Appraising constrained second-order power corrections
in HQET with
Λ
b
→
Λ
c
l
ν
Florian U. Bernlochner,
1
Michele Papucci,
2
and Dean J. Robinson
3,4
1
Physikalisches Institut der
Rheinischen Friedrich-Wilhelms-Universität Bonn
, 53115 Bonn, Germany
2
Walter Burke Institute for Theoretical Physics,
California Institute of Technology
, Pasadena, California 91125, USA
3
Ernest Orlando
Lawrence Berkeley National Laboratory
,
University of California, Berkeley
, California 94720, USA
4
Berkeley Center for Theoretical Physics, Department of Physics,
University of California, Berkeley
, California 94720, USA
(Received 17 May 2024; accepted 17 October 2024; published 3 February 2025)
We derive the
Λ
b
→
Λ
c
form factors for the Standard Model and beyond at second order in heavy quark
effective theory, applying the recently proposed residual chiral expansion to reduce the set of unknown
subsubleading hadronic functions to a single, highly constrained function, that is fully determined by
hadron mass parameters at zero recoil. We fit a form factor parametrization based on these results to all
available lattice QCD predictions and experimental data. We find that the constrained and predictive
structure of the form factors under the residual chiral expansion is in excellent agreement with lattice QCD
predictions and experimental data, as well as prior heavy-quark-effective-theory-based fits.
DOI:
10.1103/PhysRevD.111.036001
I. INTRODUCTION
Semileptonic
Λ
b
→
Λ
c
l
ν
decays provide a sensitive
laboratory for exploring the behavior of second-order
power corrections in heavy quark effective theory
(
HQET
). Unlike in the
B
→
D
ðÞ
system, in
Λ
b
→
Λ
c
transitions no new subleading Isgur-Wise functions
—
the
form factors that describe HQET matrix elements
—
enter at
first order in the heavy quark (HQ) power expansion
[1,2]
.
Thus the first-order HQ power expansion for
Λ
b
→
Λ
c
matrix elements is highly constrained, and the breaking of
the heavy quark symmetry (HQS) at second order can be
explored without
“
contamination
”
from unknown first-
order hadronic functions. References
[3,4]
exploited this
property to probe the size of
O
ð
1
=m
2
c
Þ
contributions and
obtain precise predictions for the lepton flavor universality
violation (LFUV) ratio
R
ð
Λ
c
Þ¼
Γ
½
Λ
b
→
Λ
c
τν
=
Γ
½
Λ
b
→
Λ
c
l
ν
;
l
¼
μ
;e;
ð
1
:
1
Þ
using a combined fit to LQCD calculations of the
Λ
b
→
Λ
c
form factors
[5]
and LHCb measurements of the
Λ
b
→
Λ
c
μν
differential shapes
[6]
; these predictions were
significantly more precise than prior results
[5,7
–
12]
, and
remain the most precise to date. The
“
BLRS
”
form factor
parametrization
[3]
developed therein, which we consider
here to provide the current best Standard Model (SM)
predictions, incorporated HQET power corrections up to
and including
O
ð
1
=m
2
c
Þ
, using the long-known results of
Ref.
[2]
. The combined fit demonstrated current data are
already precise enough to allow the two-parameter system
at
O
ð
1
=m
2
c
Þ
in the Bernlochner-Ligeti-Robinson-Sutcliffe
(BLRS) parametrization to be recovered at more than
2
.
5
σ
from zero
[3]
.
The role of second-order power corrections in HQET-
based descriptions of exclusive
b
→
cl
ν
semileptonic
decays is of particularly high importance. At the per-
cent-level precision (or better) anticipated in upcoming
measurements, second-order power corrections in the HQ
expansion, which naively enter at the several percent level,
may introduce sizeable sources of theoretical uncertainty in
the description of in the
B
→
D
ðÞ
l
ν
decays. This system-
atically limits (future) precision measurements of the
Cabibbo-Kobayashi-Maskawa (CKM) matrix element
j
V
cb
j
as well as precision predictions and measurements
of the LFUV ratios,
R
ð
D
Þ
and
R
ð
D
Þ
(see, e.g., the
introductory discussion in Ref.
[13]
).
In Ref.
[13]
we recently postulated a supplemental power
counting within HQET
—
the residual chiral expansion
(
RCE
)
—
that reduces the typically large and overcomplete
number of subsubleading Isgur-Wise functions entering at
second order to a smaller, highly constrained set. The RCE
Published by the American Physical Society under the terms of
the
Creative Commons Attribution 4.0 International
license.
Further distribution of this work must maintain attribution to
the author(s) and the published article
’
s title, journal citation,
and DOI. Funded by SCOAP
3
.
PHYSICAL REVIEW D
111,
036001 (2025)
2470-0010
=
2025
=
111(3)
=
036001(21)
036001-1
Published by the American Physical Society
is in effect a truncation scheme based on a conjecture that
matrix elements containing more insertions of the trans-
verse residual momentum operator acting on the HQ mass-
subtracted states are suppressed compared to those matrix
elements containing fewer such insertions. This coincides,
beyond a certain number of insertions, with those matrix
elements involving a larger number of operator product
Lagrangian insertions being suppressed compared to those
involving fewer. Truncating at second order in the RCE, at
zero recoil the second-order power corrections in HQET for
̄
B
→
D
ðÞ
transitions become constrained by hadron mass
parameters, leading to zero recoil predictions that are in
good agreement with lattice QCD (LQCD) predictions.
Reference
[13]
further showed that RCE-based parametri-
zations of these form factors achieved excellent fits to
LQCD calculations and Belle data.
In general the HQ power expansion for
Λ
b
→
Λ
c
matrix
elements is simpler than in the
̄
B
→
D
ðÞ
system because the
HQET describing these decays involve light degrees of
freedom in the simplest spin-parity
s
π
l
l
¼
0
þ
state, rather
than
s
π
l
l
¼
1
=
2
þ
. Noting the above-mentioned (and related)
absence of first-order Isgur-Wise functions in
Λ
b
→
Λ
c
matrix elements, the
Λ
b
→
Λ
c
system therefore provides
a particularly clean laboratory to explore the behavior of
the RCE, and to test whether RCE-based parametrizations
and predictions are compatible with current LQCD pre-
dictions and available experimental data, as well as prior
BLRS precision fits. (Some exploration of RCE-based
parametrizations for the more complicated
Λ
b
→
Λ
c
sys-
tem has also been conducted, with promising results
[14]
.)
In this work we show that at second order in the RCE,
the second-order power corrections are described entirely
by a single subsubleading Isgur-Wise function, that is
constrained at zero recoil by hadron mass parameters.
This holds not only at
O
ð
1
=m
2
c
Þ
but at
O
ð
1
=m
2
c;b
Þ
in the
HQ power expansion.
1
By contrast, six (three) subsublead-
ing Isgur-Wise functions enter the full HQET description
at second order (
O
ð
1
=m
2
c
Þ
), describing six (two) HQS
functions. A summary is provided in Table
I
; see also
Sec.
III C
for an explanation of this counting.
We show that the resulting zero-recoil RCE-based
predictions for the
f
1
and
g
1
SM form factors at
O
ð
α
2
s
;
α
s
=m
c;b
;
1
=m
2
c;b
Þ
are in good agreement with
LQCD calculations. Further, we show that RCE-based form
factor parametrizations at
O
ð
α
2
s
;
α
s
=m
c;b
;
1
=m
2
c;b
Þ
yield fits
in good agreement with LQCD predictions and LHCb data
as well as prior BLRS fit results, despite the RCE being
more constrained compared to the standard HQET descrip-
tion. The predictions for
R
ð
Λ
c
Þ
derived from these RCE-
based fits are similarly compatible with BLRS results. In
particular, for our
“
baseline
”
fit scenario we recover
R
ð
Λ
c
Þ¼
0
.
3251
ð
37
Þð
1
:
2
Þ
in good agreement with the BLRS result
R
ð
Λ
c
Þ¼
0
.
3233
ð
37
Þ
.
2
Supporting the prior studies in Ref.
[13]
,
the analysis in this paper serves to provide additional
evidence that the underlying conjectured truncation scheme
is not incompatible with nature.
The paper is organized as follows: In Sec.
II
we
specialize the RCE to the
Λ
b
→
Λ
c
system. In Sec.
III
and in Appendixes
A
and
B
we compute the form factors at
second order in the RCE and
O
ð
α
s
=m
c;b
;
1
=m
2
c;b
Þ
in HQET,
both in the SM and beyond. Section
III
further presents the
parametrizations of the leading and subsubleading Isgur-
Wise functions, the application of
1
S
short distance mass
scheme, and the SM zero-recoil predictions at second order
in the RCE compared to LQCD calculations. Section
IV
presents our fit scenarios and fit results to LQCD and LHCb
data. Section
V
concludes.
II. RCE PRELIMINARIES
The following specializes the general discussion in
Ref.
[13]
for
b
-to
c
-hadron transitions to the case of
Λ
b
→
Λ
c
transitions, for which the matrix elements are
described by matching onto the
s
π
l
l
¼
0
þ
HQET. These
transitions are mediated by the current
J
Γ
¼
̄
c
Γ
b
, where
Γ
denotes any Dirac operator. A full operator basis entering
the QCD matrix elements
h
Λ
c
j
̄
c
Γ
b
j
Λ
b
i
is chosen to be
J
S
¼
̄
cb;
J
P
¼
̄
c
γ
5
b;
J
V
¼
̄
c
γ
μ
b;
J
A
¼
̄
c
γ
μ
γ
5
b;
J
T
¼
̄
c
σ
μν
b;
ð
2
:
1
Þ
where
σ
μν
≡
ð
i=
2
Þ½
γ
μ
;
γ
ν
. The pseudotensor contribution
is determined by the identity
σ
μν
γ
5
≡
þð
i=
2
Þ
ε
μνρσ
σ
ρσ
,
TABLE I. Number of
Λ
b
→
Λ
c
form factors, fixed-order HQS
functions, and Isgur-Wise functions entering at each fixed order
in HQET.
Isgur-Wise functions
HQET order
Fixed-order
HQS functions
All
RCE
1
=m
0
c;b
111
1
=m
1
c;b
200
1
=m
2
c
231
1
=m
2
c;b
661
1
Here and throughout this work we use
O
ð
1
=m
2
c;b
Þ
to denote
terms of order
∼
1
=m
2
c
,
1
=m
c
m
b
, and
1
=m
2
b
.
2
An adjustment considering the impact of bin correlations on
the LHCb data normalization, along with a minor discrepancy in
the
α
2
s
terms in the
1
S
mass scheme used for the leading
renormalon cancellation, introduces slight modifications to the
BLRS fit. Consequently, the resulting BLRS prediction for
R
ð
Λ
c
Þ
differs marginally from the previously reported value of
0.3237(36)
[3]
. See Sec.
IVA
for more details.
BERNLOCHNER, PAPUCCI, and ROBINSON
PHYS. REV. D
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036001 (2025)
036001-2
which implies Tr
½
γ
μ
γ
ν
γ
ρ
γ
σ
γ
5
¼
−
4
i
ε
μνρσ
. [An opposite sign convention is usually chosen for
̄
B
→
D
ðÞ
transitions.]
Using the notation and results of Ref.
[13]
, at second order in the HQ expansion and keeping terms only to second order
in the RCE
—
i.e., at
“
O
ð
θ
2
Þ
”—
the matching between the
Λ
b
→
Λ
c
QCD and HQET matrix elements simplifies to
h
Λ
c
j
̄
c
Γ
b
j
Λ
b
i
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
Λ
c
m
Λ
b
p
≃
h
Λ
v
0
c
j
̄
c
v
0
þ
Γ
b
v
þ
j
Λ
v
b
iþ
1
2
m
c
h
Λ
v
0
c
jð
̄
c
v
0
þ
̄
J
0
1
Π
0
−
þ
L
0
1
∘
̄
c
v
0
þ
Þ
Γ
b
v
þ
j
Λ
v
b
iþ
1
2
m
b
h
Λ
v
0
c
j
̄
c
v
0
þ
Γ
ð
Π
−
J
1
b
v
þ
þ
b
v
þ
∘
L
1
Þj
Λ
v
b
i
þ
1
4
m
2
c
h
Λ
v
0
c
jð
̄
c
v
0
þ
̄
J
0
2
Π
0
−
þ
L
0
2
∘
̄
c
v
0
þ
Þ
Γ
b
v
þ
j
Λ
v
b
iþ
1
4
m
2
b
h
Λ
v
0
c
j
̄
c
v
0
þ
Γ
ð
Π
−
J
2
b
v
þ
þ
b
v
þ
∘
L
2
Þj
Λ
v
b
i
þ
1
4
m
c
m
b
h
Λ
v
0
c
j
̄
c
v
0
þ
̄
J
0
1
Π
0
−
ΓΠ
−
J
1
b
v
þ
j
Λ
v
b
i
:
ð
2
:
2
Þ
Here
Π
¼ð
1
=
v
Þ
=
2
, and
j
Λ
v
Q
i
denote the eigenstates of
the leading order HQET Lagrangian, with HQ velocity
v
.
These states are normalized such that
h
Λ
v
0
Q
ð
k
0
Þj
Λ
v
Q
ð
k
Þi ¼
2
v
0
δ
vv
0
ð
2
π
Þ
3
δ
3
ð
k
−
k
0
Þ
:
ð
2
:
3
Þ
Note this normalization choice differs from that in Ref.
[2]
,
which normalized the HQET states with respect to an HQ
mass scale,
m
Q
þ
̄
Λ
Λ
[see Eq.
(2.5)
below]. In Eq.
(2.2)
, the
Lagrangian corrections
L
1
¼
−
̄
Q
v
þ
=
D
⊥
=
D
⊥
Q
v
þ
¼
−
̄
Q
v
þ
D
2
þ
a
Q
ð
μ
Þ
g
2
σ
αβ
G
αβ
Q
v
þ
;
ð
2
:
4a
Þ
L
2
¼
̄
Q
v
þ
½
=
D
⊥
iv
·
D
=
D
⊥
Q
v
þ
¼
g
̄
Q
v
þ
½
v
β
D
α
G
αβ
−
iv
α
σ
βγ
D
γ
G
αβ
Q
v
þ
;
ð
2
:
4b
Þ
and the current corrections
J
1
¼
i
=
D
, and
J
2
¼
−
=
D
=
D
. The
conjugate forms
̄
J
n
≡
γ
0
⃖
J
†
n
γ
0
, where the arrow indicates
action of the derivatives to the left. The
∘
operator in
Eq.
(2.2)
denotes a (time-ordered) operator product, e.g.,
L
0
1
∘
̄
c
v
0
þ
ð
z
Þ¼
i
R
d
4
x
L
0
1
ð
x
Þ
̄
c
v
0
þ
ð
z
Þ
. For a discussion of the
robustness of the RCE truncation under RG evolution, see
Sec. II D of Ref.
[13]
.
In the
s
P
l
¼
0
þ
HQET, the matrix elements generated
by the chromomagnetic terms in
L
1
and
L
2
vanish (see
Appendix
A
). Related to this, matching of the QCD
correlator involving the HQET Hamiltonian
h
Λ
Q
j
̄
Q
v
þ
iv
·
DQ
v
þ
j
Λ
Q
i
onto HQET yields the hadron mass expansion
m
Λ
Q
¼
m
Q
þ
̄
Λ
Λ
þ
Δ
m
2
2
m
Q
þ
...
;
Δ
m
2
¼
−
λ
Λ
1
:
ð
2
:
5
Þ
The hadron mass parameter
̄
Λ
Λ
corresponds to the kinetic
energy of the brown muck in the HQ symmetry limit, while
λ
Λ
1
¼
−
h
Λ
v
Q
j
̄
Q
v
þ
D
2
Q
v
þ
j
Λ
v
Q
i
=
2
is generated by the kinetic
energy term in
L
1
.
Contact terms generated by derivatives acting on HQET
matrix elements give rise to Schwinger-Dyson relations
(sometimes called modified Ward identities). These relate
matrix elements entering at different orders in the HQ
expansion
(2.2)
, reducing the total number of unknown
hadronic functions entering at a particular order. In this
work, we always choose the HQ velocity to be that of the
hadron containing it, i.e.,
p
¼
m
H
v
. At leading order, one
then has the (familiar) Schwinger-Dyson relation
h
Λ
v
0
c
j
̄
c
v
0
þ
ð
z
Þ
i
⃖
D
z
μ
Γ
b
v
þ
ð
z
Þj
Λ
v
b
iþh
Λ
v
0
c
j
̄
c
v
0
þ
ð
z
Þ
Γ
i
⃗
D
z
μ
b
v
þ
ð
z
Þj
Λ
v
b
i
¼
!
̄
Λ
Λ
ð
v
−
v
0
Þ
μ
h
Λ
v
0
c
j
̄
c
v
0
þ
ð
z
Þ
Γ
b
v
þ
ð
z
Þj
Λ
v
b
i
;
ð
2
:
6
Þ
where the operator
“
¼
!
”
indicates a relation that arises
under composition of the hadron current in question with
an external operator, using integration by parts and enforc-
ing overall momentum conservation. At next-to-leading
order, and keeping only terms to
O
ð
θ
2
Þ
, the Schwinger-
Dyson relations become
h
Λ
v
0
c
j
̄
c
v
0
þ
ð
z
Þ
̄
J
0
2
Π
0
þ
Γ
b
v
þ
ð
z
Þj
Λ
v
b
i¼
!
λ
Λ
1
h
Λ
v
0
c
j
̄
c
v
0
þ
ð
z
Þ
Γ
b
v
þ
ð
z
Þj
Λ
v
b
i
;
h
Λ
v
0
c
j
̄
c
v
0
þ
ð
z
Þ
ΓΠ
þ
J
2
b
v
þ
ð
z
Þj
Λ
v
b
i¼
!
λ
Λ
1
h
Λ
v
0
c
j
̄
c
v
0
þ
ð
z
Þ
Γ
b
v
þ
ð
z
Þj
Λ
v
b
i
:
ð
2
:
7
Þ
III.
Λ
b
→
Λ
c
FORM FACTORS
A. HQET matrix elements
The
Λ
Q
ground-state baryons are formed by the tensor
product of a spin-
1
=
2
heavy quark with brown muck in the
spin-parity
s
π
l
l
¼
0
þ
definite state, and thus belong to a HQ
singlet. The particle representation of this singlet is a Dirac
spinor
U
v
, obeying the Dirac equation of motion
=
v
U
v
¼
U
v
.
APPRAISING CONSTRAINED SECOND-ORDER POWER
...
PHYS. REV. D
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036001 (2025)
036001-3
The matching of the QCD matrix elements onto HQET
in Eq.
(2.2)
then becomes
h
Λ
c
ð
p
0
Þj
̄
c
Γ
b
j
Λ
b
ð
p
Þi
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
m
Λ
c
m
Λ
b
p
¼
ζ
ð
w
Þ
̄
U
v
0
Γ
U
v
þ
X
2
n
¼
1
ε
n
c
̄
U
ð
n
Þ
ð
v
0
;v
Þ
Γ
U
v
þ
X
2
n
¼
1
ε
n
b
̄
U
v
0
Γ
U
ð
n
Þ
ð
v;v
0
Þþ
ε
c
ε
b
Tr
½
Γ
W
ð
1
;
1
Þ
ð
v;v
0
Þ
;
ð
3
:
1
Þ
choosing HQ velocities
v
¼
p=m
Λ
b
,
v
0
¼
p
0
=m
Λ
c
, and
defining
U
ð
n
Þ
ð
v; v
0
Þ¼
ˆ
K
ð
n
Þ
1
U
v
þ
ˆ
K
ð
n
Þ
2
Π
−
=
v
0
U
v
;
ð
3
:
2
Þ
W
ð
1
;
1
Þ
ð
v;v
0
Þ¼
ˆ
M
1
U
v
̄
U
v
0
þ
ˆ
M
2
Π
−
=
v
0
U
v
̄
U
v
0
þ
ˆ
M
0
2
U
v
̄
U
v
0
=
v
Π
−
þ
ˆ
M
3
Π
−
=
v
0
U
v
̄
U
v
0
=
v
Π
−
þ
ˆ
M
4
Π
−
γ
α
U
v
̄
U
v
0
γ
α
Π
−
;
ð
3
:
3
Þ
in which
ˆ
K
ð
n
Þ
1
and
ˆ
M
i
are
“
HQS functions
”
of the recoil
parameter
w
¼
v
·
v
0
¼
m
2
Λ
b
þ
m
2
Λ
c
−
q
2
2
m
Λ
b
m
Λ
c
;q
2
¼ð
p
−
p
0
Þ
2
:
ð
3
:
4
Þ
The heavy quark expansion parameters
ε
c;b
¼
̄
Λ
Λ
=
ð
2
m
c;b
Þ
,
such that all
ˆ
K
ð
n
Þ
i
,
ˆ
M
i
, and Isgur-Wise functions are
dimensionless. Note also
ˆ
M
2
¼
ˆ
M
0
2
up to
O
ð
α
s
×
1
=m
c
m
b
Þ
, i.e., third-order perturbative corrections,
which can be included separately.
In Eq.
(3.1)
we have factored out the leading Isgur-Wise
function,
ζ
ð
w
Þ
, from all terms. We use the notation that
hatted functions of
w
are normalized to
ζ
ð
w
Þ
, i.e.,
ˆ
W
ð
w
Þ
≡
W
ð
w
Þ
=
ζ
ð
w
Þ
for any Isgur-Wise function or form factor,
W
ð
w
Þ
. In the equal mass, zero-recoil limit the normaliza-
tion of the QCD matrix element for the (conserved) vector
current requires that
h
Λ
Q
j
̄
Q
γ
μ
Q
j
Λ
Q
i¼
2
m
Λ
Q
v
μ
:
ð
3
:
5
Þ
When matched onto HQET at leading order, this implies
that
ζ
ð
1
Þ¼
1
. Further zero-recoil constraints are discussed
below in Sec.
III C
.
Terms generated by current corrections in Eq.
(3.1)
are
always associated with the presence of a
Π
ð0Þ
−
projector
acting on the (anti)spinor while Lagrangian corrections are
always associated with a
Π
ð0Þ
þ
: see, e.g., Eq.
(2.2)
. Thus, we
expect
ˆ
K
ð
n
Þ
1
to contain Lagrangian corrections only, and
ˆ
K
ð
n
Þ
2
to contain only current corrections. Further, since in
the RCE at
O
ð
1
=m
c
m
b
;
θ
2
Þ
only the matrix element
generated by the product current correction
∼
̄
J
0
1
J
1
is
present, then we expect that
ˆ
M
1
and
ˆ
M
2
will vanish, but
ˆ
M
3
and
ˆ
M
4
will remain nonzero.
B. Form factor matching
Following the standard HQET-style basis for the
Λ
b
→
Λ
c
form factors, the SM matrix elements can be
represented as
[2,15,16]
h
Λ
c
ð
p
0
;s
0
Þj
̄
c
γ
ν
b
j
Λ
b
ð
p; s
Þi ¼
̄
u
ð
p
0
;s
0
Þ½
f
1
γ
μ
þ
f
2
v
μ
þ
f
3
v
0
μ
u
ð
p; s
Þ
;
h
Λ
c
ð
p
0
;s
0
Þj
̄
c
γ
ν
γ
5
b
j
Λ
b
ð
p; s
Þi ¼
̄
u
ð
p
0
;s
0
Þ½
g
1
γ
μ
þ
g
2
v
μ
þ
g
3
v
0
μ
γ
5
u
ð
p; s
Þ
;
ð
3
:
6
Þ
where again
p
¼
m
Λ
b
v
,
p
0
¼
m
Λ
c
v
0
, and the spinors are normalized to
̄
u
ð
p; s
Þ
u
ð
p; s
Þ¼
2
m
. The form factors beyond the
SM are defined via
[4]
h
Λ
c
ð
p
0
;s
0
Þj
̄
cb
j
Λ
b
ð
p; s
Þi ¼
h
S
̄
u
ð
p
0
;s
0
Þ
u
ð
p; s
Þ
;
h
Λ
c
ð
p
0
;s
0
Þj
̄
c
γ
5
b
j
Λ
b
ð
p; s
Þi ¼
h
P
̄
u
ð
p
0
;s
0
Þ
γ
5
u
ð
p; s
Þ
;
h
Λ
c
ð
p
0
;s
0
Þj
̄
c
σ
μν
b
j
Λ
b
ð
p; s
Þi ¼
̄
u
ð
p
0
;s
0
Þ½
h
1
σ
μν
þ
ih
2
ð
v
μ
γ
ν
−
v
ν
γ
μ
Þþ
ih
3
ð
v
0
μ
γ
ν
−
v
0
ν
γ
μ
Þþ
ih
4
ð
v
μ
v
0
ν
−
v
ν
v
0
μ
Þ
u
ð
p; s
Þ
:
ð
3
:
7
Þ
Matching Eqs.
(3.6)
and
(3.7)
onto Eq.
(3.1)
, the
Λ
b
→
Λ
c
form factors at
O
ð
α
s
;
1
=m
2
c;b
Þ
can be expressed in terms of the
HQS functions as
ˆ
h
S
¼
1
þ
ˆ
α
s
C
S
þ
X
Q
¼
c;b
ε
Q
ˆ
K
ð
Q
Þ
1
−
ð
w
−
1
Þ
ˆ
K
ð
Q
Þ
2
þ
ε
b
ε
c
ð
ˆ
M
1
−
2
ˆ
M
2
ð
w
−
1
Þþ
ˆ
M
3
ð
w
2
−
1
Þþ
ˆ
M
4
ð
w
þ
2
ÞÞ
;
ð
3
:
8a
Þ
ˆ
h
P
¼
1
þ
ˆ
α
s
C
P
þ
X
Q
¼
c;b
ε
Q
ˆ
K
ð
Q
Þ
1
−
ð
w
þ
1
Þ
ˆ
K
ð
Q
Þ
2
þ
ε
b
ε
c
ð
ˆ
M
1
−
2
ˆ
M
2
ð
w
þ
1
Þþ
ˆ
M
3
ð
w
2
−
1
Þþ
ˆ
M
4
ð
w
−
2
ÞÞ
;
ð
3
:
8b
Þ
BERNLOCHNER, PAPUCCI, and ROBINSON
PHYS. REV. D
111,
036001 (2025)
036001-4
ˆ
f
1
¼
1
þ
ˆ
α
s
C
V
1
þ
X
Q
¼
c;b
ε
Q
ˆ
K
ð
Q
Þ
1
−
ð
w
þ
1
Þ
ˆ
K
ð
Q
Þ
2
þ
ε
b
ε
c
ð
ˆ
M
1
−
2
ˆ
M
2
ð
w
þ
1
Þþ
ˆ
M
3
ð
w
2
−
1
Þþ
ˆ
M
4
w
Þð
3
:
8c
Þ
ˆ
f
2
¼
ˆ
α
s
C
V
2
þ
2
ε
c
ˆ
K
ð
c
Þ
2
þ
2
ε
b
ε
c
ð
ˆ
M
2
−
ˆ
M
3
ð
w
−
1
Þ
−
ˆ
M
4
Þ
;
ð
3
:
8d
Þ
ˆ
f
3
¼
ˆ
α
s
C
V
3
þ
2
ε
b
ˆ
K
ð
b
Þ
2
þ
2
ε
b
ε
c
ð
ˆ
M
2
−
ˆ
M
3
ð
w
−
1
Þ
−
ˆ
M
4
Þ
;
ð
3
:
8e
Þ
ˆ
g
1
¼
1
þ
ˆ
α
s
C
A
1
þ
X
Q
¼
c;b
ε
Q
ˆ
K
ð
Q
Þ
1
−
ð
w
−
1
Þ
ˆ
K
ð
Q
Þ
2
þ
ε
b
ε
c
ð
ˆ
M
1
−
2
ˆ
M
2
ð
w
−
1
Þþ
ˆ
M
3
ð
w
2
−
1
Þþ
ˆ
M
4
w
Þ
;
ð
3
:
8f
Þ
ˆ
g
2
¼
ˆ
α
s
C
A
2
þ
2
ε
c
ˆ
K
ð
c
Þ
2
þ
2
ε
b
ε
c
ð
ˆ
M
2
−
ˆ
M
3
ð
w
þ
1
Þ
−
ˆ
M
4
Þ
;
ð
3
:
8g
Þ
ˆ
g
3
¼
ˆ
α
s
C
A
3
−
2
ε
b
ˆ
K
ð
b
Þ
2
−
2
ε
b
ε
c
ð
ˆ
M
2
−
ˆ
M
3
ð
w
þ
1
Þ
−
ˆ
M
4
Þ
;
ð
3
:
8h
Þ
ˆ
h
1
¼
1
þ
ˆ
α
s
C
T
1
þ
X
Q
¼
c;b
ε
Q
ˆ
K
ð
Q
Þ
1
−
ð
w
−
1
Þ
ˆ
K
ð
Q
Þ
2
þ
ε
b
ε
c
ð
ˆ
M
1
−
2
ˆ
M
2
ð
w
−
1
Þþ
ˆ
M
3
ð
w
2
−
1
Þþ
ˆ
M
4
ð
w
−
2
ÞÞ
;
ð
3
:
8i
Þ
ˆ
h
2
¼
ˆ
α
s
C
T
2
þ
2
ε
c
ˆ
K
ð
c
Þ
2
þ
2
ε
b
ε
c
ð
ˆ
M
2
−
ˆ
M
3
ð
w
þ
1
Þ
−
ˆ
M
4
Þ
;
ð
3
:
8j
Þ
ˆ
h
3
¼
ˆ
α
s
C
T
3
−
2
ε
b
ˆ
K
ð
b
Þ
2
−
2
ε
b
ε
c
ð
ˆ
M
2
−
ˆ
M
3
ð
w
þ
1
Þ
−
ˆ
M
4
Þ
;
ð
3
:
8k
Þ
ˆ
h
4
¼
4
ε
b
ε
c
ˆ
M
3
:
ð
3
:
8l
Þ
Here we have defined, for convenience,
ˆ
K
ð
Q
Þ
i
¼
ˆ
K
ð
1
Þ
i
þ
ε
Q
ˆ
K
ð
2
Þ
i
;Q
¼
c; b:
ð
3
:
9
Þ
We have included here the leading perturbative corrections
in
ˆ
α
s
¼
α
s
=
π
[17]
; explicit expressions for the
C
Γ
i
func-
tions are given in Ref.
[18]
. The higher-order
ˆ
α
s
=m
c;b
corrections have been previously computed in, e.g.,
Refs.
[3,4]
, and are reproduced in Appendix
B
.
The six
b
i
HQS functions,
i
¼
1
;
...
;
6
, that encode
the second-order power corrections in the notation of
Ref.
[2]
may be expressed in terms of the six functions
ˆ
K
ð
2
Þ
1
;
2
,
ˆ
M
1
;
2
;
3
;
4
via
ˆ
b
1
=
̄
Λ
2
Λ
¼
ˆ
K
ð
2
Þ
1
−
ð
w
−
1
Þ
ˆ
K
ð
2
Þ
2
;
ˆ
b
2
=
̄
Λ
2
Λ
¼
2
ˆ
K
ð
2
Þ
2
;
ˆ
b
3
=
̄
Λ
2
Λ
¼
ˆ
M
1
−
2
ˆ
M
2
ð
w
−
1
Þþ
ˆ
M
3
ð
w
2
−
1
Þþ
ˆ
M
4
w;
ˆ
b
4
=
̄
Λ
2
Λ
¼
4
ˆ
M
2
;
ˆ
b
5
=
̄
Λ
2
Λ
¼
2
ð
ˆ
M
2
−
ˆ
M
3
ð
w
−
1
Þ
−
ˆ
M
4
Þ
;
ˆ
b
6
=
̄
Λ
2
Λ
¼
2
ð
ˆ
M
2
−
ˆ
M
3
ð
w
þ
1
Þ
−
ˆ
M
4
Þ
;
ð
3
:
10
Þ
noting the
ˆ
b
i
are dimensionful. In Refs.
[3,4]
, which
worked only to
O
ð
1
=m
2
c
Þ
, the two independent subsublead-
ing HQS functions entering at that order were expressed in
terms of
ˆ
b
1
;
2
, as done in Ref.
[2]
, rather than in terms
of
ˆ
K
ð
2
Þ
1
;
2
.
C. Summary of constrained form factors
The full derivation of the first and second-order power
corrections to the
Λ
b
→
Λ
c
form factors at
O
ð
θ
2
Þ
are
provided in Appendix
A
. As is well known, at first order,
the current corrections are fully determined by the leading
Isgur-Wise function via the leading-order Schwinger-
Dyson relation
(2.6)
and equations of motion. The first-
order chromomagnetic corrections vanish, leaving a single
subleading Isgur-Wise function associated with the
L
1
kinetic term,
χ
1
. The second-order current corrections
are only partially determined by the leading Isgur-Wise
function via the next-to-leading-order Schwinger-Dyson
relation
(2.7)
and equations of motion. At
O
ð
θ
2
Þ
in the
RCE, however, there is a single remaining Isgur-Wise
function associated with second-order current corrections,
φ
1
ð
w
Þ
. One finds that it is constrained at zero recoil such
that
ˆ
φ
1
ð
1
Þ¼
λ
Λ
1
=
ð
6
̄
Λ
2
Λ
Þ
. Analyticity of the matrix elements
ensures that the combination
½
ˆ
φ
1
ð
w
Þ
−
ˆ
φ
1
ð
1
Þ
=
ð
w
−
1
Þ
must be regular. As done in Ref.
[13]
, we define the
quotient with respect to
w
¼
1
,
ˆ
φ
♮
1
ð
w
Þ
≡
½
ˆ
φ
1
ð
w
Þ
−
ˆ
φ
1
ð
1
Þ
=
ð
w
−
1
Þ
;
ð
3
:
11
Þ
and write the form factors explicitly in terms of this regular
function. By definition
ˆ
φ
♮
1
ð
1
Þ¼
ˆ
φ
0
1
ð
1
Þ
, the gradient at
zero recoil. Finally, at
O
ð
θ
2
Þ
in the RCE there is a single
APPRAISING CONSTRAINED SECOND-ORDER POWER
...
PHYS. REV. D
111,
036001 (2025)
036001-5
Isgur-Wise function associated with Lagrangian corrections
from the
L
2
kinetic term,
β
1
.
The mass normalization condition
(3.5)
requires that in
the equal mass and zero-recoil limit the vector current
matrix elements satisfy, to all orders,
½
f
1
ð
1
Þþ
f
2
ð
1
Þþ
f
3
ð
1
Þj
m
c
¼
m
b
¼
1
:
ð
3
:
12
Þ
Thus, both the perturbative corrections and the power
corrections must vanish in this limit order by order. At
first order, it immediately follows from Eq.
(3.8)
that
2
ˆ
K
ð
1
Þ
1
ð
1
Þ¼
0
and hence from Eq.
(A3)
ˆ
χ
1
ð
1
Þ¼
0
:
ð
3
:
13
Þ
At second order, Eq.
(3.12)
implies that
2
ˆ
K
ð
2
Þ
1
ð
1
Þþ
ˆ
M
3
ð
1
Þ
−
3
ˆ
M
4
ð
1
Þ¼
0
. Applying Eqs.
(A6)
,
(A14)
,
(A16)
and the zero-recoil constraint
(A12)
one finds
ˆ
β
1
ð
1
Þ¼
λ
Λ
1
4
̄
Λ
2
Λ
:
ð
3
:
14
Þ
The three Isgur-Wise functions
β
1
,
χ
1
and
ζ
are asso-
ciated with the same HQET amplitude,
̄
U
v
0
Γ
U
v
, that
conserves heavy quark spin symmetry. Therefore within
the form factors, these three Isgur-Wise functions always
enter in the same linear combination. Defining
β
♮
1
in
the same manner as in Eq.
(3.11)
, we may reabsorb
β
♮
1
and
χ
1
into
ζ
[13]
(see also, e.g., Refs.
[3,4,16]
) via the
replacement
ζ
þ
2
ð
ε
c
þ
ε
b
Þ
χ
1
þ
2
ð
ε
2
c
þ
ε
2
b
Þð
w
−
1
Þ
β
♮
1
→
ζ
:
ð
3
:
15
Þ
Although the reabsorption of
χ
1
introduces additional
spurious
O
ð
1
=m
2
c;b
Þ
terms, these terms are
O
ð
θ
3
Þ
in the
RCE or higher, and thus may be neglected,
3
while induced
O
ð
α
s
=m
c;b
Þ
terms cancel against
O
ð
α
s
=m
c;b
Þ
perturbative
corrections involving
χ
1
. Thus, applying the RCE at
O
ð
θ
2
Þ
,
only a single subsubleading Isgur-Wise function
φ
♮
1
ð
w
Þ
enters the HQ expansion at
O
ð
1
=m
2
c;b
Þ
after redefinitions.
One finds finally (see Appendix
A
)
ˆ
K
ð
1
Þ
1
¼
0
;
ˆ
K
ð
1
Þ
2
¼
−
1
w
þ
1
;
ˆ
K
ð
2
Þ
1
¼
λ
Λ
1
2
̄
Λ
2
Λ
;
ˆ
K
ð
2
Þ
2
¼
λ
Λ
1
3
̄
Λ
2
Λ
þ
2
ð
w
−
1
Þ
ˆ
φ
♮
1
;
ð
3
:
16
Þ
and
ˆ
M
3
¼
1
w
þ
1
ð
w
−
2
Þ
λ
Λ
1
6
̄
Λ
2
Λ
þð
w
2
þ
2
Þ
ˆ
φ
♮
1
−
w
−
2
2
ð
w
þ
1
Þ
;
ˆ
M
4
¼ð
3
−
w
Þ
λ
Λ
1
6
̄
Λ
2
Λ
−
w
ð
w
−
1
Þ
ˆ
φ
♮
1
þ
w
−
1
2
ð
w
þ
1
Þ
;
ð
3
:
17
Þ
with
ˆ
M
1
¼
ˆ
M
2
¼
0
as expected. In these results we have
applied Eq.
(A13)
to Eqs.
(A6)
and
(A14)
in order to
express
ˆ
K
ð
2
Þ
2
and
ˆ
M
3
;
4
in terms of the quotient
ˆ
φ
♮
1
, and we
have applied Eq.
(3.14)
to Eq.
(A16)
to express
ˆ
K
ð
2
Þ
1
in
terms of
ˆ
β
♮
1
, which is then reabsorbed per Eq.
(3.15)
.
In Table
II
we summarize the Isgur-Wise functions
entering up to and including
O
ð
1
=m
2
c;b
Þ
in the full HQ
expansion compared to
O
ð
θ
2
Þ
in the RCE. In the former,
six Isgur-Wise functions parametrize the six second-order
HQS functions
ˆ
K
ð
2
Þ
1
;
2
and
ˆ
M
1
;
2
;
3
;
4
. Restricting to
O
ð
1
=m
2
c
Þ
only, three Isgur-Wise functions
—
two full functions and a
TABLE II. Isgur-Wise functions order by order in the HQ expansion, as they arise in the full HQ expansion
[2]
,in
the BLRS parametrization
[3,4]
, and in the RCE in this work, after redefinitions and reabsorption into lower-order
Isgur-Wise functions. Also shown are the parameter (super)sets for each parametrization, not including mass
scheme parameters. The subsubleading Isgur-Wise functions
d
2
;
3
arise from
∼
L
1
L
0
1
Lagrangian corrections
involving a double operator product, while
e
3
arises from mixed corrections
∼
L
1
J
1
(see Ref.
[2]
). At
O
ð
1
=m
2
c
Þ
only, several of these can be further reabsorbed into lower-order Isgur-Wise functions, up to the zero-recoil
contribution
e
3
ð
1
Þ
.
Expansions
1
=m
0
c;b
1
=m
c;b
1
=m
2
c
only
1
=m
2
c;b
HQET
Functions
ζ
ð
w
Þ
1
φ
♮
0
;
1
ð
w
Þ
,
e
3
ð
1
Þ
φ
♮
0
;
1
ð
w
Þ
,
e
3
ð
w
Þ
,
d
2
;
3
ð
w
Þ
,
d
1
ð
1
Þ
BLRS
[3,4]
Functions
ζ
ð
w
Þ
1
ˆ
b
1
;
2
ð
w
Þ
Parameters
ζ
0
,
ζ
00
̄
Λ
Λ
ˆ
b
1
;
2
ð
1
Þ
RCE
Functions
ζ
ð
w
Þ
1
φ
♮
1
ð
w
Þ
φ
♮
1
ð
w
Þ
Parameters
ζ
0
,
ζ
00
,
ζ
000
̄
Λ
Λ
λ
Λ
1
,
ˆ
φ
0
1
,
ˆ
φ
00
1
λ
Λ
1
,
ˆ
φ
0
1
,
ˆ
φ
00
1
3
In the full HQ expansion these induced
O
ð
1
=m
2
c;b
Þ
terms
cancel against
χ
1
terms arising from second-order Schwinger-
Dyson relations, such that
χ
1
can also be consistently reabsorbed
per Eq.
(3.15)
even at second order in the full power expansion.
BERNLOCHNER, PAPUCCI, and ROBINSON
PHYS. REV. D
111,
036001 (2025)
036001-6
zero-recoil parameter
—
parametrize the two remaining
second-order HQS functions
ˆ
K
ð
2
Þ
1
;
2
. Thus at both
O
ð
1
=m
2
c
Þ
and
O
ð
1
=m
2
c;b
Þ
the second-order HQ expansion is non-
predictive in the sense that there are at least as many Isgur-
Wise functions as there are HQS functions. Because of this,
in the BLRS approach of Refs.
[3,4]
, the two
O
ð
1
=m
2
c
Þ
independent subsubleading HQS functions
ˆ
b
1
;
2
were sim-
ply treated as independent functions. This is to be com-
pared to the constrained and predictive structure of the
O
ð
θ
2
Þ
RCE-based description of the second-order power
corrections, for which a single subsubleading Isgur-Wise
function
ˆ
φ
1
(plus hadron mass parameters) describe all six
HQS functions at
O
ð
1
=m
2
c;b
Þ
.
D. 1
S
scheme and numerical inputs
We use the
1
S
scheme
[19
–
21]
to achieve cancellation of
leading renormalon ambiguities from the mass parameter
̄
Λ
Λ
against those in the factorially growing coefficients of
the
α
s
perturbative power series
[22,23]
. In this scheme,
m
1
S
b
is defined as half of the perturbatively computed
Υ
ð
1
S
Þ
mass, such that the pole mass
m
b
ð
m
1
S
b
Þ
≃
m
1
S
b
ð
1
þ
2
α
s
ð
m
b
Þ
2
=
9
þ
...
Þ
:
ð
3
:
18
Þ
When one computes just the leading
n
f
dependence at high
orders
[24
–
26]
, the splitting of the bottom and charm quark
pole mass
δ
m
bc
≡
m
b
−
m
c
has a renormalon ambiguity
only at third order in the heavy quark expansion. Thus,
as we are working at second order in the HQ expansion, we
fix
m
c
¼
m
b
−
δ
m
bc
. The renormalization group evolution
α
s
ð
μ
Þ¼
α
s
ð
α
s
ð
m
Z
Þ
;m
Z
;
μ
Þ
computed at four-loop order
[27,28]
determines
α
s
ð
m
b
Þ
≃
0
.
215
.
Because, however,
m
1
S
b
and
δ
m
bc
are extracted numeri-
cally from fits to measurements of inclusive spectra at
O
ð
1
=m
3
Q
Þ
[29
–
31]
, third-order terms should be similarly
retained numerically in the expansion of the hadron
mass, even though we otherwise formally work to second
order. Defining
m
c
ð
m
1
S
b
Þ
≡
m
b
ð
m
1
S
b
Þ
−
δ
m
bc
, the mass
expansion
(2.5)
then becomes
m
Λ
Q
≃
m
Q
ð
m
1
S
b
Þþ
̄
Λ
Λ
−
λ
Λ
1
2
m
Q
ð
m
1
S
b
Þ
þ
ρ
Λ
1
4
½
m
Q
ð
m
1
S
b
Þ
2
:
ð
3
:
19
Þ
Here at
O
ð
θ
2
Þ
the third-order correction in the mass
expansion is proportional to the parameter
ρ
Λ
1
, in parallel
to the expansions in Ref.
[32]
for meson masses. The mass
expansion
(3.19)
for
m
Λ
b
and
m
Λ
c
may be solved simulta-
neously to yield
̄
Λ
Λ
¼
m
b
ð
m
1
S
b
Þ
m
Λ
b
−
m
c
ð
m
1
S
b
Þ
m
Λ
c
δ
m
bc
−
½
m
b
ð
m
1
S
b
Þþ
m
c
ð
m
1
S
b
Þ þ
ρ
Λ
1
4
m
b
ð
m
1
S
b
Þ
m
c
ð
m
1
S
b
Þ
;
λ
Λ
1
¼
2
m
b
ð
m
1
S
b
Þ
m
c
ð
m
1
S
b
Þ
δ
m
bc
½
m
Λ
b
−
m
Λ
c
−
δ
m
bc
þ
ρ
Λ
1
½
m
b
ð
m
1
S
b
Þþ
m
c
ð
m
1
S
b
Þ
2
m
b
ð
m
1
S
b
Þ
m
c
ð
m
1
S
b
Þ
;
ð
3
:
20
Þ
so that
̄
Λ
Λ
and
λ
Λ
1
are parametrized in terms of
m
1
S
b
,
δ
m
bc
,
and
ρ
Λ
1
.
Using the numerical inputs
m
1
S
b
¼ð
4
.
71
0
.
05
Þ
GeV
;
δ
m
bc
¼ð
3
.
40
0
.
02
Þ
GeV
ð
3
:
21
Þ
one finds, keeping
ρ
Λ
1
dependence explicit,
̄
Λ
Λ
¼ð
0
.
88
0
.
05
þ
0
.
04
½
ρ
Λ
1
=
GeV
3
Þ
GeV and
λ
Λ
1
¼ð
−
0
.
24
0
.
07
þ
0
.
49
½
ρ
Λ
1
=
GeV
3
Þ
GeV
2
. As in Ref.
[13]
we choose the
HQET to QCD matching scale
μ
bc
¼
ffiffiffiffiffiffiffiffiffiffiffiffi
m
b
m
c
p
≃
2
.
5
GeV,
such that
α
s
ð
μ
bc
Þ
≃
0
.
27
, again via the renormalization
group evolution
α
s
ð
μ
Þ¼
α
s
ð
α
s
ð
m
Z
Þ
;m
Z
;
μ
Þ
computed at
four-loop order
[27,28]
. As fit inputs we use the values in
Eq.
(3.21)
as well as
ρ
Λ
1
¼ð
−
0
.
1
0
.
2
Þ
GeV
3
;
ð
3
:
22
Þ
which corresponds to the range
λ
Λ
1
¼ð
−
0
.
3
0
.
1
Þ
GeV
2
.
This is roughly commensurate with the second-order
parameter in the expansion of the spin-averaged
B
ðÞ
and
D
ðÞ
masses from fits to inclusive spectra
[31]
. The fit input
values in Eqs.
(3.21)
and
(3.22)
should not be confused
with the central values and uncertainties recovered from
any particular fit.
At first order in the HQ expansion, the
1
S
scheme is
implemented by replacing the pole mass
m
b
ð
m
1
S
b
Þ
by
m
1
S
b
everywhere in the power corrections, because all the first-
order current corrections are fixed proportional to the
leading Isgur-Wise function by the Schwinger-Dyson
relation
(2.6)
, and because there are no (unabsorbed)
Lagrangian corrections. Thus at first order, one may
equivalently define and use an effective HQ expansion
parameter
ε
0
Q
≡
̄
Λ
1
S
Λ
2
m
1
S
Q
¼
1
2
m
1
S
Q
m
1
S
b
m
Λ
b
−
m
1
S
c
m
Λ
c
δ
m
bc
−
½
m
1
S
b
þ
m
1
S
c
þ
ρ
Λ
1
4
m
1
S
b
m
1
S
c
;
ð
3
:
23
Þ
APPRAISING CONSTRAINED SECOND-ORDER POWER
...
PHYS. REV. D
111,
036001 (2025)
036001-7