )
Repeated meson taste indices are summed as in (18). The Naab and Λabc terms above
include the diagonal σFF (B) contributions. However, the basis of Γ4 ⋊ SU(2)T eigenstates
does not include the Σabc. Instead, we consider the states of Table IV,
Naab − 1√3Σacd
and 1√
Naab +
Σacd. In terms of the Γ4 ⋊ SO(4)T basis, the corresponding diagonal
contributions to σFF (B) are
Naab − 1√3Σacd|σ
FF (B)|
Naab − 1√3Σacd〉 = 2〈Naab|σ
FF (B)|Naab〉 − 〈Σacd|σFF (B)|Σacd〉
Naab +
Σacd|σFF (B)| 1√3Naab +
Σacd〉 = 2〈Σacd|σFF (B)|Σacd〉 − 〈Naab|σFF (B)|Naab〉,
where abcd ∈ {1423, 2314, 4132, 3241}, and I used |Σabc〉 = |Σbac〉 (cf. (24b)). The off-
diagonal terms of σFF (B) are
〈Naab|σFF (B)|Λdfg〉 = B1
(δaf (ξ
bg + ξ
bd ) + 2δbdξ
− δad(ξ4νafξ4νbg + ξ4νagξ4νbf )− 2δbfξ4νadξ4νag )
(δaf (ξ
bg + ξ
bd) + 2δbdξ
− δad(ξ4afξ4bg + ξ4agξ4bf )− 2δbfξ4adξ4ag)
(δaf (ξ
bg + ξ
bd ) + 2δbdξ
− δad(ξ45afξ45bg + ξ45agξ45bf )− 2δbfξ45adξ45ag)
(δaf (ξ
bg − 2ξ4νagξ4νbd ) + δad(2ξ4νbf ξ4νag − ξ4νafξ4νbg )
+ δbfξ
ag − δbdξ4νafξ4νag + 3δag(ξ4νadξ4νbf − ξ4νafξ4νbd ))
(δaf (ξ
bg − 2ξ4agξ4bd) + δad(2ξ4bfξ4ag − ξ4afξ4bg)
+ δbfξ
ag − δbdξ4afξ4ag + 3δag(ξ4adξ4bf − ξ4afξ4bd))
(δaf (ξ
bg − 2ξ45agξ45bd ) + δad(2ξ45bf ξ45ag − ξ45afξ45bg )
+ δbfξ
ag − δbdξ45afξ45ag + 3δag(ξ45adξ45bf − ξ45afξ45bd ))
〈Naab|σFF (B)|
Nccd − 1√3Σcfg〉 = B1
(2δbc(2ξ
ad − ξ4νafξ4νag ) + 4(δadξ4νac ξ4νbc − δbdξ4νac ξ4νac )
+ δac(ξ
bg + ξ
bf − 2ξ4νac ξ4νbd − 2ξ4νadξ4νbc )
+ 4δbgξ
af − 2(δbfξ4νagξ4νac + δag(ξ4νac ξ4νbf + ξ4νafξ4νbc ))
+ δaf (ξ
bc + ξ
bg ))
(2δbc(2ξ
ad − ξ4afξ4ag) + 4(δadξ4acξ4bc − δbdξ4acξ4ac)
+ δac(ξ
bg + ξ
bf − 2ξ4acξ4bd − 2ξ4adξ4bc)
+ 4δbgξ
af − 2(δbfξ4agξ4ac + δag(ξ4acξ4bf + ξ4afξ4bc))
+ δaf (ξ
bc + ξ
(2δbc(2ξ
ad − ξ45afξ45ag) + 4(δadξ45acξ45bc − δbdξ45acξ45ac )
+ δac(ξ
bg + ξ
bf − 2ξ45acξ45bd − 2ξ45adξ45bc )
+ 4δbgξ
af − 2(δbfξ45agξ45ac + δag(ξ45acξ45bf + ξ45afξ45bc ))
+ δaf (ξ
bc + ξ
bg ))
(2(δbdξ
ac − δadξ4νac ξ4νbc ) + δbc(ξ4νafξ4νag − 2ξ4νac ξ4νad)
+ δac(4ξ
bf + 10ξ
bd − 8ξ4νbc ξ4νad − 5ξ4νafξ4νbg )
− 2δbgξ4νac ξ4νaf + δbfξ4νac ξ4νag + δag(ξ4νac ξ4νbf + ξ4νafξ4νbc )
+ δaf (4ξ
bc − 5ξ4νac ξ4νbg ))
(2(δbdξ
ac − δadξ4acξ4bc) + δbc(ξ4afξ4ag − 2ξ4acξ4ad)
+ δac(4ξ
bf + 10ξ
bd − 8ξ4bcξ4ad − 5ξ4afξ4bg)
− 2δbgξ4acξ4af + δbfξ4acξ4ag + δag(ξ4acξ4bf + ξ4afξ4bc)
+ δaf (4ξ
bc − 5ξ4acξ4bg))
(2(δbdξ
ac − δadξ45acξ45bc ) + δbc(ξ45afξ45ag − 2ξ45acξ45ad)
+ δac(4ξ
bf + 10ξ
bd − 8ξ45bc ξ45ad − 5ξ45afξ45bg )
− 2δbgξ45acξ45af + δbfξ45acξ45ag + δag(ξ45acξ45bf + ξ45afξ45bc )
+ δaf (4ξ
bc − 5ξ45acξ45bg ))
〈Λabh|σFF (B)|
Nccd − 1√3Σcfg〉 =
〈Nccd|σFF (B)|Λabh〉.
In the Weyl representation, the matrix elements of σFF (B) are consistent with the degenera-
cies and mixings parameterized in (26). The distinct, nontrivial matrix elements are listed
in Table VIII.
[1] J. B. Kogut and L. Susskind, Phys. Rev. D11, 395 (1975).
[2] E. Marinari, G. Parisi, and C. Rebbi, Nucl. Phys. B190, 734 (1981).
[3] C. Bernard, M. Golterman, and Y. Shamir, Phys. Rev. D73, 114511 (2006), hep-lat/0604017.
[4] Y. Shamir (2006), hep-lat/0607007.
[5] P. H. Damgaard and K. Splittorff, Phys. Rev. D62, 054509 (2000), hep-lat/0003017.
[6] C. Aubin and C. Bernard, Nucl. Phys. Proc. Suppl. 129, 182 (2004), hep-lat/0308036.
[7] C. Aubin and C. Bernard, Phys. Rev. D68, 034014 (2003), hep-lat/0304014.
[8] C. Aubin and C. Bernard, Phys. Rev. D68, 074011 (2003), hep-lat/0306026.
[9] C. Bernard, Phys. Rev. D73, 114503 (2006), hep-lat/0603011.
[10] D. Toussaint, Nucl. Phys. Proc. Suppl. 106, 111 (2002), hep-lat/0110010.
[11] C. T. H. Davies et al. (HPQCD, UKQCD, MILC, and Fermilab Lattice), Phys. Rev. Lett. 92,
022001 (2004), hep-lat/0304004.
[12] C. Aubin et al. (HPQCD, MILC, and UKQCD), Phys. Rev. D70, 031504 (2004), hep-
lat/0405022.
[13] C. Aubin et al. (Fermilab Lattice, MILC, and HPQCD), Phys. Rev. Lett. 94, 011601 (2005),
hep-ph/0408306.
[14] C. Aubin et al. (MILC), Phys. Rev. D70, 114501 (2004), hep-lat/0407028.
[15] C. Aubin et al. (MILC), Phys. Rev. D70, 094505 (2004), hep-lat/0402030.
[16] C. Aubin et al. (Fermilab Lattice, MILC, and HPQCD), Phys. Rev. Lett. 95, 122002 (2005),
hep-lat/0506030.
[17] W.-J. Lee and S. R. Sharpe, Phys. Rev. D60, 114503 (1999), hep-lat/9905023.
[18] C. Aubin and C. Bernard, Phys. Rev. D73, 014515 (2006), hep-lat/0510088.
[19] C. Bernard (MILC), Phys. Rev. D65, 054031 (2002), hep-lat/0111051.
[20] J. A. Bailey (2006), hep-lat/0611023.
[21] J. A. Bailey, work in progress.
[22] J. A. Bailey and C. Bernard, PoS LAT2005, 047 (2006), hep-lat/0510006.
[23] M. F. L. Golterman and J. Smit, Nucl. Phys. B255, 328 (1985).
[24] S. Scherer, Adv. Nucl. Phys. 27, 277 (2003), hep-ph/0210398.
[25] S. R. Sharpe, Nucl. Phys. Proc. Suppl. 34, 403 (1994), hep-lat/9312009.
[26] S. R. Sharpe and R. S. Van de Water, Phys. Rev. D71, 114505 (2005), hep-lat/0409018.
[27] J. N. Labrenz and S. R. Sharpe, Phys. Rev. D54, 4595 (1996), hep-lat/9605034.
[28] E. Jenkins and A. V. Manohar, Phys. Lett. B255, 558 (1991).
[29] J.-W. Chen and M. J. Savage, Phys. Rev. D65, 094001 (2002), hep-lat/0111050.
[30] H. Georgi, Weak Interactions and Modern Particle Theory (Menlo Park, Usa: Ben-
jamin/Cummings, 1984), pp. 92-95.
[31] E. Jenkins and A. V. Manohar (1991), talk presented at the Workshop on Effective Field
Theories of the Standard Model, Dobogoko, Hungary.
[32] A. Walker-Loud, Nucl. Phys. A747, 476 (2005), hep-lat/0405007.
[33] A. Manohar and H. Georgi, Nucl. Phys. B234, 189 (1984).
[34] M. Fukugita, N. Ishizuka, H. Mino, M. Okawa, and A. Ukawa, Phys. Rev. D47, 4739 (1993).
[35] J. F. Donoghue, B. R. Holstein, and B. Borasoy, Phys. Rev. D59, 036002 (1999), hep-
ph/9804281.
[36] J. A. Bailey et al. (MILC), work in progress.
	Introduction
	The Staggered Heavy Baryon Lagrangian
	O(a0) Lagrangian
	O(a2) baryon operators for analytic corrections
	L6glue and L6bilin
	L6FF(A)
	L6FF(B)
	Staggered heavy baryon Lagrangian for O(3) baryon octet masses
	The Flavor-Symmetric Nucleons
	Identifying staggered nucleons and irreps of interpolating fields
	Taste symmetry and the mass matrix
	Masses of the flavor-symmetric nucleons
	Correspondence between chiral forms and interpolating fields
	Summary and Further Directions
	Acknowledgments
	Quark-flow coefficients and taste matrices
	Four-fermion contributions and taste matrices
	References
