Heavy-Light Semileptonic Decays in Staggered Chiral Perturbation Theory
CU-TP-1177
Heavy-Light Semileptonic Decays in Staggered Chiral
Perturbation Theory
C. Aubin
Physics Department, Columbia University, New York, NY 10027
C. Bernard
Department of Physics, Washington University, St. Louis, MO 63130
Abstract
We calculate the form factors for the semileptonic decays of heavy-light pseudoscalar mesons
in partially quenched staggered chiral perturbation theory (SχPT), working to leading order in
1/mQ, where mQ is the heavy quark mass. We take the light meson in the final state to be
a pseudoscalar corresponding to the exact chiral symmetry of staggered quarks. The treatment
assumes the validity of the standard prescription for representing the staggered “fourth root trick”
within SχPT by insertions of factors of 1/4 for each sea quark loop. Our calculation is based on
an existing partially quenched continuum chiral perturbation theory calculation with degenerate
sea quarks by Bećirević, Prelovsek and Zupan, which we generalize to the staggered (and non-
degenerate) case. As a by-product, we obtain the continuum partially quenched results with
non-degenerate sea quarks. We analyze the effects of non-leading chiral terms, and find a relation
among the coefficients governing the analytic valence mass dependence at this order. Our results
are useful in analyzing lattice computations of form factors B → π and D → K when the light
quarks are simulated with the staggered action.
PACS numbers: 12.39.Fe,12.39.Hg, 11.30.Rd, 12.38.Gc
http://arxiv.org/abs/0704.0795v2
I. INTRODUCTION
Extraction of the CKM matrix elements |Vub| and |Vcs| from the experimentally measured
semileptonic decay rates for B → πℓν andD → Kℓν requires reliable theoretical calculations
of the corresponding hadronic matrix elements. Recently, there has been significant progress
in computing these matrix elements on the lattice, with good control of the systematic
uncertainties [1, 2, 3, 4]. Since computation time increases as a high power of the inverse
quark mass, the light (u, d) quark masses used in the simulations are heavier than in nature,
and a chiral extrapolation is necessary to obtain physical results. To keep systematic errors
small, the simulated u, d masses should be well into the chiral regime, giving pion masses
∼300MeV or lighter. Such masses in lattice calculations of leptonic and semileptonic heavy-
light decays are accessible with staggered quarks [5, 6, 7, 8, 9, 10]. The trade-off for this
benefit is the fact that staggered quarks do not fully remove the species doubling that occurs
for lattice fermions; for every flavor of lattice quark, there are four “tastes,” which are related
in the continuum by an SU(4) symmetry (or an SU(4)L×SU(4)R symmetry in the massless
case). The taste symmetry is broken at non-zero lattice spacing a by terms of order a2.
The breaking of taste symmetry on the lattice implies that one must take into account
taste-violations in the chiral extrapolations, leading to a joint extrapolation in both the quark
masses and the lattice spacing. Staggered chiral perturbation theory (SχPT) [11, 12, 13]
allows us to make such extrapolations systematic. For quantities with heavy quarks, one
must also incorporate Heavy Quark Effective Theory (HQET) [14, 15, 16, 17, 18] into SχPT.
This has been done in Ref. [19], and then applied to leptonic heavy-light decays. Here, we
extend the analysis of Ref. [19] to the semileptonic case.
In addition to the practical implications of taste symmetry violations for chiral extrap-
olations, the violations lead to a potentially more serious theoretical concern. Simulations
such as Refs. [5, 6, 7, 8, 9, 10, 20] take the fourth root of the staggered quark determinant
[21] in an attempt to obtain a single taste per quark flavor in the continuum limit. Were the
taste symmetry exact at finite lattice spacing, the fourth root prescription would obviously
accomplish the desired goal, since it would be equivalent to using a local Dirac operator
obtained by projecting the staggered operator onto a single-taste subspace. Because the
taste symmetry is broken, however, the fourth root is necessarily a nonlocal operation at
non-zero lattice spacing [22]. The question of whether the rooted theory is in the correct
universality class therefore becomes nontrivial. Nevertheless, there are strong theoretical
arguments [22, 23, 24, 25, 26] in the interacting theory, as well as free-theory and numerical
evidence [27] that the fourth-root trick is valid, i.e., that it produces QCD in the continuum
limit.
The current paper does not actually need to assume that the rooting procedure itself is
valid.1 Instead, like previous SχPT calculations for the rooted theory [12, 13, 19, 28], it
requires a narrower assumption: that the rooting can be represented at the chiral level by
multiplying each sea quark loop by a factor of 1/4. This can be accomplished by a quark
flow analysis [29], or, more systematically, by use of the replica trick [30]. In Ref. [24], it
was shown that the correctness of this representation of the fourth root in SχPT follows in
turn from certain — in our opinion, rather plausible — assumptions. As such, we assume
here that this representation is valid. Fitting lattice quantities to SχPT formulae (as in
Refs. [10, 20]) provides an additional empirical test of the validity of this representation.
The main purpose of the current paper is to find SχPT expressions for the form factors of
the semileptonic decay B → Pℓν, where P is some light pseudoscalar meson, which we will
refer to generically as a “pion.” We consider first the partially quenched case, and obtain the
full QCD results afterward by taking the limit where valence masses equal the sea masses.
The B is a heavy-light meson made up of a b heavy quark and a valence light quark spectator
of flavor x; we use the notation Bx when confusion as to the identity of the light spectator
could arise. The P meson (more precisely Pxy) is composed of two light valence quarks, of
flavor x and y. For simplicity we consider only the case where the outgoing pion is (flavor)
charged; in other words x 6= y. The flavor structure of the weak current responsible for the
decay is ȳγµb.
In our calculation, we take the heavy quark mass mQ to be large compared to ΛQCD and
work to leading order in the 1/mQ expansion. Our analysis also applies when the heavy
quark is a c (i.e., to D mesons), but we use B to denote the heavy-light meson to stress
the fact that only lowest order terms HQET are kept. For D mesons, of course, the higher
order terms omitted here would be more important than for B mesons.
Discretization errors coming from the heavy quark are not included in the current calcu-
lations. We assume that such errors will be estimated independently, using HQET as the
1 Of course, were the fourth root trick to prove invalid, the motivation for the current work would be lost.
effective-theory description of the lattice heavy quark [31]. It is expected that the errors from
staggered quark taste-violations, which are considered here, are significantly more important
at most currently accessible lattice spacings than the heavy-quark errors [4]. However, since
taste-violations decrease rapidly2 when the lattice spacing is reduced, this may change in
the not too distant future. In any case, the precise quantification of the total discretization
error will always require simulation at several lattice spacings.
An additional practical constraint on the current calculation is that amQ must not be
too large compared to unity. When amQ ≫ 1, the effects of the heavy quark doublers
would need to be included in the chiral theory, and the analysis would become prohibitively
complicated. A detailed discussion of this and other issues involved in incorporating heavy
quarks into SχPT appears in Ref. [19].
The calculations of interest here have been performed in continuum partially quenched
chiral perturbation theory (PQχPT) by Bećirević, Prelovsek and Zupan [33] for Nsea de-
generate sea quarks. In this paper we show how one can generalize the PQχPT formulae
to the corresponding SχPT formulae, thereby avoiding the necessity of recomputing all the
diagrams from scratch.
Some results from the current work, as well as a brief discussion of how to generalize
PQχPT to SχPT, appear in Ref. [34]. In addition, our results have already been used in
chiral fits to lattice data in Refs. [8, 9]. A related calculation for the B → D∗ and B → D
semileptonic form factors has been presented by Laiho and Van de Water [28].
The outline of this paper is as follows: We first include a brief description of heavy-light
SχPT in Sec. II. In Sec. III, we discuss the procedure for generalizing PQχPT to SχPT,
using the heavy-light form factors as examples, although the procedure can be used for
many other quantities in SχPT. Using this procedure and starting from Ref. [33], we write
down, in Sec. IV, the one-loop SχPT results for the semileptonic form factors. The partially
quenched staggered case with non-degenerate sea quarks, as well as its continuum limit, is
presented in Sec. IVA. In that section, we also discuss a method for treating — in a way
that appears to be practical for fitting lattice data — some spurious singularities which
arise in the calculations. Section IVB considers full-QCD special cases of the results from
2 Taste violations with improved staggered fermions go like α2
a2. See Fig. 1 in Ref. [32] for a test of this
relation.
Sec. IVA; while Sec. IVC discusses the analytic contributions to the form factors at this
order. In Sec. V we add in the effects of a finite spatial lattice volume. Sec. VI presents
our conclusions. We include three appendices: Appendix A gives expressions for the SχPT
propagators and vertices, as well as the corresponding continuum versions. Appendix B
lists the integrals used in the form factor calculations; while Appendix C collects necessary
wavefunction renormalization factors that were calculated in Refs. [13, 19].
II. HEAVY-LIGHT STAGGERED CHIRAL PERTURBATION THEORY
References [14, 15, 16, 17, 18] show how to incorporate heavy-light mesons into continuum
χPT; the extension to SχPT appears in Ref. [19]. Here we review the key features needed
for our calculations.
The heavy-light vector (B∗µa) and pseudoscalar (Ba) mesons are combined in the field
1 + v/
µa + iγ5Ba
, (1)
which destroys a meson. Here v is the meson velocity, and a is the “flavor-taste” index of
the light quark in the meson. For n flavors of light quarks, a can take on 4n values. Later,
we will write a as separate flavor (x) and taste (α) indices, a→ (x, α), and ultimately drop
the taste index, since the quantities we calculate will have trivial dependence on the light
quark taste. The conjugate field Ha creates mesons:
Ha ≡ γ0H†aγ0 =
µa + iγ5B
] 1 + v/
. (2)
As mentioned in the introduction, we use B to denote generic heavy-light mesons to empha-
size that we are working to leading order in 1/mQ.
Under SU(2) heavy-quark spin symmetry, the heavy-light field transforms as
H → SH ,
H → HS† , (3)
with S ∈ SU(2), while under the SU(4n)L × SU(4n)R chiral symmetry,
H → HU† ,
H → UH , (4)
with U ∈ SU(4n) defined below. We keep the flavor and taste indices implicit here.
The light mesons are combined in a Hermitian field Φ(x). For n staggered flavors, Φ is a
4n× 4n matrix given by:
U π+ K+ · · ·
π− D K0 · · ·
K− K̄0 S · · ·
. . .
. (5)
We show the n = 3 portion of Φ explicitly. Each entry in Eq. (5) is a 4×4 matrix, written
in terms of the 16 Hermitian taste generators TΞ as, for example, U =
Ξ=1 UΞTΞ. The
component fields of the flavor-neutral elements (UΞ, DΞ, . . . ) are real; the other (flavor-
charged) fields (π+Ξ , K
Ξ, . . . ) are complex. The TΞ are
TΞ = {ξ5, iξµ5, iξµν(µ < ν), ξµ, ξI} , (6)
with ξµ the taste matrices corresponding to the Dirac gamma matrices, and ξI ≡ I the 4×4
identity matrix. We define ξµ5 ≡ ξµξ5, and ξµν ≡ (1/2)[ξµ, ξν].
The mass matrix is the 4n× 4n matrix
muI 0 0 · · ·
0 mdI 0 · · ·
0 0 msI · · ·
. . .
, (7)
where the portion shown is again for the n = 3 case.
From Φ one constructs the unitary chiral field Σ = exp[iΦ/f ], with f the tree-level pion
decay constant. In our normalization, f ∼ fπ ∼= 131 MeV. Terms involving the heavy-lights
are conveniently written using use σ ≡
Σ = exp[iΦ/2f ]. These fields transform trivially
under the SU(2) spin symmetry, while under SU(4n)L × SU(4n)R we have
Σ→ LΣR† , Σ† → RΣ†L† , (8)
σ → LσU† = UσR† , σ† → Rσ†U† = Uσ†L† , (9)
with global transformations L ∈ SU(4n)L and R ∈ SU(4n)R. The transformation U, defined
by Eq. (9), is is a function of Φ and therefore of the coordinates.
It is convenient to define objects involving the σ field that transform only with U and
†. The two possibilities with a single derivative are
σ†∂µσ + σ∂µσ
, (10)
σ†∂µσ − σ∂µσ†
. (11)
Vµ transforms like a vector field under the SU(4n)L×SU(4n)R chiral symmetry and, when
combined with the derivative, can form a covariant derivative acting on the heavy-light field
or its conjugate:
Dµ)a = Hb
Dbaµ ≡ ∂µHa + iHbVbaµ ,
DµH)a =
Dabµ Hb ≡ ∂µHa − iVabµ Hb , (12)
with implicit sums over repeated indices. The covariant derivatives and Aµ transform under
the chiral symmetry as
Dµ → (H
DµH → U(
DµH) ,
Aµ → UAµU† . (13)
The combined symmetry group of the theory includes Euclidean rotations (or Lorentz
symmetry), translations, heavy-quark spin, flavor-taste chiral symmetries, and the discrete
symmetries C, P , and T . Many of these symmetries are violated by lattice artifacts and/or
light quark masses. Violations to a given order are encoded as spurions in the Symanzik
action. From these spurions, the heavy-light and light-light fields, derivatives, the heavy
quark 4-velocity vµ, and the light quark gamma matrix γµ, we can construct the chiral
Lagrangian and relevant currents order by order.
Reference [19] finds the lowest order heavy-chiral Lagrangian and left-handed current,
as well as higher order corrections. We need primarily the lowest order results here. For
convenience, we write the Lagrangian in Minkowski space, so that we can make contact with
the continuum literature.
We write the leading order (LO) chiral Lagrangian as
LLO = Lpion + LHL (14)
where Lpion is the standard SχPT Lagrangian [13] for the light-light mesons, and LHL is the
contribution of the heavy-lights. We have3
Lpion =
Tr(∂µΣ∂
µΣ†) +
µf 2Tr(MΣ+MΣ†)
(UI +DI + SI + . . .)
2 − a2V , (15)
−V = C1Tr(ξ(n)5 Σξ
†) + C3
[Tr(ξ(n)ν Σξ
ν Σ) + h.c.]
[Tr(ξ
ν5 Σξ
5ν Σ) + h.c.] + C6
Tr(ξ(n)µν Σξ
+ C2V
[Tr(ξ(n)ν Σ)Tr(ξ
ν Σ) + h.c.] + C2A
[Tr(ξ
ν5 Σ)Tr(ξ
5ν Σ) + h.c.]
+ C5V
Tr(ξ(n)ν Σ)Tr(ξ
†) + C5A
ν5 Σ)Tr(ξ
†) , (16)
LHL = −iTr(HHv·
D) + gπ Tr(HHγ
µγ5Aµ) . (17)
Here Tr denotes a trace over flavor-taste indices and, where relevant, Dirac indices. The
product HH is treated as a matrix in flavor-taste space: (HH)ab ≡ HaHb. The covariant
derivative
D acts only on the field immediately preceding it. For convenience, we work with
diagonal fields (U , D, . . . ) and leave the anomaly (m20) term explicit in Eq. (15). We can
take m20 →∞ and go to the physical basis (π0, η, . . . ) at the end of the calculation [37].
To calculate semileptonic form factors, we need the chiral representative of the left-handed
current which destroys a heavy-light meson of flavor-taste b. At LO this takes the form
γµ (1− γ5)H
σ†λ(b) , (18)
where λ(b) is a constant vector that fixes the flavor-taste: (λ(b))c = δbc, and trD is a trace on
Dirac indices only.
The power counting is a little complicated in the heavy-light case, since many scales are
available. Let mq be a generic light quark mass, and let m
π ∝ mq be the corresponding
“pion” mass, with p its 4-momentum. Further, take k as the heavy-light meson’s residual
momentum. Then our power counting assumes k2 ∼ p2 ∼ m2π ∼ mq ∼ a2, where appropriate
powers of the chiral scale or ΛQCD are implicit. The leading heavy-light chiral Lagrangian
3 There is a missing minus sign in Eq. (35) of Ref. [19].
LHL is O(k), the leading light-light Lagrangian Lpion is O(p2, mq, a2), and the leading heavy-
light current j
LO is O(1). Only these leading terms are relevant to the calculation of non-
analytic “chiral logarithms” at first non-trivial order, which give O(mq, a2) corrections to
leading expressions for semileptonic form factors.
In principle, finding the corresponding analytic corrections requires complete knowledge
of the next-order terms in the Lagrangian and current. However, since the form factors
depend only on the the valence and sea quark masses, a2, and the pion energy in the rest
frame of the B (namely v·p), the form of these corrections is rather simple and is easily
determined by the symmetries. The large number of chiral parameters that can appear in
higher-order terms in the Lagrangian and the current collapse down into relatively few free
parameters in the form factors. Unless one wants to write these free parameters in terms
of the chiral parameters, complete knowledge of the higher-order terms in the Lagrangian
and current is often unnecessary. However, one does need to know enough about the higher-
order terms to check for the possibility of relations among the free parameters that multiply
different quantities or that appear in different form factors. At the order we work here, there
is one relation among the various parameters that determine the linear dependence of the
two form factors on the valence masses. In order to be sure that this relation is valid, we
need to know all terms at next order that can contribute such linear dependence.
Fortunately, all such terms are known. For the light-light Lagrangian, Eq. (15), the
relevant terms are the standard O(p4 ∼ m2q) terms in the continuum [38]. All terms of
O(mqa2, a4), which are special to SχPT, are also available [39]. For the heavy-light La-
grangian and current, Ref. [19] lists all terms which are higher order than Eqs. (17) and (18)
by a factor of mq (most important here) or a
2. Reference [19] does not attempt a complete
catalog of the terms which are higher than Eqs. (17) and (18) by one or two powers of k,
i.e., having one or two derivative insertions. However, a sufficient number of representative
terms of this type are listed to see that the corresponding free parameters in the form factors
are all independent. We discuss the determination of the analytic terms further in Sec. IVC.
III. GENERALIZING CONTINUUM PQχPT TO SχPT
We wish to compute the decay Bx → Pxy in SχPT, where x and y are (light) flavor
labels. The taste of the light quarks in B, P and the current also needs to be specified.
We take the Pxy to be a “Goldstone pion” with taste ξ5. Let the light quark in the B
have taste α (α = 1, . . . , 4); in flavor-taste notation the light quark has index a ↔ xα.
The current, Eq. (18), has flavor-taste b ↔ yβ. Despite the existence of taste violations
at non-zero lattice spacing, the amplitude turns out to be proportional to (ξ5/2)αβ, with a
proportionality factor that is independent of the tastes α, β. We will often keep this rather
trivial taste-dependence implicit.
In Ref. [33], Bećirević, et al. have calculated the form factors for B → π and B → K
transitions in continuum PQχPT. They assume degenerate sea-quark masses, but leave Nsea,
the number of sea quarks, arbitrary. As we explain below, the Nsea dependence is a marker
for the underlying quark flow [29] within the meson diagrams. Once we have separated the
meson diagrams into their contributions from various the quark flow diagrams, we can easily
generalize the continuum PQχPT results to the staggered case, without actually having to
calculate any SχPT diagrams. To check our method, however, we have also computed many
of the diagrams directly in SχPT; the results agree.
The key feature that makes possible the generalization of continuum PQχPT results
to SχPT results is the taste-invariance of the leading-order Lagrangian for the heavy-light
mesons [19]. This means that the continuum vertices and propagators involving heavy-
light mesons are trivially generalized to the staggered case: flavor indices (which can take
Nsea values if they describe sea quarks) simply become flavor-taste indices (taking 4Nsea
sea-quark values). In one-loop diagrams, taste violations arise only from the light meson
(“pion”) propagators. Propagators and vertices for the staggered and continuum cases are
listed Appendix A.
Looking at the expressions in Appendix B of Ref. [33], we see that there are two types
of terms that can contribute to each diagram for Bx → Pxy: a term proportional to Nsea,
and a term proportional to 1/Nsea. This is the same behavior that appears, for example, in
light-light [35] or heavy-light [36] PQχPT decay constants.
The term which is proportional to Nsea comes solely from connected quark-level diagrams,
an example of which is shown in Fig. 1 (where (a) is the meson-level diagram and (b) is
the quark-level diagram).4 The appearance of the quark loop accounts for the factor of
4 By definition, the pion propagator in Fig. 1(a) is connected; the version with a disconnected propagator
is shown in Fig. 2(a).
Nsea. In detail, using Eqs. (A15) and (A16), the loop integrand is proportional to the
connected contraction
ΦijΦji′
, where the index j is repeated because the heavy-
light propagator conserves flavor. Equation (A18) then implies that the sum over j produces
a factor of Nsea when the sea quarks are degenerate. In the non-degenerate case, there is no
factor of Nsea but simply a sum over the sea-quark flavor of the virtual valence-sea pion.
In the staggered case, the internal heavy propagators, Eqs. (A1) and (A2), as well as the
vertices coupling heavy-light mesons to pions (e.g., Eq. (A3)), preserve both flavor and taste.
Therefore Fig. 1 is now simply proportional to
ΦabΦba′
Φiα,jβΦjβ,i′α′
where we have replaced the flavor-taste indices (a, b, . . . ) with separate flavor (i, j, . . . ) and
taste (α, β, . . . ) indices. From Eq. (A8), the loop integrand is then proportional to
iδii′δαα′
p2 −m2ij,Ξ + iǫ
, (19)
where the δαα′ factor shows that, despite the existence of taste violations, the loop preserves
the taste of the light quark in the heavy-light meson and is independent of that taste.
The overall factor of the SχPT diagram must be such as to reproduce the continuum
result in the a → 0 limit. Since pions come in 16 tastes, the sum over pion tastes Ξ in
Eq. (19) must come with a factor of 1/16 compared to the continuum expression. To see
this explicitly, note first that there are two factors of 1/2 relative to the continuum coming
from the vertices (compare Eqs. (A3) and (A16)), due to the non-standard normalization of
the taste generators, Eq. (6). An additional factor of 1/4 comes from the SχPT procedure
for taking into account the fourth root of the staggered determinant: This is a diagram with
a single sea quark loop.
Finally, we need to consider how such a diagram depends on the tastes α and β of the
heavy-light meson and the current. Since the taste indices flow trivially through the heavy-
light lines and vertices, and, as we have seen, through the loops, the taste dependence is
simply (ξ5/2)αβ, where the ξ5 comes from the outgoing light meson. The factor of 1/2 is due
to the normalization of the taste generators.
The net result is that terms with factors of Nsea in the continuum calculation of Ref. [33]
are converted to SχPT by the rule:
NseaF(m2M)→
(ξ5)αβ
F(m2fz,Ξ) (20)
where the sum over f is over the sea quark flavors, z is the valence flavor flowing through
the loop (either x or y), mM is the common mass of the Nsea mesons made up of a z valence
quark and the degenerate sea quarks, and F is some function of the pion masses. (For
heavy-light quantities, F is often also a function of the pion energy in the heavy-light rest
frame.) The masses of pions of various tastes and flavors (mfz,Ξ) are given in Eq. (A7).
The terms that are proportional to 1/Nsea are more subtle. They arise from diagrams with
disconnected pion propagators. The simplest example is shown at the meson level in Fig. 2(a)
and at the quark level in Fig. 2(b). The continuum form of the disconnected propagator is
given in Eq. (A21). Using the continuum values δ′ = m20/3 and m
η ≈ Nseam20/3, we see that
the disconnected propagator produces an overall factor of 1/Nsea as m0 → ∞. Equation
(A21) can then be written as a sum of residues times poles, where the residues can be rather
complicated when the sea masses are non-degenerate (see Appendix B). Thus, the final
answer after integration amounts to something of the form
R̂jF̃(m2j) , (21)
where F̃ is again a general function resulting from the loop integral, R̂j is the residue of
the pole at q2 = m2j , and j ranges over the flavor-neutral mesons involved: the sea mesons,
π0, η, . . . , and the “external” mesons in the disconnected propagator, called ii and i
′i′ in
Eq. (A21). When mi′i′ = mii, there is a double pole, and Eq. (21) should be replaced by
∂m2ii
R̂jF̃(m2j )
, (22)
where the sum over j now does not include i′i′.
When the sea quarks are degenerate, the residues simplify considerably. However, by
comparing the general forms in Eqs. (21) and (22) to the rather simple terms in Ref. [33], it
is easy to move backwards from the degenerate case and determine the form of the expressions
for non-degenerate sea quarks.
The flavor structure in the staggered case is identical to that in the continuum: Flavor
remains a good quantum number, so meson propagators in both cases can only be discon-
nected if they are flavor neutral. Because of taste violations, however, disconnected hairpin
diagrams can contribute to mesons propagators with three different tastes (singlet, vector,
and axial vector) at this order in SχPT. These three hairpin contributions are quite similar
to each other, but there are a few important differences:
• The strength of the hairpin, δ′Ξ, depends on the taste Ξ — see Eq. (A9).
• In the taste-singlet case, as in the continuum, the hairpin (4m20/3) comes from the
anomaly and makes the flavor-singlet meson heavy. Decoupling the η′I by taking
m20 → ∞ is therefore a good approximation, and we do it throughout, giving rise to
an overall factor of 1/Nsea. But in the taste-vector and taste-axial-vector cases, the
hairpins are not particularly large; indeed they are taste-violating effects that vanish
like a2 (up to logarithms) as a→ 0. So we cannot decouple the corresponding mesons,
η′V and η
A, in the taste-vector and taste-axial-vector channels.
• The taste matrices associated with the vector and axial-vector mesons, ξµ and ξµ5,
anticommute with the ξ5 coming from the outgoing Goldstone pion. Therefore the
vector and axial hairpin contributions will have an opposite sign from the singlet (and
continuum) contribution if the ξ5 needs to be pushed past a ξµ or ξµ5 to contract with
the external pion state.
Figure 3 shows the tree-level diagrams that contribute to the form factors, while Figs. 4
and 5 show all the non-vanishing one-loop diagrams. As a first example of the treatment
of diagrams with disconnected meson propagators, consider Fig. 5(b). It is not hard to see
that this diagram has only a disconnected contribution, shown as a quark-flow diagram in
Fig. 6. A connected contribution would require the contraction of the external light quark
fields x and y, which make up the outgoing pion. That is impossible since we have chosen
x 6= y.5
In our notation, the result from Ref. [33] for this diagram in the continuum partially
quenched case with Nsea degenerate sea quarks is:
(4πf)2
m2Y −m2U
m2Y −m2X
J sub1 (mY , v·p)−
m2X −m2U
m2Y −m2X
J sub1 (mX , v·p)
, (23)
where mU is the mass of any of the mesons made up of a sea quark and a sea anti-quark,
mX and mY are the masses of the flavor-neutral mesons made up of xx̄ and yȳ quarks,
respectively, and the function J sub1 , defined below in Eq. (51), is the result of the momentum
integral.
5 A similar argument will be given in more detail below when discussing Figs. 7 and 8.
The ratios of mass differences in Eq. (23) can be recognized as the residue functions (see
Appendix B) for the various poles. For example, (m2Y −m2U)/(m2Y −m2X) is the residue for
the pole at q2 = m2Y . These residues are rather simple in this case because of the degeneracy
of the sea quarks. To generalize Eq. (23) to the completely non-degenerate case, we simply
need to replace the residues by their general expressions. ForNsea non-degenerate sea quarks,
Eq. (23) is replaced by
(4πf)2
[Nsea+1,Nsea]
1 (mj , v·p)
, (24)
where the Minkowski residues R̂
[n,k]
j are defined in Eq. (B2), and the sum over j is over the
Nsea+1 mesons that make up the denominator masses in the disconnected propagator after
m20 → ∞. (See Eq. (A21) and the discussion following it.) We leave implicit, for now, the
arguments to the residues in Eq. (24); we will be more explicit in the final results below. In
addition, we will ultimately express everything in terms of Euclidean-space residues R
[n,k]
Eq. (B4), simply because those are what have been defined and used previously [13, 19].
Cases with double poles present no additional problems, since Ref. [33] shows these
explicitly as derivatives with respect to squared masses of the results of single-pole integrals.
We will therefore simply get derivatives of the usual residues, as in Eq. (B3).
As discussed above, we will need the expression before the m20 →∞ limit is taken in order
to generalize the result to the disconnected taste-vector and axial-vector cases. Equation
(A21) and the fact that m2η′ ≈ Nseam20/3 for large m0 allow us to rewrite Eq. (24) as
(4πf)2
[Nsea+2,Nsea]
1 (mj , v·p)
. (25)
The sum over j now includes the η′. The sign difference between Eqs. (24) and (25) comes
from the sign of the mass term in the Minkowski-space η′ propagator.
Generalizing Eq. (25) to the staggered case is then straightforward. For the taste-singlet
hairpin contributions, we simply replace each continuum pion mass by the mass of the
corresponding taste-singlet pion. In other words, we just let mj → mj,I in Eq. (25). Note
that, after the staggered fourth root is properly taken into account, the taste-singlet η′ mass
goes like Nseam
0/3 for large m0, as it does in the continuum, so one could reverse the process
that led to Eq. (25) and use instead Eq. (24) or even Eq. (23) (for degenerate sea-quarks),
with mj → mj,I in both cases. Just as for diagrams with connected pion propagators (see
Eq. (20)), there is also a trivial overall factor of (ξ5)αβ/2, where α and β are the tastes of
the heavy-light meson and the current, respectively, and the ξ5 is due to the pseudoscalar
(Goldstone) taste of the outgoing pion.
For the taste-vector and axial-vector disconnected contributions, a little more work is
required. We first note that the factor of m20/3 in Eq. (25) is simply δ
Ξ/4 with Ξ = I,
the strength of the taste-singlet hairpin, Eq. (A9).6 For the other tastes we then replace
δ′Ξ by the appropriate hairpin strength from Eq. (A9) and also replace the pion masses:
mj → mj,Ξ. In addition, there is an overall sign change for this diagram in going from the
singlet to the vector or axial-vector tastes. This comes from the fact that the outgoing pion
line in Fig. 5(b) lies between the two ends of the disconnected propagator. Using Eq. (A13)
and the Feynman rules for the heavy-light propagators and vertices in Appendix A, one sees
that the diagram with a taste-Ξ disconnected propagator goes like
TΞ ξ5 TΞ
. This leads
to a positive sign for Ξ = I but a negative sign for tastes that anticommute with ξ5. Finally,
the fact that there are four degenerate taste-vector (or axial-vector) pions at this order leads
to an additional overall factor of four.
When we attempt to apply the same procedure to the other diagrams in Figs. 4 and 5,
we find a further complication in diagrams Fig. 4(a), Fig. 4(b), and Fig. 5(c), where the
external pion and one or more internal pions emerge from the same vertex. The problem is
that the ordering of the taste matrices at the vertex is not determined by the meson-level
diagram (i.e., each diagram can correspond to several orderings), so we do not immediately
know the relative sign of taste-vector and axial-vector contributions relative to the singlet
contribution. Nevertheless, a quark-flow analysis allows us to identify appropriate “flags”
that signal which terms in Ref. [33] come from which orderings at the vertex.
As an example of the procedure in this case, consider Fig. 5(c). The corresponding
quark flow diagrams with disconnected pion propagators are shown Fig. 7. In Fig. 7(a),
the outgoing pion lies between the two ends of the disconnected propagator. This produces
a change in sign of the taste-vector and axial-vector hairpin contributions relative to the
taste-singlet one, just as for Fig. 5(b). In Fig. 7(b), on the other hand, the outgoing pion
is emitted outside the disconnected propagator, and all the hairpin contributions have the
6 The factor of 1/4 just comes from the different conventional normalization of the generators in the con-
tinuum and staggered cases; see Appendix A for further discussion of normalization issues.
same sign. The same is true of the reflected version of Fig. 7(b), which has the outgoing
pion emerging from the other side of the vertex.
Fortunately, Figs. 7(a) and (b) are distinguished by their flavor structure, even in the
continuum. In Fig. 7(a), the two “external” mesons in the disconnected propagator have
different flavors: The one on the left is an X meson (an xx̄ bound state); while the one on the
right is a Y meson (a yȳ bound state). In Fig. 7(b), both external mesons in the disconnected
propagator are Y mesons. Similarly, the reflected version of Fig. 7(b) has two X mesons in
the disconnected propagator. This flavor structure is immediately apparent in the results of
Ref. [33]. The parts of Fig. 5(c) that come from the quark flow of Fig. 7(a) are proportional
to the function called H1, which depends on the masses mX and mY (in our notation),
as well as the sea-meson mass. The parts of Fig. 5(c) that come from the quark flow of
Fig. 7(b) (or its reflected version) are proportional to the function called G1, which depends
only on the mass mY (or mX) and the sea-meson mass. To generalize the results of Ref. [33]
to the staggered case, we thus can use the method outlined above, and simply include an
extra minus sign for those taste-vector and axial-vector hairpin contributions proportional
to H1 (relative to the taste-singlet contributions), but not for those proportional to G1. This
approach also works for the other problematic diagrams, Figs. 4(a) and (b).
The reader may wonder why the complication associated with ordering the taste matrices
at the vertices does not occur when the internal pion propagator is connected, but only in
the disconnected case. Figure 8 shows possible quark-flow diagrams for Fig. 5(c) with a
connected pion propagator. Figure 8(a) cannot occur in our case because we have assumed
that x, the light flavor of the heavy-light meson, is different from y, the light flavor of the
weak current.7 The same reasoning is what allows us to rule out any connected contributions
to Fig. 5(b), as mentioned above. Thus all contributions with connected propagators are of
the type shown in Fig. 8(b), or its reflected version, and these never have a sign difference
between terms with different internal pion tastes.
We note that one can reproduce the SχPT results for light-light [13] and heavy-light
[19] mesons by starting from the continuum PQχPT in Refs. [35] or [36], respectively, and
following the procedure described above. The computations are in fact slightly more difficult
in those cases than in the one at hand, because Refs. [35] and [36] do not explicitly separate
7 Equivalently, we have assumed that the outgoing pion is flavor charged.
double-pole from single-pole contributions. It is therefore takes a little work to express the
answers from those references in the form of our residue functions, which is the necessary
first step before generalizing to the staggered case.
IV. FORM FACTORS FOR B → P DECAY
The standard form factor decomposition for the matrix element between a Bx meson and
a Pxy meson is
〈Pxy(p)|ȳγµb|Bx(pB)〉 =
(pB + p)µ − qµ
m2Bx −m
m2Bx −m
qµF0(q
2) , (26)
where q = pB − p is the momentum transfer. We are suppressing taste indices everywhere,
but emphasize that the light pseudoscalar Pxy is assumed to be the Goldstone meson (taste
ξ5). In the heavy quark limit, it is more convenient to write this in terms of form factors
which are independent of the heavy meson mass
〈Pxy(p)|ȳγµb|Bx(v)〉HQET = [pµ − (v·p)vµ] fp(v·p) + vµfv(v·p) , (27)
where v is the four-velocity of the heavy quark, and v·p is the energy of the pion in the
heavy meson rest frame. Recall that the QCD heavy meson state and the HQET heavy
meson state are related by
|B(pB)〉QCD =
mB|B(v)〉HQET . (28)
The form factors fp and and fv are often called f⊥ and f‖, respectively. As discussed in
Sec. III, the taste indices are left implicit in Eqs. (26) and (27), as are the trivial overall
factors of (ξ5/2)αβ in the matrix elements.
The tree-level diagrams for Bx → Pxy are shown in Fig. 3. Figure 3(a) is the tree-level
“point” contribution to fv, while Fig. 3(b) is the tree-level “pole” contribution to fp. We
f treev (v·p) =
, f treep (v·p) =
v·p+∆∗
, (29)
where ∆∗ = mB∗ − mB is the mass difference of the vector and pseudoscalar heavy-light
meson masses at leading order in the chiral expansion, i.e., neglecting all effects of light-
quark masses. As in Refs. [33, 40], we drop this splitting inside loops, but keep it in the
internal B∗ line in the tree-level diagram Fig. 3(b). This forces the tree-level pole in fp to be
atmB∗ , the physical point. Dropping ∆
∗ inside loops is consistent at leading order in HQET,
which is the order to which we are working. It would also be consistent, parametrically, to
drop ∆∗ everywhere. But this would not be convenient, since the mB∗ pole is physically
important for fp.
The non-zero diagrams that correct the form factors to one loop are shown in Fig. 4 for
fv and Fig. 5 for fp. Table I lists the correspondences between these diagrams and those
of Ref. [33]. A number of other diagrams, which can arise in principle, vanish identically
due to the transverse nature of the B∗ propagator, Eq. (A2); these additional diagrams can
be found in Ref. [33]. We do not indicate hairpin vertices explicitly in Figs. 4 and 5; the
internal pion propagators in these diagrams may be either connected or disconnected.
Before generalizing the results in Ref. [33] to SχPT, we discuss a subtle issue that affects
Fig. 5(a). If the splitting ∆∗ is dropped on internal B∗ lines in loop diagrams, as is done
in Ref. [33], this diagram has a spurious singularity (a double pole) at v · p = 0, the edge
of the physical region. The singularity arises from the presence of the two B∗ lines that
are not inside the loop integral and therefore can be on mass shell in the absence of B∗-B
splitting. Including ∆∗ on all such internal “on-shell” B∗ lines (i.e., lines not inside the loops
themselves), as is done in Ref. [40], at least pushes the unnatural double pole out of the
physical region. We will follow this prescription for including the splitting, but take it one
step further. The loop in Fig. 5(a) is a self-energy correction on the internal B∗ line. The
double pole results from not iterating the self-energy and summing the geometric series.
We will follow the more natural course and sum the series; doing so restores a standard
single-pole singularity.
There is a further one-loop contribution that can naturally be included in Fig. 5(a). The
corresponding tree-level graph, Fig. 3(b), gets two kinds of corrections that are not shown in
Fig. 5. One comes simply from the wavefunction renormalizations on the external pion and
B lines; we include those terms explicitly below. The second contribution arises from the
one-loop shift in the external meson mass. Since this mass shift depends on the flavor of the
light quark in the Bx, namely x, we call it δMx = Σx(v ·k = 0), with Σx(v ·k) the self-energy
for Bx or B
x. Note that Σx is the same for both the Bx and the B
x, since the splitting
∆∗ is dropped inside loops. When the external Bx line in Fig. 3(b) is put on mass-shell
at one loop, the denominator of the internal B∗y propagator changes from −2(v·p + ∆
∗) to
−2(v·p+∆∗ − δMx). It is convenient to define ∆∗yx as the full splitting between a B∗y and a
yx ≡MB∗y −MBx = ∆
+ δMy − δMx (30)
The internal B∗y propagator now becomes −2(v·p+∆
yx − δMy). The contribution from the
mass shift may then be combined with the tree-level and Fig. 5(a) contributions to give:
f selfp (v·p) =
v·p+∆∗yx +D(v·p)
, (31)
D(v·p) ≡ Σy(v·p)− Σy(0) , (32)
where the subtraction in D comes from the effect of putting the Bx on mass shell, via
Eq. (30).
The main difference between the approach taken to the spurious singularity of Fig. 5(a)
and that of Bećirević et al. [33] is that they work to first order in the self-energy in the
corresponding diagram (their diagram (7)). Expanding Eq. (31), we find that D is related
in the continuum limit to what Ref. [33] calls δf
D(v·p) = −v·p δf (7)p . (33)
Thus we can find the staggered D(v·p) simply by applying the methods of Sec. III to δf (7)p .
We can now write down the expressions for the form factors for Bx → Pxy decay. For the
point form factor, fv, we have
fBx→Pxyv = f
1 + δfBx→Pxyv + c
xmx + c
ymy + c
sea(mu +md +ms)
+ cv1(v·p) + cv2(v·p)2 + cvaa2
, (34)
where f treev is given by Eq. (29), and the analytic coefficients c
y, . . . arise from next-to-
leading order (NLO) terms in the heavy-light chiral Lagrangian (see Sec. IVC). The non-
analytic pieces, which come from the diagrams shown in Fig. 4 as well as the wavefunction
renormalizations, are included in δf
Bx→Pxy
δfBx→Pxyv = δf
v + δf
δZBx +
δZPxy . (35)
The wavefunction renormalization terms, δZBx and δZPxy , have been calculated previously
[13, 19] in SχPT and are listed in Appendix C.
For the fp form factor, we write
fBx→Pxyp = f
p + f̃
δfBx→Pxyp + c
xmx + c
ymy + c
sea(mu +md +ms)
1(v·p) + c
2(v·p)2 + cpaa2
. (36)
where f selfp is defined in Eq. (31), and
f̃ treep (v·p) ≡
v·p+∆∗xy
. (37)
Non-analytic contributions are summarized in the function D(v·p) in f selfp , Eq. (32), and
Bx→Pxy
p , which comes from Figs. 5(b)-(d) and wavefunction renormalizations. Explicitly,
δfBx→Pxyp = δf
p + δf
p + δf
δZBx +
δZPxy . (38)
For simplicity, we do not include the superscript Bx → Pxy on the individual diagrams in
Eqs. (35) and (38).
Using f̃ treep , which includes the full B
y–Bx splitting ∆
yx, rather than f
p , Eq. (29),
changes Eq. (36) only by higher-order terms. However, it is convenient to keep the same
splitting in both f selfp and the other terms in Eq. (36). Note that it is also consistent at this
order to use the alternative form
fBx→Pxyp = f
1 + δfBx→Pxyp + c
xmx + c
ymy + c
sea(mu +md +ms)
1(v·p) + c
2(v·p)2 + cpaa2
, (39)
The analytic terms in fv and fp are not all independent. As mentioned in Sec. II, there
is one relation among the terms that control the valence mass dependence:
cpx + c
x = c
y + c
y (40)
We show that this relation follows from the higher order terms in the Lagrangian and current
in Sec. IVC. All other NLO parameters in Eqs. (34) and (36) are independent.
A. Form factors for 3-flavor partially quenched SχPT
First we display the results for the individual diagrams shown in Figs. 4 and 5 for the
fully non-degenerate case with three dynamical flavors (the “1+1+1” case). This means
that we have already taken into account the transition from 4 to 1 tastes per flavor. Indeed,
our method of generalizing the partially quenched continuum expressions to the staggered
case automatically includes this adjustment. We detail below the minor changes needed to
obtain 2+1 results from those in the 1+1+1 case.
We first define sets of masses which appear in the numerators and denominators of the
disconnected propagators with taste labels implicit (see Appendix B):
µ(3) = {m2U , m2D, m2S} , (41)
M(3,x) = {m2X , m2π0 , m2η} , (42)
M(4,x) = {m2X , m2π0 , m2η, m2η′} , (43)
M(4,xy) = {m2X , m2Y , m2π0, m2η} , (44)
M(5,xy) = {m2X , m2Y , m2π0, m2η, m2η′} . (45)
For the mass sets (42) and (43), there are also corresponding sets with x→ y and X → Y .
When we show explicit taste subscripts such as I or V on the mass sets µ orM, it means
that all the masses in the set have that taste.
The functions that appear in the form factors are8
I1(m) = m
, (46)
I2(m,∆) = −2∆2 ln
− 4∆2F
+ 2∆2 , (47)
J1(m,∆) =
−m2 + 2
(∆2 −m2)F
m2 , (48)
F (x) =
1− x2 tanh−1
1− x2
, 0 ≤ x ≤ 1
x2 − 1 tan−1
x2 − 1
, x ≥ 1 .
The main difference in these formulae with those of Bećirević et al. [33] is that they keep
the divergence pieces, while we have renormalized as in Refs. [13, 19]. To convert to our
form, replace the MS scale µ in Ref. [33] with the chiral scale Λ and set their quantity ∆̄ to
zero, where
∆̄ ≡ 2
− γ + ln(4π) + 1 , (50)
8 For ease of comparison to Ref. [33], we use I1(m) instead of ℓ(m
2) (as in Refs. [13, 19]) for the chiral
logarithm.
with d the number of dimensions. F (x) is only needed for positive x; so we use the simpler
form given in Ref. [40], rather than the more general version worked out in Ref. [41] and
quoted in Ref. [33]. We do not list the function J2, which appears in the integral J µν of
Eq. (B8) but does not enter the final answers.
We also define a “subtracted” J1 function by
J sub1 (m,∆) ≡ J1(m,∆)−
. (51)
The subtraction term cancels the singularity when ∆→ 0. The function J sub1 enters naturally
in the expression for the self energy correction D(v·p) because of the the subtraction in
Eq. (32). It also turns out to arise from the integral in Fig. 5(b) — see Eq. (26) in Ref. [40].
For the point corrections in the 1+1+1 case, we have
δf4(a)v
)Bx→Pxy
1+1+1
2(4πf)2
I1(myf,Ξ) + 2I2(myf,Ξ, v·p)
j∈M(4,xy)
[4,3]
M(4,xy)I ;µ
[I1(mj,I) + 2I2(mj,I , v·p)]
∂m2Y,I
j∈M(3,y)
[3,3]
M(3,y)I ;µ
[I1(mj,I) + 2I2(mj,I , v·p)]
+ a2δ′V
∂m2Y,V
j∈M(4,y)
[4,3]
M(4,y)V ;µ
[I1(mj,V ) + 2I2(mj,V , v·p)]
j∈M(5,xy)
[5,3]
M(5,xy)V ;µ
[I1(mj,V ) + 2I2(mj,V , v·p)]
+ [V → A]
, (52)
δf4(b)v
)Bx→Pxy
1+1+1
= − 1
6(4πf)2
[I1(mxf,Ξ) + I1(myf,Ξ)]
∂m2Y,I
j∈M(3,y)
[3,3]
M(3,y)I ;µ
I1(mj,I)
∂m2X,I
j∈M(3,x)
[3,3]
M(3,x)I ;µ
I1(mj,I)
j∈M(4,xy)
[4,3]
M(4,xy)I ;µ
I1(mj,I)
+ a2δ′V
∂m2Y,V
j∈M(4,y)
[4,3]
M(4,y)V ;µ
I1(mj,V )
∂m2X,V
j∈M(4,x)
[4,3]
M(4,x)V ;µ
I1(mj,V )
j∈M(5,xy)
[5,3]
M(5,xy)V ;µ
I1(mj,V )
+ [V → A]
. (53)
Those that correct the pole form factors are
Bx→Pxy
1+1+1 = −
3g2πv·p
(4πf)2
J sub1 (myf,Ξ, v·p)
j∈M(3,y)
∂m2Y,I
[3,3]
M(3,y)I ;µ
J sub1 (mj,I , v·p)
+ a2δ′V
j∈M(4,y)
∂m2Y,V
[4,3]
M(4,y)V ;µ
J sub1 (mj,V , v·p)
+ [V → A]
, (54)
δf5(b)p
)Bx→Pxy
1+1+1
(4πf)2
j∈M(4,xy)
[4,3]
M(4,xy)I ;µ
J sub1 (mj,I , v·p)
+a2δ′V
j∈M(5,xy)
[5,3]
M(5,xy)V ;µ
J sub1 (mj,V , v·p) + [V → A]
,(55)
δf5(c)p
)Bx→Pxy
1+1+1
6(4πf)2
[I1(mxf,Ξ) + I1(myf,Ξ)]
j∈M(3,y)
∂m2Y,I
[3,3]
M(3,y)I ;µ
I1(mj,I)
j∈M(3,x)
∂m2X,I
[3,3]
M(3,x)I ;µ
I1(mj,I)
j∈M(4,xy)
[4,3]
M(4,xy)I ;µ
I1(mj,I)
+ a2δ′V
j∈M(4,y)
∂m2Y,V
[4,3]
M(4,y)V ;µ
I1(mj,V )
j∈M(4,x)
∂m2X,V
[4,3]
M(4,x)V ;µ
I1(mj,V )
j∈M(5,xy)
[5,3]
M(5,xy)V ;µ
I1(mj,V )
+ [V → A]
, (56)
δf5(d)p
)Bx→Pxy
1+1+1
= − 1
2(4πf)2
I1(myf,Ξ)
j∈M(3,y)
∂m2Y,I
[3,3]
M(3,y)I ;µ
I1(mj,I)
+a2δ′V
j∈M(4,y)
∂m2Y,V
[4,3]
M(4,y)V ;µ
I1(mj,V )
+ [V → A]
.(57)
In Eqs. (52) through (57), the explicit factors of 1/3 in front of terms involving the taste-
singlet (I) mesons come from the factors of 1/Nsea in Ref. [33].
To get the full corrections for both fv and fp, we need to add in the wavefunction renor-
malizations, given in Appendix C in Eqs. (35) and (38). Putting these together with the
analytic terms and (for fp) theD term, Eqs. (34) and (36) give the complete NLO expressions
for the form factors in SχPT.
The above 1+1+1 results are expressed in terms of the Euclidean residue functions R
[n,k]
Eq. (B4). In the 2+1 case, there is a cancellation in the residues between the contribution
of the U or D in the numerator and that of the π0 in the denominator. Thus, to obtain the
2+1 from the 1+1+1 case, one must simply reduce by one all superscripts on the residues,
i.e., R[n,k] → R[n−1,k−1], and remove mπ0 and (say) mD from the mass sets:
µ(3) → {m2U , m2S} , (58)
M(3,x) → {m2X , m2η} , (59)
M(4,x) → {m2X , m2η, m2η′} , (60)
M(4,xy) → {m2X , m2Y , m2η} , (61)
M(5,xy) → {m2X , m2Y , m2η, m2η′} . (62)
We also write here the expressions for three non-degenerate dynamical flavors in contin-
uum PQχPT, which to our knowledge do not appear in the literature. These expressions
can be obtained either by returning to Ref. [33] and using the residue functions to generalize
to the non-degenerate case, or simply by taking the continuum limit of the above equations.
Either way, the results for fv are
δf4(a),contv
)Bx→Pxy
2(4πf)2
I1(myf) + 2I2(myf , v·p)
j∈M(4,xy)
[4,3]
M(4,xy);µ(3)
[I1(mj) + 2I2(mj , v·p)]
j∈M(3,y)
[3,3]
M(3,y);µ(3)
[I1(mj) + 2I2(mj , v·p)]
δf4(b),contv
)Bx→Pxy
= − 1
6(4πf)2
[I1(mxf) + I1(myf )]
j∈M(3,y)
[3,3]
M(3,y);µ(3)
I1(mj)
j∈M(3,x)
[3,3]
M(3,x);µ(3)
I1(mj)
j∈M(4,xy)
[4,3]
M(4,xy);µ(3)
I1(mj)
, (63)
while those for fp are
Dcont
)Bx→Pxy
= − 3g
(4πf)2
J sub1 (myf , v·p)
j∈M(3,y)
[3,3]
M(3,y);µ(3)
J sub1 (mj , v·p)
δf5(b),contp
)Bx→Pxy
(4πf)2
j∈M(4,xy)
[4,3]
M(4,xy);µ(3)
J sub1 (mj , v·p)
δf5(c),contp
)Bx→Pxy
6(4πf)2
[I1(mxf) + I1(myf )]
j∈M(3,y)
[3,3]
M(3,y);µ(3)
I1(mj)
j∈M(3,x)
[3,3]
M(3,x);µ(3)
I1(mj)
j∈M(4,xy)
[4,3]
M(4,xy);µ(3)
I1(mj)
δf5(d),contp
)Bx→Pxy
2(4πf)2
I1(myf )
j∈M(3,y)
[3,3]
M(3,y);µ(3)
I1(mj)
. (64)
Corresponding continuum-limit results for the wave-function renormalizations are given in
Appendix C.
B. Full QCD Results
Adding together the complete results for the “full QCD” case is straightforward. For
simplicity, we specialize to case mu = md (i.e., 2 + 1). For B → π, the complete corrections
(including wave-function renormalization) are:
DB→π = − 3g
(4πf)2
2J sub1 (mπ,Ξ, v·p) + J sub1 (mK,Ξ, v·p)
J sub1 (mπ,I , v·p) +
J sub1 (mη,I , v·p)
j∈{π,η,η′}
(−a2δ′V )R
[3,1]
j ({mπ,V , mη,V , mη′,V }; {mS,V }) J sub1 (mj,V , v·p)
V → A
, (65)
δfB→πp =
(4πf)2
1 + 3g2π
[2I1(mπ,Ξ) + I1(mK,Ξ)]
1 (mπ,I , v·p) +
1 (mη,I , v·p) +
1 + 3g2π
3I1(mπ,I)− I1(mη,I)
j∈{π,η,η′}
a2δ′VR
[3,1]
j ({mπ,V , mη,V , mη′,V }; {mS,V })
1 (mj,V , v·p) +
1 + 3g2π
I1(mj,V )
+ [V → A]
, (66)
δfB→πv =
(4πf)2
1− 3g2π
[2I1(mπ,Ξ) + I1(mK,Ξ)]
+ 2I2(mπ,Ξ, v·p) + I2(mK,Ξ, v·p)
1 + 3g2π
I1(mπ,I)−
I1(mη,I)
j∈{π,η,η′}
a2δ′VR
[3,1]
j ({mπ,V , mη,V , mη′,V }; {mS,V })
3(g2π − 1)
I1(mj,V )− 2I2(mj,V , v·p)
+ [V → A]
. (67)
For B → K,9 we have
DB→K = −
3g2π(v·p)
(4πf)2
2J sub1 (mK,Ξ, v·p) + J sub1 (mS,Ξ, v·p)
J sub1 (mη,I , v·p)− J sub1 (mS,I , v·p)
j∈{S,η,η′}
(−a2δ′V )R
[3,1]
j ({mS,V , mη,V , mη′,V }; {mπ,V })J sub1 (mj,V , v·p)
V → A
, (68)
δfB→Kp =
(4πf)2
−2 + 3g
I1(mK,Ξ)−
I1(mS,Ξ)− 3g2πI1(mπ,Ξ)
1 (mη,I , v·p) +
I1(mπ,I)−
4 + 3g2π
I1(mη,I) +
I1(mS,I)
+ a2δ′V
−m2η,V
J sub1 (mη,V , v·p)− J sub1 (mη′,V , v·p)
j∈{π,η,η′}
[3,1]
j ({mπ,V , mη,V , mη′,V }; {mS,V }) I1(mj,V )
j∈{S,η,η′}
[3,1]
j ({mS,V , mη,V , mη′,V }; {mπ,V }) I1(mj,V )
+ [V → A]
, (69)
9 The transition B → K occurs through penguin diagrams; D → K is a standard semileptonic decay due
to the current in Eq. (18). We keep the notation B → K however to stress that we are working to lowest
order in the heavy quark mass.
δfB→Kv =
(4πf)2
2− 3g2π
I1(mK,Ξ)− 3g2πI1(mπ,Ξ) +
I1(mS,Ξ)
+ 2I2(mK,Ξ, v·p) + I2(mS,Ξ, v·p)
I1(mS,I) +
I1(mπ,I) +
8− 3g2π
I1(mη,I) + I2(mη,I , v·p)− I2(mS,I , v·p)
+ a2δ′V
I1(mη′,V )− I1(mη,V ) + I2(mη′,V , v·p)− I2(mη,V , v·p)
−m2η,V
j∈{S,η,η′}
[3,1]
j ({mS,V , mη,V , mη′,V }; {mπ,V })
I1(mj,V ) + I2(mj,V , v·p)
j∈{π,η,η′}
[3,1]
j ({mπ,V , mη,V , mη′,V }; {mS,V }) I1(mj,V )
+ [V → A]
. (70)
C. Analytic terms
From the power counting discussed in Sec. II, as well as interchange symmetry among the
sea quark masses, the form factors at the order we are working can only depend only on the
valence quark masses mx and my, the sum of the sea quark masses mu +md +ms, the pion
momentum (through v·p), and the lattice spacing, a. The last must appear quadratically,
since the errors of the staggered action are O(a2). Recall that we do not include any
discretization errors coming from the heavy quark in our effective theory.
Thus we expect to have the analytic terms shown in Eqs. (34) and (36) with coefficients
i and c
i . (Here i = {x, y, sea, 1, 2, a}.) We then can examine, one by one, the known NLO
terms in the Lagrangian and current to check for the existence of relations among the c
and/or cvi . As soon as a sufficient number of terms are checked to ensure that the parameters
are independent, we are done. It is therefore not necessary in all cases to have a complete
catalog of NLO terms. Unless otherwise indicated, all NLO terms discussed in this section
come from Ref. [19].
Note first of all that we do not need to include explicitly the effects of mass-
renormalization terms in the NLO heavy-light Lagrangian, such as
2λ1Tr
+ 2λ′1Tr
, (71)
where we define
M± = 1
σMσ ± σ†Mσ†
. (72)
The effect of the terms in Eq. (71) is absorbed into the B∗y -Bx mass difference ∆
yx, Eq. (30),
just like the one-loop contribution to the mass. Corresponding O(a2) term in the Lagrangian,
which can be obtained by replacing M+ above by various taste-violating operators, can
likewise be ignored here.
We now consider the discretization corrections parametrized by cpa and c
a. There are a
large number of O(a2) corrections to the Lagrangian and the current that can contribute to
these coefficients, so it is not surprising that they are independent. For example, consider
the following terms in the NLO heavy-light Lagrangian
cA3,k Tr
HHγµγ5{Aµ,OA,+k }
, (73)
where the OA,+k are various taste-violating operators, similar to those in Eq. (16) above.
These terms do not contribute to cva, but only to c
a, though corrections to the B-B
vertex in Fig. 3(b). On the other hand, there are many terms that contribute both to cva
and to cpa. An example is the following correction to the current
rA1,k trD (γ
µ(1− γ5)H)OA,+k σ
†λ(b) , (74)
which contributes equally to cva and c
a. Additional examples are provided by those terms
with two derivatives in the O(mqa2) pion Lagrangian [39], which correct both coefficients
though their effect on the pion wave-function renormalization.
We consider the v·p and (v·p)2 terms next, namely cv1, cv2, c
1, and c
2. This is a case
where a complete catalog of Lagrangian and current corrections does not exist. However, it
is easy to find corrections that contribute only to fv or only to fp. As in the previous case,
corrections to the B-B∗-π vertex in Fig. 3(b) only affect fp at the order we are working.
Thus,
(v · →DHH −HHv · ←D) γµγ5Aµ
contributes to c
1 only; while
HHγµγ5(v ·
D )2Aµ
contributes to c
2 only. Similarly, only fv is affected, though Fig. 3(a), by any correction to
the current whose expansion in terms of pion fields starts at linear order (i.e., corrections of
schematic form H( iΦ
+ · · · ), with · · · denoting higher order terms in Φ). Thus,
γµ (1−γ5)H
v · A σ†λ(b) (77)
contributes to cv1 only; while
γµ (1−γ5)H
v · →Dv · A σ†λ(b) (78)
contributes to cv2 only. Since there is at least one Lagrangian or current term that contributes
to each of cv1, c
1, and c
2 exclusively, these coefficients are independent.
The argument for the independence of the sea-quark mass terms, i.e., the coefficients cvsea
and cpsea, is similar. The Lagrangian correction
HHγµγ5A
Tr(M+) (79)
contributes to cpsea only; while the current correction
ρ2 trD (γ
µ(1− γ5)H)σ†λ(b)Tr(M+) (80)
contributes equally to both cpsea and c
sea. These two observations are enough to guarantee
that cvsea and c
sea are independent.
We now turn to the coefficients that control the valence quark mass dependence of the
form factors: cvx, c
x, and c
y. At first glance, it would seem unlikely that there could
be any constraint among these parameters, since there are seven terms in the Lagrangian
and current in Ref. [19] that could generate valence mass dependence.10 However, three of
these terms are immediately eliminated, either because they could only contribute to flavor-
neutral pions (with x = y), or because they produce no fewer than two pions. There are
then two remaining corrections to the heavy-light Lagrangian,
ik1Tr
HHv·←DM+ − v·→DHHM+
+ k3Tr
HHγµγ5{Aµ,M+}
and two corrections to the current,
ρ1 trD (γ
µ(1− γ5)H)M+σ†λ(b) + ρ3 trD (γµ(1− γ5)H)M−σ†λ(b) (82)
10 There are additional terms involving Tr(M+), as in Eqs. (79) and (80), that only give sea quark mass
dependence at this order.
The k3 term in Eq. (81) contributes only to fp, through the B-B
∗-π vertex. However,
because of the anticommutator, its contribution is proportional to mx+my, so it gives equal
contributions to cpx and c
y. Similarly, because the one-pion term in M− is proportional to
ΦM +MΦ, the ρ3 term contributes equally to cvx and cvy (but not at all to cpx and cpy).
Further, sinceM+ creates only even number of pions, we can replace it byM in Eq. (82).
The ρ1 term can then easily be seen to contribute equally to c
y and c
x, since the current
needs to annihilate a B∗y in the fp case and a Bx in the fv case.
The contributions of the k1 term in Eq. (81) are the most non-trivial. It contributes to
both cvx and c
x through wave function renormalization on the external Bx line, but it also
contributes to cpy through an insertion on the internal B
y line in Fig. 3(b). However, since
wave-function renormalization effects on external lines go like
Z, the contributions of this
term to both cvx and c
x are exactly half of its contribution to c
y. Thus, all four terms in
Eqs. (81) and (82) are consistent with the relation given in Eq. (40).
We still need to worry about valence mass dependence generated by the standard O(p4)
pion Lagrangian [38] through wave function renormalization of the external pion. Such
contributions do exist (from L5), but the x ↔ y symmetry of the pion guarantees they are
proportional to mx +my in both fp and fv, and hence do not violate Eq. (40).
A consistency check of the relation, Eq. (40), as well as of the claimed independence of the
other analytic terms, can be performed by considering the change in the chiral logarithms in
Eqs. (52) through (57) and Eqs. (C1) and (C2) under a change in chiral scale. To simplify
the calculation, it is very convenient to use the conditions obeyed by sums of residues, which
are given in Eq. (38) of the second paper in Ref. [13]. We find that such a scale change can
be absorbed by parameters that obey Eq. (40) but are otherwise independent.
In the continuum limit, cpsea and c
sea remain independent, as do c
1, and c
2. We
disagree on these points with Ref. [33], which found cpsea = c
sea, and did not consider analytic
terms giving v·p dependence. The difference can be traced to the inclusion here of the effects
of the complete set of NLO mass-dependent terms, as well as a sufficient number of higher
derivative terms (Eqs. (75) through (78)). In particular, the independence of cpsea and c
sea can
be traced to the existence of the Lagrangian correction, Eq. (79), which was not considered
in Ref. [33]. On the other hand, the relation among the valence mass coefficients, Eq. (40),
is obeyed by the expressions for these coefficients found in Ref. [33]. This occurs because
the contributions of the terms proportional to k3 and ρ3 in Eqs. (81) and (82), which were
not considered in Ref. [33], are proportional to mx +my and automatically obey Eq. (40).
Note, finally, that the relation in Eq. (40) is almost certain to be violated at next order
in HQET. This is because the contributions from operators like the k1 term in Eq. (81)
will affect the B and the B∗ differently at O(1/mQ), destroying the cancellation that made
Eq. (40) possible.
V. FINITE VOLUME EFFECTS
In a finite volume, we must replace the integrals in Eqs. (B5) through (B8) by discrete
momentum sums. We assume that the time direction is large enough to be considered
infinite (this is the case in MILC simulations), and that each of the spatial lengths has
(dimensionful) size L.
The correction to Eq. (B5) is given explicitly in Ref. [12]. In finite volume, we need only
make the replacement
I1(m)→ I fv1 (m) = I1(m) +m2δ1(mL) . (83)
Here δ1 is a sum over modified Bessel functions
δ1(mL) =
~r 6=0
K1(rmL)
, (84)
where ~r is a 3-vector with integer components, and r ≡ |~r |.
Arndt and Lin [42] have worked out the finite volume correction to Eq. (B6). In our
notation, the function I2(m,∆) is replaced by its finite volume form, I
2 (m,∆),
I2(m,∆)→ I fv2 (m,∆) = I2(m,∆) + δI2(m,∆, L) , (85)
where the correction δI2(m,∆, L) is given simply in terms of the function JFV(m,∆, L)
defined in Eq. (44) of Ref. [42]:11
δI2(m,∆, L) = −(4π)2∆ JFV(m,∆, L)
JFV(m,∆, L) ≡
~r 6=0
ωq(ωq +∆)
sin(qrL)
, (86)
11 We have added the L argument to JFV for consistency with our notation
with ωq =
q2 +m2.
The asymptotic form of JFV(m,∆, L) for large mL is useful for practical applications,
where typically mL > 3, and often mL > 4 [8, 9]. Arndt and Lin have found [42]:
JFV(m,∆, L) =
~r 6=0
e−rmLA , (87)
A = e(z2) [1− Erf(z)] +
1− Erf(z)
1− Erf(z)
(rmL)3
, (88)
where
. (89)
Computing higher orders in the 1/(mL) expansion is possible if greater precision is needed.
Since the functions I1(m) and I2(m,∆) arise from the integral Iµ3 (m,∆) in Eq. (B7), as
well as from Eqs. (B5) and (B6), which serve to define them, it is necessary to check that
the finite volume corrections coming from Eq. (B7) are just those given by Eqs. (83) and
(85) above. This is easily seen to be true in the rest frame of the heavy quark, in which
we are working. It is a consequence of the facts that: (1) in the rest frame, only the µ = 0
component of Iµ3 (m,∆) is non-zero, and (2) the integral over dq0 is unaffected by finite
volume, since we assume large time-extent of the lattices. The finite volume integral then
splits into I fv1 (m) and I
2 (m,∆) pieces, just as in infinite volume.
Finally, we have to examine the finite volume corrections to the integral J µν , Eq. (B8).
Since the function J2(m,∆) does not enter our final results, we need only evaluate
J ≡ (gνµ − vνvµ)J µν =
(2π)4
i(gνµ − vνvµ)qµqν
(v·q −∆+ iǫ)(q2 −m2 + iǫ)
(2π)4
(v·q −∆+ iǫ)(q2 −m2 + iǫ)
(4π)2
J1(m,∆) , (90)
where q is the spatial 3-vector part of qµ. In the last line, the arrow refers to the fact
that the function J1 arises after regularization and renormalization of the integral. A useful
regulator in the present context is given by the insertion of a factor of exp(−ωq/Λ0), where
Λ0 is a cutoff. After performing the contour integral over q
(2π)3
2ωq(ωq +∆)
(2π)3
(2π)3
(2π)3
∆2 −m2
2ωq(ωq +∆)
. (91)
The first term is a pure divergence with no m or ∆ dependence. It is thus the same in
finite volume or infinite volume [12]. The correction to the middle term is proportional to
the correction to I1, since the same integral appears after performing the q
0 integration in
Eq. (B5). Similarly, the integral in the third term is proportional to that arising from the
q0 integration in Eq. (B6), and the correction is therefore already known. We have
J1(m,∆)→ J fv1 (m,∆) = J1(m,∆) + δJ1(m,∆, L) , (92)
where
δJ1(m,∆, L) =
m2 −∆2
δI2(m,∆, L)−
δ1(mL) (93)
The correction to J sub1 , Eq. (51), is
δJ sub1 (m,∆, L) = δJ1(m,∆, L) +
16π2m2
JFV(m, 0, L) , (94)
where JFV(m, 0, L) is the same as Eq. (87) with A = 1.
With the expressions in this section, it is straightforward to incorporate the corrections
to I1, I2, and J1 numerically into fits to finite-volume lattice data.
VI. CONCLUSIONS
We have presented the NLO expressions in partially quenched SχPT for the form factors
associated with B → Pxy semileptonic decays, for both infinite and finite volume. Using
a quark flow analysis, we have obtained these results by generalizing the NLO PQχPT
expressions calculated in the continuum in Ref. [33]. The main subtlety in applying this
technique is due to the appearance of taste matrices inside the Feynman diagrams, since
non-trivial signs can arise from the anticommutation relations of the taste generators. We
have shown that these signs can be accounted for by a careful analysis of the relevant quark
flow diagrams.
The SχPT expressions are generally necessary for performing chiral fits to lattice simu-
lations where staggered light quarks are used. For simpler quantities than the form factors,
SχPT has been seen to be essential [10, 20] in order to get reliable extrapolations both to
the continuum limit and to the physical quark mass values. For form factors, the lattice
data in Ref. [9] was not yet sufficiently precise for the SχPT expressions to be required (over
continuum forms) for acceptable fits. However, we expect that the forms derived here will
become more and more important as the lattice data improves.
Our results are valid to lowest order in HQET; in general, we neglect 1/mB corrections.
We do however include the B∗-B splitting ∆∗ on internal B∗ lines that are not in loops.
This prescription allows the form factor fp to have the physical m
B pole structure. Our
treatment of the B∗-B splitting is similar, but not identical, to that of Refs. [33, 40]. Unlike
those authors, we iterate self-energy contributions, namely Fig. 5(a) and the effect of the
one-loop mass shift of the B, to all orders. This seems to us to be a natural choice, and
also makes the one-loop corrections better behaved. Indeed, with the values of light quark
masses and momenta typically used in staggered simulations [8, 9], the one-loop B mass
shift can dominate other one-loop corrections, so summing such self-energy contributions
to all orders seems entirely appropriate. The final answers are then expressed in terms of
the splitting ∆∗yx ≡ MB∗y −MBx . In fitting lattice data, we suggest using the actual lattice
values of this mass difference (at the simulated light quark mass values and lattice spacings),
rather than applying a one-loop formula for the mass shifts.
Our primary results for the staggered, partially quenched case with three non-degenerate
sea quarks are found in Sec. IVA. The form factor fv (also known as f‖) is given by
Eq. (34) in terms of quantities defined in Eqs. (35), (52) and (53), as well as the wave
function renormalization factors δZPxy and δZBx that are listed in Eqs. (C1) and (C2) of
Appendix C. Similarly, the form factor fp (also known as f⊥), is given by Eq. (36) in terms
of quantities defined in Eqs. (31), (37), (38), and (54) through (57), as well as the wave
function renormalization factors. We have also found a single relation, Eq. (40), among the
parameters that control the analytic valence mass dependence. While this relation is also
satisfied by the parameters written down in Ref. [33], it is important to know that it persists
even in the presence of the complete NLO forms of the Lagrangian and current.
Appropriate limits of our expressions can be taken for various relevant cases, including
the case of full (unquenched) staggered QCD (in Sec. IVB) and the case of continuum
PQχPT with non-degenerate sea quark masses [Eqs. (63), (64), (C3), and (C4)]. Despite
the fact that the latter are continuum results, they have not, to our knowledge, appeared
in the literature before. Finally, our expressions can be corrected for finite volume effects
using the results of Sec. V.
ACKNOWLEDGMENTS
We thank J. Bailey, B. Grinstein, A. Kronfeld, P. Mackenzie, S. Sharpe and our colleagues
in the MILC collaboration for helpful discussions. We also are grateful to D. Lin for dis-
cussions on finite volume corrections and for sharing with us the Mathematica code used to
make the expansions in Ref. [42]. This work was partially supported by the U.S. Department
of Energy under grant numbers DE-FG02-91ER40628 and DE-FG02-92ER40699.
APPENDIX A: FEYNMAN RULES
In this appendix we list the SχPT propagators and (some of) the vertices in Minkowski
space [19], as well as the corresponding continuum versions.
In SχPT, the propagators for the heavy-light mesons are
(k) =
2(v·k + iǫ)
, (A1)
B∗µaB
(k) =
−iδab(gµν − vµvν)
2(v·k −∆∗ + iǫ)
. (A2)
Here a, b indicate the flavor-taste of the light quarks, and ∆∗ is the B∗-B splitting in the
chiral limit, which we often neglect since we work to leading order in HQET.
The BB∗π vertex is:
B†a B
µb −B∗†µa Bb
∂µΦba , (A3)
where repeated indices are summed. Other needed vertices come from the expansion of the
LO current, Eq. (18). We have:
LO = κB
δac −
ΦabΦbc + · · ·
+ κvµBa
Φac + · · ·
, (A4)
where repeated indices are again summed and · · · represents terms involving higher numbers
of pions, as well as contributions from the axial vector part of the current, which are not
relevant to the form factors.
If desired, each flavor-taste index can be replaced by a pair of indices representing flavor
and taste separately. We use Latin indices in the middle of the alphabet (i, j, . . . ) as pure
flavor indices, which take on the values 1, 2, . . . , Nsea in full QCD. Greek indices at the
beginning of the alphabet (α, β, γ, . . . ) are used for quark taste indices, running from 1 to
4. Thus we can replace a→ iα and write, for example,
(k) =
iδijδαβ
2(v·k + iǫ)
. (A5)
As in Refs. [13, 19], pion propagators are treated most easily by dividing them into
connected and disconnected pieces, where the disconnected parts come from insertion (and
iteration) of the hairpin vertices. The connected propagators are
ΦΞijΦ
(p) =
iδii′δjj′δΞΞ′
p2 −m2ij,Ξ + iǫ
, (A6)
where Ξ is one of the 16 meson tastes [as defined after Eq. (5)], and mij,Ξ is the tree-level
mass of a taste-Ξ meson composed of quarks of flavor i and j:
m2ij,Ξ = µ(mi +mj) + a
2∆Ξ. (A7)
Here ∆Ξ is the taste splitting, which can be expressed in terms of C1, C3, C4 and C6 in
Eq. (16) [13]. There is a residual SO(4) taste symmetry [11] at this order, implying that
the mesons within a given taste multiplet (P , V , T , A, or I) are degenerate in mass. We
therefore usually use the multiplet label to represent the splittings.
Since the heavy-light propagators are most simply written with flavor-taste indices, as in
Eqs. (A1) and (A2), it is convenient to rewrite Eq. (A6) in flavor-taste notation also:
ΦabΦb′a′
(p) ≡
Φiα,jβΦj′β′,i′α′
(p) =
iδii′δjj′T
p2 −m2ij,Ξ + iǫ
, (A8)
where TΞ are the 16 taste generators, Eq. (6).
For flavor-charged pions (i 6= j), the complete propagators are just the connected propa-
gators in Eq. (A6) or (A8). However, for flavor-neutral pions (i = j), there are disconnected
contributions coming from one or more hairpin insertions. At LO, these appear only for taste
singlet, vector, or axial-vector pions. Denoting the Minkowski hairpin vertices as −iδ′Ξ, we
have [13]:
δ′Ξ =
a2δ′V , TΞ ∈ {ξµ} (taste vector);
a2δ′A, TΞ ∈ {ξµ5} (taste axial-vector);
4m20/3, TΞ = ξI (taste singlet);
0, TΞ ∈ {ξµν , ξ5} (taste tensor or pseudoscalar)
δ′V (A) ≡
(C2V (A) − C5V (A)) . (A10)
The disconnected pion propagator is then
ΦΞijΦ
(p) = δijδj′i′δΞΞ′DΞii,i′i′ , (A11)
where [13]
DΞii,i′i′ = −iδ′Ξ
(p2 −m2ii,Ξ)
(p2 −m2
i′i′,Ξ)
(p2 −m2U,Ξ)(p2 −m2D,Ξ)(p2 −m2S,Ξ)
(p2 −m2
)(p2 −m2η,Ξ)(p2 −m2η′,Ξ)
. (A12)
For concreteness we have assumed that there are three sea-quark flavors: u, d, and s; the
generalization to Nsea flavors is immediate. Here mU,Ξ ≡ muu,Ξ is the mass of a taste-Ξ
pion made from a u and a ū quark, neglecting hairpin mixing (and similarly for mD,Ξ and
mS,Ξ), mπ0,Ξ, mη,Ξ, and mη′,Ξ are the mass eigenvalues after mixing is included, and the iǫ
terms have been left implicit. When specifying the particular member of a taste multiplet
appearing in the disconnected propagator is unnecessary, we abuse this notation slightly
following Eq. (A7) and refer to DVii,i′i′ , DAii,i′i′ , or DIii,i′i′. In flavor-taste notation we have:
ΦabΦb′a′
(p) ≡
Φiα,jβΦj′β′,i′α′
(p) = δijδj′i′
TΞαβT
β′α′DΞii,i′i′ (A13)
For comparison, we now describe the continuum versions of the Feynman rules [18]. Since
taste violations do not appear in LHL, Eq. (17), the continuum-theory version of Eqs. (A1)
and (A2) are unchanged except that flavor-taste indices are replaced by pure flavor indices
(i, j):
(k) =
2(v·k + iǫ)
[continuum], (A14)
B∗µiB
(k) =
−iδij(gµν − vµvν)
2(v·k + iǫ)
[continuum]. (A15)
Similarly, the continuum BB∗π [18] and current vertices are identical to those in SχPT,
aside from the redefinition of the indices and a factor of 2 for each Φab field due to the
non-standard normalization of the generators in the SχPT case, Eq. (6). The continuum
version of Eq. (A3) is
µi Bj − B
∂µΦji [continuum]; (A16)
while the continuum version of Eq. (A4) is
LO = κB
δℓk −
ΦℓiΦik + · · ·
+ κvµBℓ
Φℓk + · · ·
[continuum]. (A17)
Because of taste-violations in the SχPT pion sector, the differences between the propaga-
tors Eqs. (A6), (A11) and (A12) and their continuum versions are slightly less trivial. The
continuum connected propagator is
ΦijΦj′i′
(p) =
iδii′δjj′
p2 −m2ij + iǫ
[continuum], (A18)
m2ij = µ(mi +mj) [continuum]. (A19)
The continuum disconnected propagator is
ΦijΦj′i′
(p) = δijδj′i′Dii,i′i′ [continuum], (A20)
where [13]
Dii,i′i′ = −iδ′
(p2 −m2ii)
(p2 −m2
(p2 −m2U )(p2 −m2D)(p2 −m2S)
(p2 −m2
)(p2 −m2η)(p2 −m2η′)
[continuum], (A21)
with now δ′ = m20/3.
Note the difference in normalization between δ′ and the SχPT taste-singlet hairpin, δ′I ,
Eq. (A9). This arises from the fact that m20/3 is defined to be the strength of the hairpin
vertex when one has a single species of quark on each side of the vertex [12]. In the staggered
case, each normalized taste-singlet field is made out of four species (tastes), for example
φI = 1
(φ11+φ22+φ33+φ44), where φ is flavor neutral, and only taste indices are shown. In
the disconnected propagator of two such fields, there are 16 terms, and a factor of (1/2)2 from
the normalization, so there is an overall factor of 4 relative to a single-species disconnected
propagator, such as that of φ11 with φ22. At one loop, the “external” fields in this propagator
are always valence fields, so the normalization issue has nothing directly to do with the
fourth root trick for staggered sea quarks. (The normalization is in fact compensated by the
extra factors of 2 in the continuum vertices.) The rooting does however affect the η′I mass
that appears in denominator of Eq. (A21), which comes from iterations of the hairpin and
therefore involves sea quarks. The end result is that m2η′,I ≈ Nseam20/3 (for large m0), rather
than ≈ 4Nseam20/3, the value in the unrooted theory [13]. In the continuum, we also have
m2η′ ≈ Nseam20/3.
APPENDIX B: INTEGRALS
Here we collect the integrals needed in evaluating the diagrams for the semileptonic form
factors [19, 33].
The disconnected propagators can be written as a sum of single or double poles using
the (Euclidean) residue functions introduced in Ref. [13] or their Minkowski-space versions.
We define {m} ≡ {m1, m2, . . . , mn} as the set of masses that appear in the denominator of
Eq. (A12), and {µ} ≡ {µ1, µ2, . . . , µk} as the numerator set of masses. Then, for n > k and
all masses distinct, we have:
I [n,k] ({m};{µ}) ≡
i=1(q
2 − µ2i )
j=1(q
2 −m2j + iǫ)
[n,k]
j ({m};{µ})
q2 −m2j + iǫ
, (B1)
where the Minkowski space residues R̂
[n,k]
j are given by
[n,k]
j ({m};{µ}) ≡
i=1(m
j − µ2i )
r 6=j(m
j −m2r)
. (B2)
If there is one double pole term for q2 = m2ℓ (where mℓ ∈ {m}), then
I [n,k]dp (mℓ; {m};{µ}) ≡
i=1(q
2 − µ2i )
(q2 −m2
+ iǫ)
j=1(q
2 −m2j + iǫ)
[n,k]
j ({m};{µ})
q2 −m2j + iǫ
. (B3)
In the end we want to write the results in terms of the Euclidean-space residues R
[n,k]
because they are ones we have used previously [13, 19]. In Euclidean space the sign of each
factor in Eq. (B2) is changed. We therefore have
[n,k]
j ({m};{µ}) ≡
i=1(µ
i −m2j )
r 6=j(m
r −m2j)
= (−1)n+k−1R̂[n,k]j ({m};{µ}) . (B4)
The integrals needed for the form factors are ([17, 33])
I1 = µ4−d
(2π)d
q2 −m2 + iǫ
(4π)2
I1(m) , (B5)
I2 = µ4−d
(2π)d
(v·q −∆+ iǫ)(q2 −m2 + iǫ)
(4π)2
I2(m,∆) , (B6)
Iµ3 = µ4−d
(2π)d
(v·q −∆+ iǫ)(q2 −m2 + iǫ)
(4π)2
[I2(m,∆) + I1(m)] , (B7)
J µν = µ4−d
(2π)d
iqµqν
(v·q −∆+ iǫ)(q2 −m2 + iǫ)
(4π)2
[J1(m,∆)g
µν + J2(m,∆)v
µvν ] ,
where the arrows represent the fact that the r.h.s. of these expressions have already been
renormalized (unlike the corresponding equations in Ref. [33]).
APPENDIX C: WAVEFUNCTION RENORMALIZATION FACTORS
The one loop chiral corrections to the wave function renormalization factors ZB and ZP
are are [13, 19]
δZPxy =
3(4πf)2
[I1 (mxf,Ξ) + I1 (myf,Ξ)]
j∈M(3,x)
∂m2X,I
[3,3]
M(3,x)I ;µ
I1(mj,I)
j∈M(3,y)
∂m2Y,I
[3,3]
M(3,y)I ;µ
I1(mj,I)
j∈M(4,xy)
[4,3]
M(4,xy)I ;µ
I1(mj,I)
+ a2δ′V
j∈M(4,x)
∂m2X,V
[4,3]
M(4,x)V ;µ
I1(mj,V )
j∈M(4,y)
∂m2Y,V
[4,3]
M(4,y)V ;µ
I1(mj,V )
j∈M(5,xy)
[5,3]
M(5,xy)V ;µ
I1(mj,V )
V → A
, (C1)
δZBx =
−3g2π
(4πf)2
I1(mxf,Ξ)
j∈M(3,x)
∂m2X,I
[3,3]
M(3,x)I ;µ
I1(mj,I)
+ a2δ′V
j∈M(4,x)
∂m2X,V
[4,3]
M(4,x)V ;µ
I1(mj,V )
+ [V → A]
, (C2)
where f runs over the sea quarks (u, d, s).
For the continuum result in partially quenched χPT, we can simply set a = 0 and ignore
taste splittings. In the 1+1+1 case, we get
δZcontPxy =
3(4πf)2
[I1 (mxf) + I1 (myf )]
j∈M(3,x)
[3,3]
M(3,x);µ(3)
I1(mj)
j∈M(3,y)
[3,3]
M(3,y);µ(3)
I1(mj)
j∈M(4,xy)
[4,3]
M(4,xy);µ(3)
I1(mj)
, (C3)
δZcontBx =
−3g2π
(4πf)2
I1(mxf)
j∈M(3,x)
[3,3]
M(3,x);µ(3)
I1(mj)
Returning to a 6= 0, and taking the valence quark masses to be mx = my = mu = md, we
have the 2+1 full QCD pion result in SχPT:
δZπ =
3(4πf)2
[4I1(mπ,Ξ) + 2I1(mK,Ξ)]
+(−4a2δ′V )
(m2SV −m
(m2ηV −m2πV )(m
−m2πV )
I1(mπV ) +
(m2SV −m
(m2πV −m2ηV )(m
−m2ηV )
I1(mηV )
(m2SV −m
(m2πV −m
)(m2ηV −m
I1(mη′
V → A
. (C5)
Taking the valence quark masses to be mx = mu = md and my = ms gives the 2+1 full
QCD kaon result:
δZK =
3(4πf)2
(2I1(mπ,Ξ) + 3I1(mK,Ξ) + I1(mS,Ξ))
I1(mπI ) +
I1(mηI )− I1(mSI )
+(−a2δ′V )
(m2SV +m
− 2m2ηV )
(m2πV −m2ηV )(m
−m2ηV )(m
−m2ηV )
I1(mηV )
(m2SV +m
− 2m2
(m2πV −m
)(m2SV −m
)(m2ηV −m
I1(mη′
m2SV −m
(m2ηV −m2πV )(m
−m2πV )
I1(mπV ) +
m2πV −m
(m2ηV −m
−m2SV )
I1(mSV )
V → A
. (C6)
Setting mx = mu = md in Eq. (C2) results in the 2+1 full QCD result for the B
wavefunction renormalization:
δZB =
(4πf)2
[2I1(mπ,Ξ) + I1(mK,Ξ)] +
I1(mπI )−
I1(mηI )
+ a2δ′V
(m2SV −m
(m2ηV −m2πV )(m
−m2πV )
I1(mπV ) +
(m2SV −m
(m2πV −m2ηV )(m
−m2ηV )
I1(mηV )
(m2SV −m
(m2πV −m
)(m2ηV −m
I1(mη′
+ [V → A]
. (C7)
Finally, putting mx = ms and mu = md in Eq. (C2), we obtain the full QCD Bs renormal-
ization factor in the 2+1 case:
δZBs =
(4πf)2
[I1(mS,Ξ) + 2I1(mK,Ξ)] + I1(mSI )−
I1(mηI )
+ (−a2δ′V )
(m2SV −m
(m2SV −m2ηV )(m
I1(mSV ) +
(m2ηV −m
(m2ηV −m
)(m2ηV −m
I1(mηV )
−m2πV )
−m2SV )(m
−m2ηV )
I1(mη′
+ [V → A]
. (C8)
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TABLE I: Connecting the one-loop diagrams from Ref. [33] (left column) and this paper (right
column).
Ref. [33] This work
(4) Fig. 4(a)
(7) Fig. 5(a)
(9) Fig. 5(b)
(12) Fig. 5(c)
(13) Fig. 5(d)
(14) Fig. 4(b)
(a) (b)
FIG. 1: Example of a connected one-loop form factor diagram at (a) the meson level and (b) the
quark level. For the meson diagram, the double line is a heavy-light meson while the single line is
a pion. For the quark-level diagram, the solid line is a heavy quark and the dashed line is a light
quark. The internal sea quark loop is required by the (quark-flow) connected pion propagator;
purely valence diagrams are only possible with a disconnected pion propagator. Therefore this
diagram gives rise to a factor of Nsea in the degenerate case.
(a) (b)
FIG. 2: Example of a disconnected one-loop form factor diagram at (a) the meson level and (b)
the quark level. The cross in the meson diagram represents the two-point interactions in χPT,
and is represented by the “hairpin” in the quark-level diagram. There are no factors of Nsea but
instead factors of 1/Nsea coming from the decoupling of the η′.
(a) (b)
FIG. 3: Tree level diagrams for (a) fv and (b) fp. The double line is the heavy-light meson and
the single line is the pion.
(a) (b)
FIG. 4: One-loop fv diagrams. The internal light meson lines may in general be connected or
disconnected: possible hairpin insertions are not shown explicitly.
FIG. 5: One-loop fp diagrams. The internal light meson lines may in general be connected or
disconnected: possible hairpin insertions are not shown explicitly.
FIG. 6: The quark-flow diagram for Fig. 5(b), omitting the heavy quark line for clarity. The
mesons in the loop are X and Y mesons, flavor-neutral mesons made up of x and y quarks. Note
that even though only a single hairpin insertion is shown explicitly, the figure should be interpreted
as representing all diagrams with one or more hairpins.
(a) (b)
FIG. 7: Possible quark-flow diagrams for Fig. 5(c) with a disconnected meson propagator in the
loop. The solid rectangle encloses the 5-point vertex of Fig. 5(c). The heavy quark line has been
omitted for clarity. A “reflected” version of diagram (b), with the outgoing pion on the other side
of the vertex, is also possible.
(a) (b)
FIG. 8: Possible quark-flow diagrams for Fig. 5(c) with a connected meson propagator in the loop.
The solid rectangle encloses the 5-point vertex of Fig. 5(c). The heavy quark line has been omitted
for clarity. Since we have assumed that x and y are different flavors, diagram (a) cannot occur in
our case. Diagram (b) can occur, as can a “reflected” version with the outgoing pion on the other
side of the vertex.
	Introduction
	Heavy-Light Staggered Chiral Perturbation Theory
	Generalizing Continuum PQ0.4exPT to S0.4exPT
	Form Factors for BP Decay
	Form factors for 3-flavor partially quenched S0.4exPT
	Full QCD Results
	Analytic terms
	Finite Volume Effects
	Conclusions
	Feynman Rules
	Integrals
	Wavefunction Renormalization Factors
	References
