Radiative $B$ to axial-vector meson decays at NLO in Soft-Collinear Effective
Theory
Arslan Sikandar1, M. Jamil Aslam1, Ishtiaq Ahmed2, Saba Shafaq3
1Department of Physics, Quaid-i-Azam University, Islamabad 45320, Pakistan.
2National Centre for Physics, Quaid-i-Azam University, Islamabad, Pakistan.
3 Department of Physics, International Islamic University, Islamabad,
Pakistan.

Abstract

The rare decay $B\rightarrow A\gamma$, with $A$ representing axial-vector
mesons such as $K_{1}(1270),\;K_{1}(1400),\;b_{1}(1300),\;a_{1}(1260)$, is
studied at next-to-leading order (NLO) in soft collinear effective theory
(SCET). The large outgoing meson energy encourages the study of the decay with
an appropriate factorization scheme that separates the factorizable and non-
factorizable parts systematically. We have analyzed the leading-power and
$\mathcal{O}(\alpha_{s})$ diagrams that contribute to matching to SCETI. The
new intermediate theory is matched onto SCETII and the running of SCETI
operators is performed to sum large perturbative logarithms. The values of
soft-overlap function $\zeta_{\perp}$ for $K_{1}(1270,\;1400),a_{1}$ and
$b_{1}$ mesons are estimated from the light cone-sum-rules (LCSR), and later
using it the corresponding branching fractions for
$B\to\left(K_{1}(1270,\;1400),\;a_{1},\;b_{1}\right)\gamma$ decays are
calculated. We find that in case of $B\to K_{1}(1270,\;1400)\gamma$ decays the
results are in good agreement with their experimental measurements. Also the
estimated values of the branching ratios of the $B\to(b_{1},\;a_{1})\gamma$
decays are potentially large to be measured at the LHCb and future
B-factories.

##  I Introduction

The rare decays involving $b\to s,\;d$ transitions are suppressed in the
Standard Model (SM) due to Glashow-Iliopoulos-Maiani (GIM) mechanism. These
flavor changing neutral current (FCNC) transitions make the rare decays
important for the precision tests of the SM predictions and also to hunt for
the physics beyond it, i.e., the new physics (NP). In this regard, the
corresponding exclusive radiative process $B\rightarrow V\gamma$ (V is a
vector meson) had been studied extensively both in SM [1, 2, 3] and in various
NP scenarios [4, 5]. Similarly, motivated by the above fact and the
complementary confirmation the exclusive radiative decays $B\rightarrow
A\gamma$, $A$ is any of the the axial-vector mesons
$K_{1}(1270),\;K_{1}(1400),\;b_{1}(1235))$ and $a_{1}(1260)$, have also been
studied in the context of large energy effective theory (LEET) in [6, 7, 8, 9,
10]. However, it is well known that among other effective field theories the
power of SCET lies in the fact that it provides a platform to study the decays
at different momentum scales. Along with this, it is a systematic field-
theoretic approach that provides a freedom to study beyond leading order
corrections, powers corrections (PC) as well as the running of the operators
to sum the large logarithms. Therefore, we expect that the results obtained in
this case will be more precise as compared to the other approaches, and also
they will help us to get a clear understanding of the underlying physics. With
this motivation, the present work is dedicated to the study of $B\rightarrow
A\gamma$ decays in SCET which at a hadronic level corresponds to the heavy-to-
light transitions.

In case of heavy-to-light decays, the standard SCET factorization is already
developed [see for example [11, 12] for details and its proof]. The
factorization relation for the matrix elements of the dimension-six operators
$Q_{i}$ (c.f. Eq. (5)) at leading power in the expansion $\Lambda/E$ (where
$E$ is the final meson energy) and to all orders in $\alpha_{s}$ is

$\langle A\gamma|Q_{i}|B\rangle=C_{i}^{I}\zeta^{B\rightarrow
A_{\perp}}(E)+\int_{0}^{\infty}\frac{d\omega}{\omega}\phi_{B}(\omega)\int_{0}^{1}du\phi_{A_{\perp}}(u)C_{i}^{II}(\omega,u)$
(1)

where $C_{i}^{I}$ and $C_{i}^{II}$ are the Wilson coefficients (WCs) of the
effective theory that we need to calculate using perturbation theory. The
hadronic quantities $\phi_{A_{\perp}}$ and $\phi_{B}$ are the light-cone
distribution amplitudes (LCDAs) for axial-vector and $B$ mesons, respectively.
In the large recoil limit, a single function $\zeta^{\perp}(E)$ \- often
called the soft overlap function is enough to describe the decay form factors
of $B\rightarrow A_{\perp}\gamma$ up to factorizable corrections as shown in
[13]. Along with this overlap function, we also require $\zeta^{\parallel}(E)$
in the semi-leptonic decays where $\perp$ and $\parallel$ correspond to the
perpendicular and parallel polarizations of the final state meson,
respectively.

The general proof of writing the relation (1) is based on the separation of
soft and hard-scattering contributions (which are also known as the soft and
hard modes) to the form factors, the convergence of convolution integral, and
the absence of the contribution from non-valence Fock states at leading order.
The soft contributions in the overlap functions encapsulated in the matrix
elements of SCET operators between initial and final state mesons that obey
the spin-symmetry relations at the large-recoil. Such operators (in SCET) are
not simple and lead to ”non-factorizable” matrix elements that are sensitive
to the end-point singularities and non-valence Fock states. Therefore, the
validity of Eq. (1) lies in the fact that any non-factorizable part of the
decay amplitude can be represented by $\zeta(E)$.

It is worthy to show that the diagrams in which the emitted photon from one of
the quarks have the same structure as those of the diagrams of heavy-to-light
form factors [14]. This proof gets complicated when the emitted photon is from
the spectator quark of $B-$meson; because in such a scenario, some quarks and
gluons will be collinear to the photon that can lead to the singularities in
the momentum. To cope with this issue, one needs to introduce new soft-
collinear modes [15] in theory and has to show that one can decouple such
modes in the low-energy theory at a leading- power. It is important to mention
that a finite mass of $s-$quark would not spoil our factorization formula and
contribute only in $\phi_{A}^{\perp}$ and $\zeta_{\perp}^{B\rightarrow
A_{\perp}}$. Here, we will ignore these contributions, which can be taken into
account by following the method outlined in [16] for the explicit calculation
of $\zeta_{\perp}^{B\rightarrow(K_{1},a_{1},b_{1})}$ and
$\phi_{(K_{1},a_{1},b_{1})}^{\perp}$.

In SCET, we deal with different momentum modes of quark and gluon fields, and
the definition of physical scales becomes important for matching the effective
theory. Therefore in this formalism, one can define the momentum modes
according to Eq. (6) in soft, soft-collinear, collinear, hard, and hard-
collinear modes. Some of these modes are integrated-out while the others are
required to preserve Eq. (1). That is a two-step procedure accommodated by the
expansion parameter $\lambda\sim\Lambda/m_{b}$. At a first step, we need to
match the QCD to some intermediate effective theory, i.e., $\text{SCET}_{I}$
at hard scale $\mu_{h}\sim m_{b}$ to ensure that the hard modes $(1,1,1)m_{b}$
are integrated out. The next step is to match SCETI to a final theory called
SCETII at an intermediate scale $\mu_{i}\sim\sqrt{m_{b}\Lambda}$. This ensures
that the hard-collinear modes $(\lambda,1,\lambda^{1/2})m_{b}$ are also
integrated out and we are left with the final theory of soft
$(\lambda,\lambda,\lambda)m_{b}$, collinear $(\lambda^{2},1,\lambda)m_{b}$ and
soft-collinear $(\lambda^{2},\lambda,\lambda^{3/2})m_{b}$ modes. The final
thing that needed to be proven is whether the soft-collinear modes spoil the
validity of Eq. (1) or not. We have discussed this issue at length in Sec.
III.

In the two-step procedure mentioned above, it is pertinent to discuss the
operators required at a given order in power counting. In this work, we are
concerned with contributions to an-order of $\mathcal{O}(\alpha_{s})$. It
requires three types of operators satisfying Eq. (1): the A-type operators
that are quark-antiquark two-particle currents along with a photon emission
from the quark-antiquark. The B-type operators are three particle quark-
antiquark-gluon currents with photon emission from quark-antiquark, and the
C-type operators are four fermion operators with a photon emitting from the
spectator quark-antiquark. In our forthcoming study, we will discuss these
three types of operators briefly.

To bridge the matching between SCETI and SCETII, the notion of hard
$(\mu_{h})$ and intermediate ($\mu_{i}$) scales are very useful. It will make
it easy to sum the large perturbative logarithms in going from hard
$\mu_{h}\sim m_{b}$ to an intermediate-scale $\mu_{i}\sim\sqrt{m_{b}\Lambda}$
by Renormalization Group (RG) analysis of SCETI. Similarly, the RG-analysis of
SCETII can be done in going from $\mu_{i}$ to a hadronic scale
$\mu\sim\Lambda$ at which we wish to study the mesons. It will also require
the running of meson distribution amplitude that in itself is a complicated
procedure - but the intermediate scale, in this case, is already close to
final state masses, and hence it is avoided.

Regarding the meson in the final state, in the study presented here, we are
interested in the axial-vector mesons $(K_{1B},\;b_{1})$ and
$(K_{1A},\;a_{1})$ which are ${}^{1}P_{1}$ states with $J^{PC}=1^{+-}$ and
$1^{3}P_{1}$ states with $J^{PC}=1^{++}$, respectively. We also know that the
physical axial-vector meson $(K_{1}(1270),\;K_{1}(1400))$ states arise as a
result of mixing of $1^{1}P_{1}(K_{1A})$ and $1^{3}P_{1}(K_{1B})$ states.
Consequently, these axial-vector mesons are relatively massive (around $1.3$
GeV) as compared to the vector meson states $(K^{*},\omega,\rho)$. However,
they are not heavy enough to rule out the heavy-quark-large-energy expansion
that we use for the heavy-to-light decays. Recently, the branching fractions
of $B\to K_{1}(1270,1400)\gamma$ decays are reported to be [17]:

$\displaystyle\mathcal{B}(B^{+}\rightarrow
K_{1}^{+}(1270)\gamma)=(4.4^{+0.7}_{-0.6})\times 10^{-5},$
$\displaystyle\mathcal{B}(B^{+}\rightarrow
K_{1}^{+}(1400)\gamma)=(10^{+5}_{-4})\times 10^{-6}.$ (2)

In the case of the final state axial-vector mesons $(b_{1},\;a_{1})$, the
experimental values of the branching ratios are still unknown. However, by
looking at the CKM matrix elements involved in these decay, one can expect
their branching ratios to be two orders of magnitude smaller than $B\to
K_{1}(1270)\gamma$ decay, which is also the case for their corresponding
vector meson states $(\omega)$ and $(\rho)$ [18, 19]

$\displaystyle\mathcal{B}(B\rightarrow\omega\gamma)$ $\displaystyle=$
$\displaystyle(5.4\pm 0.23(\text{stat.})\pm 0.21(\text{syst.}))\times
10^{-7},$ $\displaystyle\mathcal{B}(B^{+}\rightarrow\rho^{+}\gamma)$
$\displaystyle=$ $\displaystyle(6.8\pm 0.36(\text{stat.})\pm
0.31(\text{syst.}))\times 10^{-7}.$ (3)

The work performed here is organized as follows: In Sec. II we introduce the
SCETI operators for (A, B, C)-type currents that are relevant to the radiative
decays under consideration. In the same section, we work out the matching
calculation from QCD to the SCETI. In Sec. III, the SCETII operators are given
as four-quark operators to match with the SCETI operators. The relevant WCs,
known as the jet-functions, are presented in the same section. We calculate
the matrix elements for $B\rightarrow A\gamma$ in Sec. IV, and for their
completeness, we add the contributions of large perturbative logarithms. Sec1.
V is devoted to this discussion. Section VI presents the results for the
branching ratios along-with the calculations of soft overlap function using
light-cone sum rules. Finally, Sec. VII provides a summary of our main
results. The appendix contains our discussion of the matching contributions of
the quark loop operator $Q_{1}$ and chromomagnetic operator $Q_{8}$, given in
Eq. (5).

##  II SCETI Operators and QCD$\rightarrow$ SCETI matching

The effective Hamiltonian corresponding to the FCNC $b\rightarrow(s,d)$
transition can be written as [20]

$H_{eff}=-\frac{G_{F}}{\sqrt{2}}V_{tp}^{*}V_{tb}\sum_{i=1}^{8}C_{i}(\mu)Q_{i}(\mu),$
(4)

with $p=s$ for $K_{1}$ and $d$ for $(b_{1},\;a_{1})$ mesons. Here,
$C_{i}(\mu)$ and $Q_{i}(\mu)$ with $i=1,...,8$ are the WCs and their
corresponding operators, respectively. The operators $Q_{i}(\mu)$ are
expressed as

$\displaystyle Q_{1}$ $\displaystyle=$
$\displaystyle\bar{s}\gamma^{\mu}(1-\gamma_{5})c\quad\bar{c}\gamma_{\mu}(1-\gamma_{5})b,$
$\displaystyle Q_{2}$ $\displaystyle=$
$\displaystyle\bar{s}^{\alpha}\gamma^{\mu}(1-\gamma_{5})c^{\beta}\quad\bar{c}^{\beta}\gamma_{\mu}(1-\gamma_{5})b^{\alpha},$
$\displaystyle Q_{3}$ $\displaystyle=$
$\displaystyle\bar{s}\gamma^{\mu}(1-\gamma_{5})b\sum_{q}(\bar{q}\gamma_{\mu}(1-\gamma_{5})q),$
$\displaystyle Q_{4}$ $\displaystyle=$
$\displaystyle\bar{s}^{\alpha}\gamma^{\mu}(1-\gamma_{5})b^{\beta}\sum_{q}(\bar{q}^{\beta}\gamma_{\mu}(1-\gamma_{5})q^{\alpha}),$
$\displaystyle Q_{5}$ $\displaystyle=$
$\displaystyle\bar{s}\gamma^{\mu}(1-\gamma_{5})b\sum_{q}(\bar{q}\gamma_{\mu}(1+\gamma_{5})q),$
$\displaystyle Q_{6}$ $\displaystyle=$
$\displaystyle\bar{s}^{\alpha}\gamma^{\mu}(1-\gamma_{5})b^{\beta}\sum_{q}(\bar{q}^{\beta}\gamma_{\mu}(1-\gamma_{5})q^{\alpha}),$
$\displaystyle Q_{7}$ $\displaystyle=$
$\displaystyle-\frac{e}{8\pi^{2}}m_{b}\bar{s}\sigma^{\mu\nu}(1+\gamma_{5})bF_{\mu\nu},$
$\displaystyle\ Q_{8}$ $\displaystyle=$
$\displaystyle-\frac{g}{8\pi^{2}}m_{b}\bar{s}\sigma^{\mu\nu}(1+\gamma_{5})T^{a}bG_{\mu\nu}^{a},$
(5)

where $\alpha$ and $\beta$ represent the color indices and $T^{a}$ is a
$SU(3)$ color matrix. The WCs in Eq. (4) are known at next-to-next leading
logarithm (NNLL) [21, 22]. Though the matrix elements for the QCD penguin
operators $Q_{3}-Q_{6}$ arise at $\mathcal{O}(\alpha_{s})$, which is
compatible with our study - their relevant coefficients are too small to
contribute and hence we can safely ignore them in our forthcoming analysis.
The operator $Q_{2}$ contributes at $\mathcal{O}(\alpha_{s}^{2})$, so its
contribution omitted too. Therefore, among the eight operators defined in Eq.
(5), the phenomenologically relevant operators for the radiative decay
$B\rightarrow A\gamma$ at $\mathcal{O}(\alpha_{s})$ are $Q_{1},Q_{7}$ and
$Q_{8}$.

Before we proceed to the formal calculations, it is useful to adopt the
general way of writing the momenta of quark or gluon fields in terms of light-
like vectors, i.e.,

$p^{\mu}=n\cdot p\frac{\bar{n}^{\mu}}{2}+\bar{n}\cdot
p\frac{n^{\mu}}{2}+p_{\perp}^{\mu}=p_{+}^{\mu}+p_{-}^{\mu}+p_{\perp}^{\mu},$
(6)

Defining $P^{\mu}=m_{B}v^{\mu}$, so that the four-vector $v^{\mu}$ is in the
direction of motion of $B-$meson. As the axial-vector meson(s) is
significantly lighter compared to $B-$meson, therefore, the light-like vector
$n^{\mu}$ is in its direction of motion with the properties
$n^{2}=\bar{n}^{2}=0$ and $n\cdot\bar{n}=2$ along with $v_{\perp}=0$. The
different scaling of the set $(p_{+},p_{-},p_{\perp})$ depicts different
momentum modes which were already explained in the Sec. I.

In SCET, we can define the moment mode for each quark and gluon fields. In
this regard, the SCETI is a theory that contains soft and hard-collinear
modes. The hard-collinear quark and gluon fields along with heavy-quark and
soft fields scale as

$\displaystyle\xi_{hc},\xi_{\bar{hc}}\sim\lambda^{1/2},\quad
A_{hc}^{\mu}\sim(\lambda,1,\lambda^{1/2}),\quad h\sim\lambda^{3/2},\quad
A_{s}\sim(\lambda,\lambda,\lambda).$ (7)

To obtain the gauge-invariant operators for the fields defined in Eq. (5), we
require light-like Wilson lines which connect them at different points.
Therefore, in SCETI, the hard-collinear fields should also be invariant under
hard-collinear gauge transformations, which requires to introduce of a hard-
collinear Wilson line

$\mathcal{W}_{hc}(x)=\text{Pexp}\left(ig\int_{-\infty}^{0}ds\bar{n}\cdot
A_{hc}(x+s\bar{n})\right),$ (8)

where the gauge-invariant hard-collinear fields are defined as

$\displaystyle\chi_{hc}(x)$ $\displaystyle=$ $\displaystyle
W^{\dagger}_{hc}(x)\xi_{hc}(x),$ $\displaystyle\mathcal{A}_{hc}^{\mu}(x)$
$\displaystyle=$ $\displaystyle
W^{\dagger}_{hc}[iD_{hc}^{\mu}(x)W_{hc}(x)]+\frac{\bar{n}^{\mu}}{2}[W^{\dagger}_{hc}(x)gn\cdot
A_{s}(x_{-})W_{hc}(x)-gn\cdot A_{s}(x_{-})].$ (9)

In above equations, it can be noticed that the position variable, $x$, is
defined in a similar way as the light-like momentum, i.e., $x\equiv
x_{+}+x_{-}+x_{\perp}$ and $s$ is the light-ray variable in Eq. (8). To
construct operators at leading power using the above fields, it is pertinent
to show that they are invariant under first reparmeterization conditions,
i.e., $n^{\mu}\rightarrow(1+\alpha)n^{\mu}$ and
$\bar{n}^{\mu}\rightarrow(1-\alpha)\bar{n}^{\mu}$. The tree level coefficients
for A-type operators of the form $\bar{\chi}_{hc}\Gamma h$ with $\Gamma=I$ for
scalar, $\Gamma_{V_{i}}=(\gamma^{\mu},v^{\mu},n^{\mu})$ for vector currents,
$\Gamma_{T_{j}}=(\gamma^{[\mu}\gamma^{\nu]},\gamma^{[\mu}v^{\nu]},\gamma^{[\mu}n^{\nu]},v^{[\mu}n^{\nu]})$
for tensor currents [23, 24]. The corresponding tree level coefficients are

$\displaystyle C_{S}^{A}$ $\displaystyle=$ $\displaystyle 1,\quad
C_{V1}^{A}=1,\quad C_{V2,3}^{A}=0,\quad C_{T1}^{A}=1,\quad C_{T2,3,4}^{A}=0.$
(10)

The coefficients of pseudo-scalar A- type operators are similar to the scalar
ones while the coefficients of axial-vector A-type operators are related to
the vector ones as $C_{V1,V2}^{A}=C_{A1,A2}^{A}$ and $C_{V3}^{A}=-C_{A3}^{A}$.
As the operators in the above basis mix up B-type operators of the form
$\bar{\chi}_{hc}\Gamma\not{\mathcal{A}}_{hc}h$, the matching results are
obtained by using a new basis (for details see ref. [23]). The B-type
operators for vectors
$\Gamma_{i}^{\mu}=(\gamma_{\perp}^{\mu},v^{\mu},n^{\mu})$ and tensors
$\Gamma_{j}^{\mu\nu}=(\gamma^{[\mu}_{\perp}\gamma^{\nu]}_{\perp},\gamma^{[\mu}_{\perp}v^{\nu]},\gamma^{[\mu}_{\perp}n^{\nu]},v^{[\mu}n^{\nu]})$
in new basis along with their tree level coefficients read as follows

$\displaystyle J_{S}^{B^{\prime}}$ $\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})\not{\mathcal{A}}_{\perp}(r\bar{n})h(0),\qquad\qquad\qquad\
\ \ \ C_{S}^{B^{\prime}}=-1,$ $\displaystyle J_{Vi}^{B^{\prime}\mu}$
$\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})\not{\mathcal{A}}_{\perp}(r\bar{n})\Gamma_{i}^{\mu}h(0),\qquad\qquad\qquad
C_{V1}^{B^{\prime}}=1,C_{V2}^{B^{\prime}}=-2,C_{V3}^{B}=1-z,$ $\displaystyle
J_{V4}^{B^{\prime}\mu}$ $\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})\gamma_{\perp}^{\mu}\not{\mathcal{A}}_{\perp}(r\bar{n})h(0),\qquad\qquad\qquad
C_{V4}^{B}=0,$ $\displaystyle J_{Tj}^{B^{\prime}\mu}$ $\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})\not{\mathcal{A}}_{\perp}(r\bar{n})\Gamma_{j}^{\mu\nu}h(0),\qquad\qquad\qquad\
C_{T1}^{B^{\prime}}=-1,C_{T2}^{B^{\prime}}=-4,C_{T3}^{B^{\prime}}=2,C_{T4}^{B^{\prime}}=2,$
$\displaystyle J_{T5}^{B^{\prime}\mu}$ $\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})\mathcal{A}_{\perp\alpha}(r\bar{n})\gamma_{\perp}^{[\alpha}\gamma^{\mu}_{\perp}\gamma^{\nu]}_{\perp}h(0),\qquad\
\ \ \ \ \ C_{T5}^{B^{\prime}}=0,$ $\displaystyle J_{T6}^{B^{\prime}\mu}$
$\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})v^{[\mu}\gamma_{\perp}^{\nu]}\not{\mathcal{A}}_{\perp}(r\bar{n})h(0),\qquad\qquad\qquad
C_{T6}^{B^{\prime}}=0,$ $\displaystyle J_{T7}^{B^{\prime}\mu}$
$\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})n^{[\mu}\gamma_{\perp}^{\nu]}\not{\mathcal{A}}_{\perp}(r\bar{n})h(0),\qquad\qquad\qquad
C_{T7}^{B^{\prime}}=2z,$ (11)

where $z=2E/m_{b}$ with $E$ being the energy of the final state quark and
$\gamma_{\perp}^{\mu}=\gamma^{\mu}-n^{\mu}\not{\bar{n}}/2-\bar{n}^{\mu}\not{n}/2$.
For B-type operators, the coefficients of scalar and psuedoscalar currents
have opposite sign whereas the coefficients of axial-vector currents are
related with the vector one as
$C^{B^{\prime}}_{V1,V4}=-C^{B^{\prime}}_{A1,A4}$ and
$C^{B^{\prime}}_{V2,V3}=C^{B^{\prime}}_{A2,A3}$.

To construct the operators in the SCETI for $B\rightarrow A\gamma$ decay; it
is required to have one $n$-hard-collinear field (axial-vector meson in our
case), one $\bar{n}$-hard-collinear field ($\gamma$ in our case), and a heavy
quark field. So at leading power in $\lambda$, the relevant operator and WC
for A-type operator is

$\displaystyle J^{A}(s,a)$ $\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})(1+\gamma_{5})\not{\mathcal{A}}_{\bar{hc}\perp}^{(em)}h(0),$
$\displaystyle C^{A}(E,E_{\gamma})$ $\displaystyle=$ $\displaystyle\int ds\int
dae^{is\bar{n}\cdot P}e^{ian\cdot P_{\gamma}}\tilde{C}^{A}(s,a),$ (12)

where $E=\bar{n}\cdot P/2$ and $E_{\gamma}\equiv n\cdot P_{\gamma}/2$. Also,
$\bar{n}\cdot P$ is the large component of the total outgoing $n$-hard-
collinear momentum and $n\cdot P_{\gamma}$ is the outgoing photon momentum.
Similarly, the relevant SCETI operators for B-type currents (three particle
current) and their corresponding WCs are

$\displaystyle J^{B}_{1}(s,r,a)$ $\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})(1+\gamma_{5})\not{\mathcal{A}}_{\bar{hc}\perp}^{(em)}(an)\not{\mathcal{A}}_{hc\perp}(r\bar{n})h(0),$
(13) $\displaystyle J^{B}_{2}(s,r,a)$ $\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})(1+\gamma_{5})\not{\mathcal{A}}_{hc\perp}(r\bar{n})\not{\mathcal{A}}_{\bar{hc}\perp}^{(em)}(an)h(0),$
(14) $\displaystyle C^{B}_{j}(E,E_{\gamma},u)$ $\displaystyle=$
$\displaystyle\int ds\int dr\int dae^{i(us+\bar{u}r)\bar{n}\cdot P}e^{ian\cdot
P_{\gamma}}\tilde{C}_{j}^{B}(s,r,a).$ (15)

The index $j=1,\;2$ in B-type WCs refers to the two types of operators written
in Eqs. (13, 14). The variables $u$ and $\bar{u}$ denote the fraction of the
large component of the $n$-hard-collinear momentum carried by the quark and
the gluon fields, respectively. The relevant SCETI operator for the photon
emission from the spectator quark (C-type matching) is a four-quark operator.
Along with the corresponding WC, it becomes

$\displaystyle J^{C}(s,r,a)$ $\displaystyle=$
$\displaystyle\bar{\chi}_{hc}(s\bar{n})(1+\gamma_{5})\frac{\not{\bar{n}}}{2}\chi_{hc}(r\bar{n})\bar{\chi}_{\bar{hc}}(an)(1+\gamma_{5})\frac{\not{n}}{2}h(0),$
$\displaystyle C_{k}^{C}(u)$ $\displaystyle=$ $\displaystyle\int ds\int dr\int
dae^{i(us+\bar{u}r)\bar{n}\cdot P}e^{ian\cdot
P_{\gamma}}\tilde{C}_{k}^{C}(s,r,a).$ (16)

While matching QCD to SCETI at order $\alpha_{s}$, one needs to calculate the
WCs at one-loop for the A-type operators while at tree level for the B-type
currents. Let us denote $\Delta_{i}C_{j}^{(A,B,C)}$ for the purpose of
matching the results of weak effective operators $Q_{i}$ to the SCETI
operators $J_{j}^{A,B,C}$. Thus, we can write the total matching coefficients
as

$\displaystyle\mathcal{C}^{(A,C)}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\sum_{p=u,c}V_{ps}^{*}V_{pb}\left[C_{1}(\mu_{{}_{QCD}})\Delta_{1}^{p}C^{(A,C)}(\mu_{{}_{QCD}},\mu)+C_{7}(\mu_{{}_{QCD}})\Delta_{7}C^{(A,C)}(\mu_{{}_{QCD}},\mu)+C_{8}(\mu_{{}_{QCD}})\Delta_{8}C^{(A,C)}(\mu_{{}_{QCD}},\mu)\right],$
$\displaystyle\mathcal{C}^{B}$ $\displaystyle=$
$\displaystyle\frac{G_{F}}{\sqrt{2}}\sum_{p=u,c}V_{ps}^{*}V_{pb}\left[C_{1}(\mu_{{}_{QCD}})\Delta_{1}^{p}C^{B}_{1,2}(\mu_{{}_{QCD}},\mu)+C_{7}(\mu_{{}_{QCD}})\Delta_{7}C^{B}_{1,2}(\mu_{{}_{QCD}},\mu)+C_{8}(\mu_{{}_{QCD}})\Delta_{8}C^{B}_{1,2}(\mu_{{}_{QCD}},\mu)\right],$
(17)

where $\mu_{{}_{QCD}}=\mu_{h}\sim m_{b}$ is the hard scale at which QCD is
matched on to SCETI and $\mu\sim\sqrt{m_{b}\Lambda}$ is an intermediate scale.
The tree-level matching of $Q_{7}$ is trivial as can be seen from Eq. (11).
The matching coefficient at order $\mathcal{O}(\alpha_{s})$ can be obtained
via matching the QCD diagram 1(a) at leading power to SCETI current $J^{A}$ as
given in [24, 25]

(a)

(b)

(c)

Figure 1: The diagrams (a), (b) and (c) for matching $Q_{7}$ onto $J^{A}$ ,
$J_{1}^{B}$ and $J_{2}^{B}$, respectively.

(a)

(b)

(c)

Figure 2: The diagrams (a), (b) and (c) for matching $Q_{1}$ onto $J^{A}$ ,
$J_{1}^{B}$ and $J_{2}^{B}$, respectively.

(a)

(b)

Figure 3: The diagrams (a), (b) for matching $Q_{8}$ onto SCETI operators.

$\displaystyle\Delta_{7}C^{A}$ $\displaystyle=$
$\displaystyle\frac{e\bar{m}_{b}E_{\gamma}}{2\pi^{2}}\left\\{1+\frac{\alpha_{s}(m_{b})C_{F}}{4\pi}\left[-2\text{ln}^{2}\frac{\mu}{2E}-5\text{ln}\frac{\mu}{2E}-2\text{ln}\frac{\mu_{{}_{QCD}}}{2E}-2\text{Li}_{2}(1-\frac{2E}{m_{b}})-6-\frac{\pi^{2}}{12}\right]+\mathcal{O}(\alpha_{s}^{2})\right\\},$
(18)

where $m_{b}$ is the pole mass of $b-$quark and $\bar{m}_{b}$ is the running
mass at $\mathcal{O}(\alpha_{s})$ in $\overline{MS}$ scheme at the scale
$\mu_{QCD}$, i.e.,

$\displaystyle\overline{m}_{b}(\mu_{QCD})$ $\displaystyle=$ $\displaystyle
m_{b}\left[1+\frac{\alpha_{s}C_{F}}{4\pi}\left(3\text{ln}\frac{m_{b}^{2}}{\mu^{2}_{QCD}}-4\right)\right].$

The B-type operators defined in equations (13) and (14) are power suppressed
as compared to A-type - but they will contribute at leading power when matched
to the SECTII operators. For the coefficients of operator $Q_{7}$ one could
match the QCD diagrams 1(b) and 1(c) to currents $J_{1}^{B}$ and $J_{2}^{B}$,
respectively. Making use of the tensor coefficients for B-type currents in Eq.
(11), the matching gives

$\Delta_{7}C_{1}^{B}=\frac{e\bar{m}_{b}E_{\gamma}}{4\pi^{2}m_{b}}\qquad,\qquad\Delta_{7}C_{2}^{B}=0+\mathcal{O}(\alpha_{s}).$
(19)

For the operator $Q_{1}$ with a $c-$quark loop (setting $m_{u}=0$), the QCD
diagram that contributes at leading order for the A-type current matching is
given in Fig. 2(a). It is a vertex diagram from which hard modes have to be
integrated out along with the contribution of the fermion loop when matching
to SCETI. Figs. 2(b) and 2(c) are bremsstrahlung diagrams with an on-shell
photon. The corresponding coefficients in this case are

$\displaystyle\Delta_{1}^{c}C^{A}$ $\displaystyle=$
$\displaystyle\frac{\alpha_{s}C_{F}}{4\pi}G_{i}(x_{c})\Delta_{7}C^{A},$
$\displaystyle\Delta_{1}^{c}C^{B}_{1}(u)$ $\displaystyle=$
$\displaystyle-\Delta_{1}^{q}C_{2}^{B}(u)=\frac{2e}{3}f\left(\frac{\bar{m}_{c}^{2}}{4\bar{u}EE_{\gamma}}\right)\Delta_{7}C_{1}^{B},$
(20)

where $x_{c}=\bar{m}_{c}^{2}/m_{b}^{2}$ and the functions $G_{1}(x_{c})$ and
$f(\frac{m_{c}^{2}}{4EE_{\gamma}\bar{u}})$ are given in the Appendix.

Ignoring the penguin operators $Q_{3}-Q_{6}$, one is left with the gluon
chromomagnetic operator $Q_{8}$. The diagrams shown in Figs. Figs. 3(a) and
3(b) are required to match the SCETI A-type and B-type operators,
respectively. Contrary to this, the diagrams drawn in Fig. 3, where the photon
is emitted from the $b-$quark are suppressed therefore
$\Delta_{8}C_{2}^{B}(u)\simeq 0$. The coefficients for the A- and B- type
currents matching are [26]

$\displaystyle\Delta_{8}C^{A}\ \ \
=\frac{\alpha_{s}C_{F}}{4\pi}G_{8}\Delta_{7}C^{A}\qquad;\qquad\Delta_{8}C_{1}^{B}(u)=\frac{\bar{m}_{b}}{4\pi^{2}}\frac{e}{3}\frac{\bar{u}}{u}.$
(21)

Now the photon emission from the spectator quark may contribute to the leading
power. In Figs. 4(a) and 4(b), the photon emission from anti-quark contributes
at sub-leading power and hence is suppressed [27, 28]. In Figs. 4(c) and 4(d),
the photon-emission from spectator quark contributes at leading power when
matched onto SCETII \- but it vanishes when we calculate the matrix elements
for the transversely polarized axial-vector mesons using Eq. (35).

(a)

(b)

(c)

(d)

Figure 4: Diagrams with photon emission from the spectator anti-quark matched
on to $J^{C}$.

##  III SCETII Operators and SCETI $\rightarrow$ SCETII matching

We know that the SCETII is a low energy theory of soft, collinear, and soft-
collinear modes. The next step in a two-step matching procedure is to go from
an intermediate scale to the hadronic scale achieved by matching SCETI
operators with SCETII operators. It requires the construction of SCETII
operators at leading power scaling in a way we constructed them in the SCETI.
In these operators, the relevant WCs have known to be the Jet functions.
Furthermore, at a leading-power, the soft-collinear modes could be de-coupled
via field redefinition in soft and collinear Lagrangians. To obtain the gauge-
invariant operators, again we need to define soft and collinear Wilson lines
as we wrote them for hard-collinear fields in Eq. (8)

$\displaystyle\mathcal{W}_{c}(x)$ $\displaystyle=\text{P
exp}\left(ig\int_{-\infty}^{0}ds\bar{n}\cdot A_{c}(x+s\bar{n})\right),$
$\displaystyle\mathcal{S}(x)$ $\displaystyle=\text{P
exp}\left(ig\int_{-\infty}^{0}ds\bar{n}\cdot A_{s}(x+s\bar{n})\right).$ (22)

It is useful to introduce the gauge-invariant quark and gluon fields which
scales as follows

$\displaystyle\bar{\chi}_{c}$
$\displaystyle=\mathcal{W}_{c}^{\dagger}\xi_{c}\sim\lambda,\qquad
Q_{s}=\mathcal{S}^{\dagger}q_{s}\sim\lambda^{3/2},\qquad\mathcal{H}=\mathcal{S}^{\dagger}h\sim\lambda^{3/2},$
$\displaystyle\mathcal{A}_{c}^{\mu}$
$\displaystyle=\mathcal{W}_{c}^{\dagger}(iD_{c}^{\mu}\mathcal{W}_{c})\sim(\lambda^{2},0,\lambda),\qquad\mathcal{A}_{s}^{\mu}=\mathcal{S}^{\dagger}(iD_{s}^{\mu}\mathcal{S})\sim(\lambda^{2},0,\lambda).$
(23)

We are not required to perform SCETI to SCETII matching for A-type currents as
the matrix elements of $J^{A}$ already give the non-factorizable part, i.e.,
soft-overlap function. Therefore we only need to define B-type operators for
the matching. Recall, there were two B-type currents, so they match onto as

$\displaystyle O_{1}^{B}(s,t)$ $\displaystyle=$
$\displaystyle\bar{\chi}_{c}(s\bar{n})(1+\gamma_{5})\not{\mathcal{A}}_{\bar{c}\perp}^{(em)}(0)\frac{\bar{\not{n}}}{2}\chi_{c}(0)\bar{\mathcal{Q}}_{s}(tn)(1-\gamma_{5})\frac{\not{n}}{2}\mathcal{H}_{s}(0),$
$\displaystyle O_{2}^{B}(s,t)$ $\displaystyle=$
$\displaystyle\bar{\chi}_{c}(s\bar{n})(1+\gamma_{5})\frac{\bar{\not{n}}}{2}\chi_{c}(0)\bar{\mathcal{Q}}_{s}(tn)(1+\gamma_{5})\frac{\not{n}}{2}\not{\mathcal{A}}_{\bar{c}\perp}^{(em)}(0)\mathcal{H}_{s}(0).$
(24)

The photon field is in the $\bar{n}$ direction and is collinear. The first
operator in Eq. (24) matches SCETI operator with photon emission from
$b-$quark and it requires opposite chirality of the spectator quark field,
$\bar{\mathcal{Q}}_{s}$. On the other hand, the second operator matches SCETI
operator with photon emission from $s-$quark.

We can see that at leading power, the SCETII operators have a soft and a
collinear part only. Therefore, the matrix elements would be found
independently in terms of LCDA of $B-$ and final state axial-vector mesons.
The decoupling of soft-collinear modes made our factorization successful.
Moreover, if they did not decouple they would have appeared as end-point
divergences in convolution integrals (1), which is in contrast to the LEET as
discussed in [29]. To mention, the LEET does not correctly produce the
infrared divergences; therefore, we left with end-point divergences arising
from the convolution integral at a leading twist. These end-point divergences
were then assumed to be absorbed in the soft-form factor in the LEET. Hence,
it would suggest the difference between soft-form factors calculated in both
theories.

With this note, the corresponding WC in momentum space is given as

$D_{1,2}^{B}(\omega,u)\equiv\int ds\int dte^{-i\omega n\cdot
vt}e^{ius\bar{n}\cdot P}\tilde{D}_{1,2}^{B}(s,t),$ (25)

Figure 5: SCETI diagram matching onto SCETII 4-quark operator. The dashed
gluon is hard-collinear

.

which is actually a convolution of the SCETI WC, $C_{i}^{B}$, with a jet
function $\mathcal{J}_{\perp,\parallel}$, i.e.,

$\displaystyle
D_{1,2}^{B}(\omega,u,\mu_{i})=\frac{1}{\omega}\int_{0}^{1}dy\mathcal{J}_{\perp}\left(u,y,ln\frac{2E\omega}{\mu_{i}^{2}},\mu_{i}\right)C_{1,2}^{B}(y,\mu_{1}),$
(26)

where $\mu_{i}\sim\sqrt{2E\Lambda_{QCD}}$ is an intermediate scale. At tree
level, the matching of $J_{1}^{B}$ onto $O_{1}^{B}$ is trivial (c.f. Fig. 5)
and the corresponding jet function is given as

$\displaystyle\mathcal{J}_{\perp}(u,v)$ $\displaystyle=$
$\displaystyle\mathcal{J}_{\parallel}(u,v)=-\frac{4\pi
C_{F}\alpha_{s}}{N}\frac{1}{2E\bar{u}}\delta(u-v),$ (27)

while the similar matching for $J_{2}^{B}$ is suppressed at a leading power.

##  IV Matrix Elements

The quark and gluon fields defined so far are in different momentum regions.
It will help us to conveniently match the effective theories at different
scales and summing large logarithms. We will discuss it in detail in Sec. V.
As we have already mentioned that the $B-$meson is a bound state of two soft
quarks, and the final state axial-vector meson is that of two collinear
quarks; therefore, their four-momenta can be defined as

$\displaystyle p_{B}^{\mu}$ $\displaystyle=m_{B}v^{\mu},\quad\quad
p^{\mu}_{A}=En^{\mu}+\frac{m_{A}^{2}}{4E}\bar{n}^{\mu}\quad\quad
p_{\gamma}^{\mu}=(E-\frac{m^{2}_{A}}{4E})\bar{n}^{\mu},$ (28)

where $E$ is the off-shell energy with $p_{A}^{2}=m_{A}^{2}$ and
$p_{\gamma}^{2}=0$. One can notice that the large component of the axial-
vector meson is in the $n-$direction while the emitted photon in
$\bar{n}-$direction. Similarly, the on-shell energy, the three-momentum of the
axial-vector meson, and the energy of an on-shell photon are defined as

$\displaystyle
E_{F}=\frac{m_{B}^{2}+m_{A}^{2}}{2m_{B}},\quad\quad|\mathbb{\Delta}|=\sqrt{E_{F}^{2}-m_{A}^{2}},\quad\quad
E_{\gamma}=\frac{m^{2}_{B}-m^{2}_{A}}{2m_{B}},$ (29)

where $E_{F}\sim E\sim\Delta=m_{B}/2$ is the energy of the final state meson
at maximum recoil. Let $\eta^{*}$ define the polarization vector of the axial-
vector meson so that the transversality condition $\eta^{*}\cdot p_{A}=0$
gives

$\displaystyle\eta^{*}\cdot
n=-\frac{m_{{}_{A}}^{2}}{4E\Delta}\eta^{*}\cdot\bar{n}.$ (30)

The photon has polarization $\varepsilon^{*}$ and hence $\varepsilon^{*}\cdot
p_{\gamma}=\varepsilon^{*}\cdot\bar{n}=0$. The $B\rightarrow A$ form factors
can be parameterized in terms of soft-overlap function $\zeta_{A\perp}$ by
using SCETI A-type operator as

$\displaystyle\langle A(p_{A})|\bar{\chi}_{{}_{hc}}\Gamma h|B(v)\rangle$
$\displaystyle=-2E_{F}\zeta_{A}(E_{F})\text{tr}[\bar{\mathcal{M}}_{A\perp}(n)\Gamma\mathcal{M}_{B}(v)],$
(31)

where the projection operators $\mathcal{M}_{A\perp}(n)$ and
$\mathcal{M}_{B}(v)$ are

$\displaystyle\mathcal{M}_{B}(v)$
$\displaystyle=-\frac{1+\not{v}}{2}\gamma_{5},\qquad\mathcal{\bar{M}}_{A\perp}(n)=-\not{\eta}_{\perp}^{*}\gamma_{5}\frac{\not{\bar{n}}\not{n}}{4}.$
(32)

Substituting these projectors from Eq. (32) in Eq. (31) with
$\Gamma=(1+\gamma_{5})$, we get

$\displaystyle\langle
A(p_{F},\eta^{*})\gamma(\varepsilon^{*})|J^{A}|B(v)\rangle=\mathcal{C}^{A}E_{F}\zeta_{A\perp}(E_{F})\left[2(\varepsilon^{*}\cdot\eta^{*})-(n\cdot\eta^{*})(\bar{n}\cdot\varepsilon^{*})\right.$
$\displaystyle\left.-(n\cdot\varepsilon^{*})(\bar{n}\cdot\eta^{*})+i\epsilon^{\alpha\beta\mu\nu}n_{\alpha}\bar{n}_{\beta}\varepsilon^{*}_{\mu}\eta^{*}_{\nu}\right].$

As $\bar{n}\cdot\varepsilon^{*}=0$ and because both photon and axial-vector
meson are left circularly polarized, $\varepsilon^{*}\cdot\eta^{*}=1$. Making
use of Eq. (30) and $\epsilon^{0123}=-1$, we get
$\epsilon^{\alpha\beta\mu\nu}n_{\alpha}\bar{n}_{\beta}\varepsilon^{*}_{\mu}\eta^{*}_{\nu}=-2i$.
This leads to

$\displaystyle\langle
A(p_{F})\gamma|J^{A}|B(v)\rangle=4\mathcal{C}^{A}E_{F}\left(1+\frac{m^{2}_{A}}{8E\Delta}\right)\zeta_{A\perp}(E_{F}).$
(33)

The matrix elements of B-type currents $O_{1,2}^{B}$ of SCETII, with
independent soft and collinear parts can be written as a convolution of the
LCDA’s of the respective meson given here as

$\displaystyle\langle
0|\bar{\mathcal{Q}}_{s}(tn)\frac{\not{n}}{2}\Gamma\mathcal{H}(0)|\bar{B}(v)\rangle$
$\displaystyle=$
$\displaystyle-\frac{i\sqrt{m_{B}}F(\mu)}{2}\text{Tr}\left(\frac{\not{n}}{2}\Gamma\frac{1+\not{v}}{2}\gamma_{5}\right)\times\int_{0}^{\infty}d\omega
e^{-i\omega tn\cdot v}\phi_{B}(\omega,\mu),$ (34) $\displaystyle\langle
A_{\perp}(p_{F},\eta^{*})|\bar{\chi}_{c}(s\bar{n})\Gamma\frac{\bar{n}}{2}\chi_{c}(0)|0\rangle$
$\displaystyle=$ $\displaystyle-\frac{if_{A_{\perp}}(\mu)}{4}\bar{n}\cdot
p_{F}\text{Tr}\left(\not{\eta}^{*}_{\perp}\gamma_{5}\Gamma\frac{\not{\bar{n}}\not{n}}{4}\right)\times\int_{0}^{\infty}due^{ius\bar{n}\cdot
p_{F}}\phi_{A_{\perp}}(u,\mu),$ (35)

where $\phi_{B}$ and $\phi_{A\perp}$ are the distribution amplitudes for $B-$
and $A-$mesons, respectively, and $f_{A\perp}$ is the axial-vector meson decay
constant which depends upon the scale of the theory. The scale-dependent
quantity $F(\mu)$ is related to $B-$meson decay constant $f_{B}$ at NLO as

$\displaystyle f_{B}\sqrt{m_{B}}$
$\displaystyle=F(\mu)\left(1+\frac{C_{F}\alpha_{s}(\mu)}{4\pi}\left(3\text{ln}\frac{m_{b}}{\mu}-2\right)\right).$

For $\Gamma=(1-\gamma_{5})$ the $B-$meson LCDA is $n\cdot v$ and for
$\Gamma=(1+\gamma_{5})\gamma_{\perp}^{\mu}$ it vanishes. Contrary to this, for
$\Gamma=(1+\gamma_{5})$ axial-vector meson LCDA vanishes, whereas for
$\Gamma=(1+\gamma_{5})\gamma_{\perp}^{\mu}$ it is $(1+m_{A}^{2}/8E\Delta)$ and
hence $O_{1}^{B}$ will effectively contributes. Collecting the results from
Eqs. (33, 34) and Eq. (35) to get the complete matrix elements at leading
power and at an order $\mathcal{O}(\alpha_{s})$, we have

$\displaystyle\langle
A_{\perp}(p_{F},\eta^{*})\gamma(\varepsilon^{*})|\mathcal{H}_{W}|B(v)\rangle$
$\displaystyle=$ $\displaystyle
4\left(1+\frac{m^{2}_{A}}{8E\Delta}\right)\left[\mathcal{C}^{A}E_{F}\zeta_{A\perp}(E_{F})-\frac{m_{B}^{3/2}F(\mu)}{2}\int_{0}^{\infty}\frac{d\omega}{\omega}\phi_{B}(\omega,\mu)\right.$
(36) $\displaystyle\times$
$\displaystyle\left.\int_{0}^{1}duf_{A\perp}\phi_{A_{\perp}}(u,v)\int_{0}^{1}dv\mathcal{J}_{\perp}\left(u,v,\text{ln}\frac{m_{B}\omega}{\mu^{2}}\right)\mathcal{C}_{1}^{B}(v,\mu)\right].$

##  V Resummation

The matching coefficients calculated in the previous section are reliable and
can be trusted up-to a hard-scale $\mu_{h}\sim m_{b}$. However, one needs to
re-sum large logarithms in the corrected coefficients in going from hard to an
intermediate scale such as $\mu_{h}\rightarrow\mu_{i}$ or a hadronic scale
$\mu_{i}\rightarrow\mu\sim\Lambda$. In principle, one needs to separately re-
sum the logarithms in the matching calculation of QCD to SCETI and then from
SCETI to SCETII. In this context, the A-type coefficients $\mathcal{C}^{A}$
are obtained at a hard-scale along with the soft-overlap function
$\zeta_{A\perp}$. Hence, we could neglect the running of A-type matching
coefficients. The running of B-type operators requires the calculation of the
one-loop anomalous dimension by keeping the UV divergent terms appearing in
the SCETI loop diagrams [c.f. Fig. 3 of [23]]. As we are dealing with axial-
vector meson(s) having mass around $1.3$ GeV that lies very close to the
intermediate scale ($1.5$ GeV), therefore, the resummation of SCETII operators
is not needed in our case.

The evolution of the B-type coefficients read as

$\displaystyle\frac{d}{dln\mu}\mathcal{C}^{B}_{j}(E,v)$ $\displaystyle=$
$\displaystyle\gamma^{B}_{ij}(u,v)\mathcal{C}^{B}_{i}(E,u),$ (37)

where $\gamma^{B}$ is an anomalous dimension and it depends upon the variable
$u$ that denotes the fraction of momentum carried by hard-collinear quark or
gluon. The solution of the evolution equation (37) is

$\displaystyle\mathcal{C}^{B^{\prime}}_{j}(E,u,\mu)$ $\displaystyle=$
$\displaystyle\left(\frac{2E_{F}}{\mu_{h}}\right)^{a(\mu_{h},\mu)}e^{S(\mu_{h},\mu)}\int_{0}^{1}dvU_{\perp,\parallel}(u,v,\mu_{h},\mu)\mathcal{C}_{j}^{B^{\prime}}(E,v,\mu_{h}),$
(38)

where the prime notation is adopted to make it consistent with our earlier
tree level coefficients in Eq. (11). The functions $a(\mu_{h},\mu)$ and
$S(\mu_{h},\mu)$ that appeared in Eq. (38) are defined as [30, 31]

$\displaystyle S(\mu_{h},\mu)$ $\displaystyle=$
$\displaystyle\frac{\Gamma_{0}}{4\beta_{0}^{2}}\left[\frac{4\pi}{\alpha_{s}(\mu_{h})}\left(1-\frac{1}{r_{1}}-\text{ln}r_{1}\right)+\frac{\beta_{1}}{2\beta_{0}}\text{ln}^{2}r_{1}-\left(\frac{\Gamma_{1}}{\Gamma_{0}}-\frac{\beta_{1}}{\beta_{0}}(r_{1}-1-\text{ln}r_{1})\right)\right],$
$\displaystyle a(\mu_{h},\mu)$ $\displaystyle=$
$\displaystyle-\frac{\Gamma_{0}}{2\beta_{0}}\text{ln}r_{1},$ (39)

where $\Gamma_{0}=4C_{F}$ and

$\displaystyle\Gamma_{1}$ $\displaystyle=$ $\displaystyle
4C_{F}\left[\left(\frac{67}{9}-\frac{\pi^{2}}{3}\right)C_{A}-\frac{20}{9}T_{F}n_{f}\right],\beta_{0}=\frac{11}{3}C_{A}-\frac{4}{3}T_{F}n_{F},$
$\displaystyle\beta_{1}$ $\displaystyle=$
$\displaystyle\frac{34}{3}C_{A}^{2}-\frac{20}{3}C_{A}T_{F}n_{f}-4C_{F}T_{F}n_{f},$
(40)

along with $r_{1}=\alpha_{s}(\mu)/\alpha_{s}(\mu_{h})$. To find the evolution
function $U_{\perp,\parallel}$, we employed the method used in [23] to solve
the following RG equation using the initial condition
$U_{\perp,\parallel}(u,v,\mu_{h},\mu_{h})=\delta(u-v)$

$\displaystyle\frac{d}{d\text{ln}\mu}\frac{U_{\perp,\parallel}(u,v,\mu_{h},\mu)}{\bar{u}}$
$\displaystyle=$
$\displaystyle\int_{0}^{1}dyy\bar{y}^{2}\frac{V_{\perp,\parallel}}{\bar{y}\bar{u}}\frac{U_{\perp,\parallel}(u,v,\mu_{h},\mu)}{\bar{y}}+w(u)\frac{U_{\perp,\parallel}(u,v,\mu_{h},\mu)}{\bar{u}}.$
(41)

Figure 6: Evolution function with orange being $U_{\perp}$ and blue being
$U_{\parallel}$ using 80 basis functions. The increase in the number of basis
functions will decrease the amplitude of oscillations and smooth the curves. A
slope can be fitted on these curves as a function of $u$. However, this will
not affect the results calculated here.

The functions $V_{\perp,\parallel}$ and $w(u)$ are defined in [23] which are
actually the anomalous dimension contributions. A basis function comprising of
Jacobi Polynomials (obtained by solving eigenvalue equation for
$V_{\perp,\parallel}$) is used to construct the solution for Eq. (41). This
solution is plotted in Fig. 6 by taking a set of 80 basis functions. We can
notice from the figure that the parallel evolution function $U_{\parallel}$ is
almost constant at a value of $0.98$ for all values of $u$, which is
represented by blue lines. The perpendicular evolution function, $U_{\perp}$,
on the other hand can be parametrized as a linear function of the momentum
fraction variable $u$.

The RG-improved amplitude can now be written in a short form as

$\displaystyle\mathcal{M}=\mathcal{C}^{A}(E,\mu)\zeta_{A\perp}(E)$
$\displaystyle-$
$\displaystyle\frac{\sqrt{m_{B}}}{4}\left(\frac{2E_{F}}{\mu_{h}}\right)^{a(\mu_{h},\mu_{i})}e^{S(\mu_{h},\mu_{i})}\int_{0}^{\infty}\frac{d\omega}{\omega}\phi_{B}(\omega,\mu)\int_{0}^{1}duf_{A\perp}\phi_{A_{\perp}}(u,v)$
(42) $\displaystyle\times$
$\displaystyle\int_{0}^{1}dv\mathcal{J}_{\perp}\left(u,v,\text{ln}\frac{m_{B}\omega}{\mu^{2}}\right)U_{\perp}(u,v,\mu_{h},\mu_{i})\mathcal{C}_{1}^{B}(v,\mu).$

##  VI Soft-overlap function and Branching Fractions

The RG-improved amplitude given in Eq. (42) is useful to study the branching
fraction of $B\rightarrow(K_{1},b_{1},a_{1})\gamma$ decay, which is an
important observable for experimental searches. In the case of decays under
consideration, we can write

$\displaystyle Br(B^{+}\rightarrow(K_{1}^{+},b_{1}^{+},h_{1}^{+})\gamma)$
$\displaystyle=\frac{\tau_{B}m_{B}}{4\pi}\left(1+\frac{m_{A}^{2}}{8E_{F}\Delta}\right)\left(1-\frac{m_{A}^{2}}{m_{B}^{2}}\right)|\mathcal{M}|^{2},$
(43)

where $m_{A}$ denotes the mass of the axial-vector meson. The coefficients
$\mathcal{C}^{A}$ and $\mathcal{C}_{1}^{B}$ used in Eq. (42) are assembled as

$\displaystyle\mathcal{C}^{A}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V^{*}_{cp}V_{cb}\left[C_{7}+\frac{C_{F}\alpha_{s}}{4\pi}\left(C_{1}G_{1}(x_{c})+C_{8}G_{8}\right)\right]\nabla_{7}C^{A},$
(44) $\displaystyle\mathcal{C}_{1}^{B}$
$\displaystyle=\frac{G_{F}}{\sqrt{2}}V^{*}_{cp}V_{cb}\left[C_{7}+C_{1}\frac{1}{3}f\left(\frac{\bar{m}_{c}^{2}}{4\bar{u}E_{F}E_{\gamma}}\right)+C_{8}\frac{\bar{u}}{3u}\right]\nabla_{7}C^{B}_{1},$
(45)

with $p=s$ and $d$ for $K_{1}$ and $(b_{1},a_{1})$ mesons, respectively. To
find the branching fraction, we require the soft-overlap function
$\zeta_{\perp}(E_{F})$. We could approximate this non-factorizable part by
using the light-cone-sum-rules (LCSR) or SCET sum rules [16]. In this
particular case, we will rely on the LCSR values of the axial-vector form
factor $\mathcal{A}$, and we can write it in terms of the soft-overlap
function

$\mathcal{A}(E_{F})=\left(1+\frac{m_{A}}{m_{B}}\right)\frac{E_{F}}{\Delta}\mathcal{C}^{A}_{A_{1}}\zeta_{A}^{\perp}(E_{F}),$
(46)

Our choice of $\mathcal{A}(E_{F})$ is such that the factorizable part of
$\mathcal{A}(E_{F})$ is $\mathcal{O}(\alpha_{s})$ suppressed compared to other
form factors. This is consistent to our previous LEET results for form factors
[29].

Table 1: The values of input parameters used in our numerical analysis. Values
---
$m_{B}$ 5.28 GeV |  $G_{F}$ $1.16\times 10^{-6}$ GeV |  $m_{K_{1A}}$ 1.31GeV
$m_{K_{1B}}$ 1.34 GeV |  $m_{a_{1}}$ 1.23GeV |  $m_{b_{1}}$ 1.21GeV
$f_{K_{1A}\perp}$ 0.25$\pm$ 0.013 GeV |  $f_{K_{1B}\perp}$ 0.19$\pm$ 0.01 GeV |  $f_{a_{1}}$ 0.234GeV
$f_{b_{1}\perp}$ 0.18$\pm$0.008GeV |  $a_{1\perp}^{(a_{1})}(1GeV)$ -1.04$\pm$0.34 |  $a_{2\perp}^{(b_{1})}(1GeV)$ 0.03$\pm 0.19$
$f_{B}$ 0.2GeV |  $a_{0\perp}^{K_{1A}}(1GeV)$ 0.26+0.03-0.22 |  $a_{\perp 1}^{(K_{1A})}(1GeV)$ -1.08$\pm$0.48
$a_{2\perp}^{(K_{1A})}(1GeV)$ 0.02$\pm$0.2 |  $a_{1\perp}^{(K_{1B})}(1GeV)$ $0.3+0.0-0.31$ |  $a_{\perp 2}^{(K_{1B})}(1GeV)$ $-0.02\pm 0.22$
$\lambda_{B,+}$ 0.35GeV |  $\tau_{B}$ 1.6ps |  $C_{1}(m_{b})$ 1.108
$C_{7}^{(eff)}(m_{b})$ -0.311 |  $|V^{*}_{tb}V_{td}|$ $5\times 10^{-3}$ |  $|V^{*}_{cs}V_{cb}|$ $4\times 10^{-2}$
$\theta_{K_{1}}$ -34${}^{0}\pm 13^{0}$ |  $C_{8}^{(eff)}(m_{b})$ -0.151 |

The $C^{\text{A}}_{A_{1}}$ is the coefficient obtained by matching A-type
operators with axial-vector current $A_{1}=\gamma^{\mu}\gamma_{5}$ in the full
theory. It is given as

$C^{\text{A}}_{A_{1}}=1-\frac{\alpha_{s}C_{F}}{4\pi}\left[\left(\frac{1}{1-x}-3\right)\text{ln}(x)+2\text{ln}^{2}(x)+2\text{ln}^{2}\left(\frac{\mu_{i}}{m_{b}}\right)^{2}+2\text{Li}_{2}(1-x)-(4\text{ln}(x)-5)\text{ln}\left(\frac{\mu_{i}}{m_{b}}\right)+\frac{\pi^{2}}{12}+6\right],$
(47)

with $x=2E_{F}/m_{b}$. As physical $K_{1}$ meson is a mixture of
${}^{1}P_{1}(K_{1B})$ and ${}^{3}P_{1}(K_{1A})$ states therefore, their
corresponding form factors will mix too. The two physical mass states
$K_{1}(1270)$ and $K_{1}(1400)$ depend upon the mixing angle $\theta_{K_{1}}$
and the corresponding form-factors are given by

$\displaystyle\mathcal{A}^{(1270)}(E_{F})$ $\displaystyle=$
$\displaystyle\frac{m_{B}+m_{K{{}_{1}(1270)}}}{m_{B}+m_{K{{}_{1A}}}}\mathcal{A}^{K_{1A}}(E_{F})\text{sin}\theta_{K_{1}}+\frac{m_{B}+m_{K{{}_{1}(1270)}}}{m_{B}+m_{K}{{}_{1B}}}\mathcal{A}^{K_{1B}}(E_{F})\text{cos}\theta_{K_{1}},$
$\displaystyle\mathcal{A}^{(1400)}(E_{F})$ $\displaystyle=$
$\displaystyle\frac{m_{B}+m_{K{{}_{1}(1400)}}}{m_{B}+m_{K{{}_{1A}}}}\mathcal{A}^{K_{1A}}(E_{F})\text{cos}\theta_{K_{1}}-\frac{m_{B}+m_{K{{}_{1}(1400)}}}{m_{B}+m_{K{{}_{1B}}}}\mathcal{A}^{K_{1B}}(E_{F})\text{sin}\theta_{K_{1}}.$
(48)

These form-factors are given as a function of final state meson’s energy with
maximum recoil, i.e., $q^{2}=0$. The LCDAs for final state axial-vector mesons
at twist-2 for $1^{1}P_{1}$ $(K_{1B},b_{1})$ and $1^{3}P_{1}$ $(K_{1B},a_{1})$
states are calculated in [32] and these are given as

$\displaystyle\Phi_{\perp}^{1^{1}\text{P}_{1}}(u)$ $\displaystyle=$
$\displaystyle
6u\bar{u}\left[1+3a_{1}^{\perp}\Upsilon+a_{2}^{\perp}\frac{3}{2}(5\Upsilon^{2}-1)\right],$
$\displaystyle\Phi_{\perp}^{1^{3}\text{P}_{1}}(u)$ $\displaystyle=$
$\displaystyle
6u\bar{u}\left[a_{0}^{\perp}+3a_{1}^{\perp}\Upsilon+a_{2}^{\perp}\frac{3}{2}(5\Upsilon^{2}-1)\right],$
(49)

with $\Upsilon=2u-1$. The LCDA for $1^{1}\text{P}_{1}$ and $1^{3}\text{P}_{1}$
states are normalized as $\int_{0}^{1}du\phi_{\perp}(u)/u=1$ and
$\int_{0}^{1}du\phi_{\perp}(u)/u=a_{0}^{\perp}$, respectively. The Gagenbaur
moments and decay constants at a scale $\mu=1$GeV are given in Table 1 .
Ideally, one needs to RG run these parameters from intermediate scale to lower
scale - but RG-improvement of these hadronic parameters to find branching
fractions is not helpful as the uncertainties associated with their values at
present are large. For physical $K_{1}$ meson states these LCDAs mixes just
like the form factors given in Eq. (48) and these can be written as [33]

$\displaystyle\Phi^{K_{1}(1270)}_{\perp}(u)$ $\displaystyle=$
$\displaystyle\frac{f^{K_{1A}}_{\perp}}{f^{K_{1}(1270)}_{\perp}}\text{sin}\theta_{K_{1}}\Phi^{K_{1A}}_{\perp}(u)+\frac{f^{K_{1B}}_{\perp}}{f^{K_{1}(1270)}_{\perp}}\text{cos}\theta_{K_{1}}\Phi^{K_{1B}}_{\perp}(u),$
$\displaystyle\Phi^{K_{1}(1400)}_{\perp}(u)$ $\displaystyle=$
$\displaystyle\frac{f^{K_{1A}}_{\perp}}{f^{K_{1}(1400)}_{\perp}}\text{cos}\theta_{K_{1}}\Phi^{K_{1A}}_{\perp}(u)-\frac{f^{K_{1B}}_{\perp}}{f^{K_{1}(1400)}_{\perp}}\text{sin}\theta_{K_{1}}\Phi^{K_{1B}}_{\perp}(u).$
(50)

Many different models deals with the $B-$meson distribution amplitudes see
e.g. [34]. Recently, the BABAR experiment [35] has constrained the value of
$\lambda_{B}$ in range $0.30-0.67$ GeV. In ref. [36], Beneke et al. have
further improved this range by considering power- and the radiative
corrections. In the present analysis, we used the optimal value
$\lambda_{B}\sim 0.35$GeV. The decay constants for the mesons are taken from
[32].

From Eq. (46), by taking the values of $\mathcal{A}_{K_{1A}}(0)=0.45\pm 0.08$
and $\mathcal{A}_{K_{1B}}(0)=-0.37\pm 0.06$ for $K_{1}(1270)$ and
$K_{1}(1400)$ mixed $K_{1}$ states and $\mathcal{A}_{a_{1}}(0)=0.48\pm 0.08$,
$\mathcal{A}_{b_{1}}(0)=-0.025\pm 0.05$, the values of soft form factor
$\zeta_{\perp}(E_{F})$ for different axial-vector states are obtained to be
[32]

$\displaystyle\zeta^{K_{1}(1270)}_{\perp}(E_{F})$ $\displaystyle=$
$\displaystyle-0.47\pm 0.03,$
$\displaystyle\zeta^{K_{1}(1400)}_{\perp}(E_{F})$ $\displaystyle=$
$\displaystyle 0.14\pm 0.06,$
$\displaystyle\zeta^{a_{1}(1260)}_{\perp}(E_{F})$ $\displaystyle=$
$\displaystyle 0.37\pm 0.08,$
$\displaystyle\zeta^{b_{1}(1235)}_{\perp}(E_{F})$ $\displaystyle=$
$\displaystyle-0.22\pm 0.04,$ (51)

where we took $\theta_{K_{1}}=-34^{o}$, and the uncertainties are associated
to the LCSR value of the axial-vector form factor $\mathcal{A}$ for different
axial-vector mesons. It is important to emphasis that the value of
$\zeta_{K_{1}}$ for $K_{1}(1400)$ is highly sensitive to the value of the
mixing angle. We can see that its central value lies in the range
$[0.03,0.24]$ when we vary $\theta_{K_{1}}\in[-46^{o},-22^{o}]$, i.e. $\mp
12^{o}$ around $\theta_{K_{1}}=-34^{o}$. However, the values of the form
factor for $K_{1}(1270)$ do not change significantly and they lie in the range
$[-0.49,-0.45]$.

In Eq. (51), we can see that the value of $\zeta_{K_{1}(1400)}$ is almost
three times smaller than $\zeta_{K_{1}(1200)}$, and this suppression of the
form factor due to the mixing angle for the physical state $K_{1}(1400)$ makes
the convolution integral almost equally important. Also, the hadronic
uncertainties arising due to the spectator part affect the branching ratio of
$K_{1}(1400)$ more compared to $K_{1}(1270)$. The LCSR value
$\mathcal{A}(0)=0.45$ for $a_{1}$ is significantly different from the one
found in [37] i.e. $\mathcal{A}(0)=0.26$. Hence, to have a reliable estimate
of the branching fraction for the radiative $B\to a_{1}\gamma$ decay, we
desire to have a more accurate value of $\mathcal{A}(0)$. To do so, one would
probably have to rely on SCET sum rules similar to [16, 38] or use lattice QCD
results to fix the overlap function. The other option for its fixing is to use
an experimentally measured branching fraction when available. Nevertheless,
these branching fractions calculated here and tabulated in Table 2 are within
a measurable range of LHCb and the future $B$-factories.

We compared the theoretical values of the branching fractions for $B\to
K_{1}(1270,1400)\gamma$ decay obtained here with the corresponding results
obtained using ADS/QFT approach [39], LCQCD Sum Rules [40] and the
experimental values in Table2. The uncertainty in branching fraction for our
work in Table 2 is due to LCSR. The result for $K_{1}(1270)$ is less impacted
by hadronic uncertainties and is well within the range of observed
experimental value.

Keeping in view that the branching ratio of $B\to K_{1}(1270)\gamma$ is
already measured experimentally, it will be useful to give the following
ratios

$\displaystyle\mathcal{R}[\mathcal{B}(B\rightarrow
K_{1}(1270)\gamma)/(\mathcal{B}(B\rightarrow a_{1}(1260)\gamma)]$
$\displaystyle=$ $\displaystyle 68.6,$
$\displaystyle\mathcal{R}[\mathcal{B}(B\rightarrow
K_{1}(1270)\gamma)(\mathcal{B}(B\rightarrow b_{1}(1235)\gamma)]$
$\displaystyle=$ $\displaystyle 178.$ (52)

Table 2: The calculated values of the branching fractions of $B\to\left(b_{1},\;a_{1},\;K_{1}(1270,1400)\right)\gamma$ decays. Using $\theta_{K_{1}}=-34^{o}$ the results of the branching fractions of $B\to\left(K_{1}(1270,1400)\right)\gamma$ calculated in this work are compared with ADS/QFT [39], LCQCD Sum Rules [40], and the experimental values [17]. All the results are of the order $10^{-6}$. Decay Channel | ADS/QFT Approach [39] | LCQCD Sum Rules [40] | This work | Exp. values [17]
---|---|---|---|---
$K_{1}^{+}(1270)\gamma$ | $71\pm 23$ | $66^{+21+30+2+6}_{-12-24-4-12}$ | $48\pm 6$ | $44^{+7}_{-6}$
$K_{1}^{+}(1400)\gamma$ | $15.6\pm 10.4$ | $6.5^{+4+2.6+0.1+11.9}_{-2.2-0-0.2-5.9}$ | $5.4\pm 4.8$ | $10^{+5}_{-4}$
$a_{1}^{+}(1260)\gamma$ | – | – | $0.7\pm 0.037$ | –
$b_{1}^{+}(1235)\gamma$ | – | – | $0.27\pm 0.04$ | –

It is tempting to find the branching fractions for other $1^{1}P_{1}$ states
$(h_{1}(1170),\;h_{1}(1380))$ and $1^{3}P_{1}$ states
$(f_{1}(1285),\;f_{1}(1420))$. These mass states arise due to the mixing of
the singlet and octet states, e.g.,

$\displaystyle|h_{1}(1170)\rangle$
$\displaystyle=|h_{1}\rangle\text{cos}\theta_{1^{1}P_{1}}+|h_{8}\rangle\text{sin}\theta_{1^{1}P_{1}},$
$\displaystyle|h_{1}(1380)\rangle$
$\displaystyle=-|h_{1}\rangle\text{sin}\theta_{1^{1}P_{1}}+|h_{8}\rangle\text{sin}\theta_{1^{1}P_{1}}.$
(53)

Similarly, the $f_{1}$ states mixes with angle $\theta_{1^{3}P_{1}}$.
Previously, the singlet states showed the anomalous behavior in their masses,
and branching fraction, e.g., the branching fraction of $B\rightarrow
K^{+}\eta^{\prime}$ is almost six times larger than that of $B\rightarrow
K^{+}\eta$. For radiative $\phi$ decay, the branching ratio
$\mathcal{R}[\phi\rightarrow f_{0}\gamma/\phi\rightarrow a_{0}\gamma]\sim 6.1$
[17]. We also expect such an enhancement in the branching fraction of the
singlet states in radiative $B-$decays. To take into account the singlet-
amplitude, a new set of operators can be introduced at leading power to
account, e.g., from Fig. 7 of [31]. This would be like $b\rightarrow
sq\bar{q}$ where $q\bar{q}$ hadronizes to $(|f_{1}\rangle,|h_{1}\rangle)$
states. For the partons that carry small momentum, there arise soft-collinear
modes that interact with the collinear and soft parts at leading order. These
contributions appear as end-point singularities and spoil our factorization.
Therefore, the estimated values of the branching fractions for singlet-states
would not be precise enough. It may be possible to improve this estimate by
introducing a new non-perturbative parameter in the factorization relation,
which is on the line of QCD factorization scheme presented in [41]. However,
it lies beyond the scope of the present study.

##  VII Summary

The radiative $B-$ to axial-vector meson decays are studied at next-to-leading
order in $\alpha_{s}$ and at the leading-power of $1/m_{b}$ in the SCET. The
effective theory approach and power counting helped us in collecting diagrams
that contribute to the matching calculations. These diagrams are matched onto
SCETI operators at one-loop order for A-type operators and the tree level for
the B-type operators. The diagrams with photon emission from the spectator
anti-quark contribute to leading power - but vanish when the matrix elements
are calculated for the transversely polarized axial-vector meson. The large
perturbative logarithms are resummed for the case of B-type operators using
one-loop SCETI diagrams. Estimating the soft-overlap functions a.k.a. soft
form factors from LCSR, the branching fractions of
$B\to(K_{1}(1270),K_{1}(1400),b_{1}(1235),a_{1}(1260))\gamma$ decays are
calculated. We find that the value of the form factor for $K_{1}(1400)$
depends greatly on the mixing angle, and so does the corresponding branching
fraction. The results for both physical $K_{1}$ mesons are within the measured
experimental uncertainty. For $(a_{1},b_{1})$ mesons the values lie within the
reach of LHCb and the future B-factories. Hence, we hope that the future
developments in the experimental and theoretical side will help us in having a
precise estimate of the values of the soft overlap functions of $b_{1}$ and
$a_{1}$ mesons.

##  VIII Appendix

For the matching of $Q_{1}$ and $Q_{8}$ on A-type operators, the functions
$G_{1}$ and $G_{8}$ were required. These are taken from [42], and their
explicit form is

$\displaystyle G_{1}(x)$ $\displaystyle=$
$\displaystyle\frac{-104}{27}\text{ln}\frac{\mu}{m_{b}}-\frac{833}{162}-\frac{20\pi
i}{27}+8\frac{8\pi^{2}}{9}x^{3/2}$ $\displaystyle+$
$\displaystyle\frac{2}{9}\left[48+30\pi i-5\pi^{2}-36\xi(3)+(36+6\pi
i-9\pi^{2})\text{ln}(x)\right.$ $\displaystyle\left.\qquad\qquad+(3+6\pi
i)\left(\text{ln}(x)\right)^{2}+\left(\text{ln}(x)\right)^{3}\right]x$
$\displaystyle+$
$\displaystyle\frac{2}{9}\left[18+2\pi^{2}-2\pi^{3}+(12-6\pi^{2})\text{ln}x+6\pi
i\left(\text{ln}(x)\right)^{2}+\left(\text{ln}(x)\right)^{3}\right]x^{2}$
$\displaystyle+$ $\displaystyle\frac{1}{27}\left[-9+112\pi
i-14\pi^{2}+(182-48\pi
i)\text{ln}(x)-126\left(\text{ln}(x)\right)^{2}\right]x^{3}+\mathcal{O}(x^{4}),$
$\displaystyle G_{8}$ $\displaystyle=$
$\displaystyle\frac{8}{3}\text{ln}\left(\frac{\mu}{m_{b}}\right)+\frac{11}{3}+\frac{2\pi
i}{3}-\frac{2\pi^{2}}{9},$ (54)

here $x=m_{c}^{2}/m_{b}^{2}$. The function $f(z)$ where
$z=\frac{m_{q}^{2}}{4EE_{\gamma}\bar{u}}$ given in the matching of $Q_{1}$
operators to B and C type currents is given as

$\displaystyle
f(z)=\left\\{\begin{array}[]{@{}ll@{}}1+4z\left(\text{arctanh}(\sqrt{1-4z})-i\frac{\pi}{2}\right)^{2},&\text{for}\
z<1/4\\\ \\\ 1-4z\ \text{arctan}^{2}\frac{1}{\sqrt{4z-1}},&\text{for}\
z>1/4.\end{array}\right.$ (58)

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# Rain Sensing Automatic Car Wiper Using AT89C51 Microcontroller

Abhishek Das  Aditya Desai  Vivek Dhuri

##  1 Abstract

The turn of the century has seen a tremendous rise in technological advances
in the field of automobiles. With 5G technology on its way and the development
in the IoT sector, cars will start interacting with each other using V2V
communications and become much more autonomous. In this project, an effort is
made to move in the same direction by proposing a model for an automatic car
wiper system that operates on sensing rain and snow on the windshield of a
car. We develop a prototype for our idea by integrating a servo motor and
raindrop sensor with an AT89C51 Microcontroller.

##  2 Introduction

Today’s car wipers are manual systems that work on the principle of manual
switching. So here we propose an automatic wiper system that automatically
switches ON on detecting rain and stops when the rain stops. Our project
brings forward this system to automate the wiper system not need manual
intervention. For this purpose, we use a rain sensor along with a
microcontroller to drive the wiper motor. Our system uses a rain sensor to
detect rain, this signal is then processed by a microcontroller to take the
desired action. The rain sensor works on the principle of using water for
completing its circuit, so when rain falls on it, the circuit gets completed
and sends out a signal to the microcontroller. The microcontroller now
processes this data and controls the motor. This system is equally useful for
Aircraft and a smaller version of this can be used by motorbikers in their
helmets so that they can drive easily in rains. Figure 1 shows the block
diagram for our proposed idea.

We use Assembly Language Coding using arm Keil $\mu$Vision 5 interface. For
PCB designing, we use EAGLE software (Monk & Amos, 2017) and for circuit
simulation, we have used Proteus Design Suite (Su & Wang, 2010).

Figure 1: Block Diagram of Proposed Model

In what follows, we discuss the related prior work for such a problem in the
next section (3), followed by defining the problem statement (4) and
discussing our novel approaches in section (5). We then present our
Experimental setup in section (6) followed with its results and discussion in
section (7). Finally, we end the discussion with conclusion and future
directions in the last section (8). The scripts and circuit designs are
publicly available here.

##  3 Related Work

In the current scenario, only high-end vehicles employ intelligent rain-
sensing automatic wiper systems. Our system is modeled to demonstrate how
useful is an automatic wiper system that adjusts speed itself based on
rainfall intensity. Such a system improves the safety of a ride. There are
many instances of accidents occurring during heavy rainfall due to lack of
proper vision. In many cases, these accidents were due to manual errors (for
example: not increasing the speed of the wiper) from the driver. An automatic,
intelligent system like ours removes any manual errors. Our system adjusts
wiper speed according to the intensity of rainfall and hence improves safety.
Nowadays some models of Ford and Hyundai are also implementing an automatic
wiper system in their vehicles[1].

Figure 2: 8051 Microcontroller pin configuration

Now, we discuss some prior work in this area. Since the rain sensor used in
the automatic wiper system is expensive (Kulkarni & Holalad, ) designed a
semi-automatic rain wiper that could be installed in economic vehicles. Their
semi-automatic rain wiper had Cup sensor which was based on the principle of
rate of water flow and volume of water. The cup sensor was made up of a
conical cup with probes at different levels of height. These levels of the
probe were used to increase the wiper speed. Therefore, depending on the rain
intensity the wiper system could change the speed. Their design was economical
and had three different stages of rain intensity.

Similarly, (Ashik & Basavaraju, 2014) designed automatic wipers with mist
control that worked with three different rain intensities, which are
drizzling, medium rain and heavy rain. The automatic wiper and internal wiper
uses the combination of a sensor, microcontroller and the wiper motor. The
external sensor and internal sensor is based on the principle of conductance.
The microcontroller actuates the speed of the wiper motor by measuring the
rain intensity as detected by the external motor. Similarly, the internal mist
controllers are placed on the windshield which detects the mist signalling the
controller to actuate the internal wiper motor.

##  4 Experimental Setup

###  4.1 AT89C51 Microcontroller

The AT89C51 (Mazidi et al., 2005) is a low-power, high-performance CMOS 8-bit
microcomputer with 4K bytes of Flash programmable and erasable read-only
memory. The on-chip Flash allows the program memory to be reprogrammed in-
system or by a conventional non-volatile memory programmer. By combining a
versatile 8-bit CPU with Flash on a monolithic chip, the Atmel AT89C51 is a
powerful microcomputer that provides a highly-flexible and cost-effective
solution to many embedded control applications. Figure 2 shows the pin
configuration of 8051 Microcontroller.

Figure 3: Rain Sensor Module

###  4.2 Rain Sensor Module

A rain sensor module is an easy tool for rain detection (Gupta et al., ). It
can be used as a switch when a raindrop falls through the raining board and
for measuring rainfall intensity. Figure 3 shows a depiction of a typical Rain
Sensor Module. Due to its compact design and light weight, it can be easily
attached into any system. The module features, a rain board, and the control
board that is separate for more convenience, a power indicator LED, and
sensitivity adjustable through a potentiometer. A raindrop sensor is a board
coated with nickel in the form of lines. It works on the principle of ohms
law. When there is no raindrop on board. Resistance is high so we get high
voltage according to V=IR. When raindrop present it reduces the resistance
because water is a conductor of electricity and the presence of water connects
nickel lines in parallel so reduced resistance and the reduced voltage drop
across it.

Figure 4: Servo Motor

Figure 5: Operation of Servo based on Pulse Width Modulation

###  4.3 Servo Motor

Servo motors (Sachin & Gaonkar, 2013) are self-contained mechanical devices
that are used to control the machines with great precision. . Usually the
servo motor is used to control the angular motion from 0° to 180° and 0° to
90°. The servo motor can be moved to a desired angular position by sending
Pulse Width Modulated (Holtz, 1992) signals on the control wire. The servo
understands the language of pulse position modulation. A pulse of width
varying from 1 millisecond to 2 milliseconds in a repeated time frame is sent
to the servo around 50 times in a second. The width of the pulse determines
the angular position. For example, a pulse of 1 millisecond moves the servo
towards 0°, while a 2 milliseconds wide pulse would take it to 180°. The pulse
width for in-between angular positions can be interpolated accordingly. Thus a
pulse of width 1.5 milliseconds will shift the servo to 90°. It must be noted
that these values are only approximations. The actual behavior of the servos
differs based on their manufacturer. A sequence of such pulses (50 in one
second) is required to be passed to the servo to sustain a particular angular
position. When the servo receives a pulse, it can retain the corresponding
angular position for the next 20 milliseconds. So a pulse in every 20
millisecond time frame must be fed to the servo. Figure 4 shows an example of
the servo motor we have used in our implementation, while Figure 5 shows the
operation of servo motor based on Pulse Width Modulated signals.

Figure 6: Flowchart

Figure 7: PROTEUS Simulation Diagram

Figure 8: Layout of our Printed Circuit Board in EAGLE software

###  4.4 Circuit Simulation and PCB Designing

Proteus Design Suite by Labcenter Electronics provides a simple interface to
design and simulate various circuits. It has a variety of electronic
components and settings for each of them to choose from and is an efficient
method to test the initial circuits for sanity checks before implementation.
It has the option of adding various switches and connect and visualize the
flow in real-time, providing error logs and failure cases. Figure 7
demonstrates the simulation of our circuit design. After selecting all the
components and verifying the simulation on the software, we then start
implementing the actual PCB designing process which includes steps like
printing a layout from EAGLE PCB design software, etching the PCB, drilling,
integrating and soldering all the components and finally testing the
prototype. Figure 8 shows the Layout of our printed circuit board on the EAGLE
Software and Figure 9 shows the board after etching process.

##  5 Results and Conclusion

Thus, we have implemented a model that senses rains and automatically switches
on the wiper and adjusts its speed according to the intensity of the rain. As
the intensity of the rain increases, the speed of the wiper increases to a
certain level. Figure 6 shows the workflow for our proposal. The
microcontroller checks for the digital pin and analog pin inputs of the rain
sensor. When there is slight water on the sensor, the digital pin is set to
logic ’0’. This is used to detect presence of rain water. To check the
intensity of rain, we monitor the analog pin output of the rain sensor, whose
threshold can be adjusted manually through an attached Potentiometer to
indicate how much water should be considered as high rain. According to our
observations, the wiper takes 2.2 seconds when a drop of water is poured on
the sensor, while it takes only 1.4 seconds when the sensor is submerged in a
glass of water. We learned how to interface servo motor with AT89C51
Microcontroller and the rain sensor module interfacing with AT89C51
Microcontroller. Figure 10 shows the prototype we have developed to
demonstrate our idea.

Figure 9: Printed Circuit Board after Etching process

Figure 10: Working prototype

##  6 Future Scope and Market Potential

The world will one day move in self-driving cars is already evident in a
series of functions that today’s cars have begun to perform without human
intervention. Even in the models sold in India, some cars tell you the route
and journey time, park on their own, start the wipers if it is raining, switch
on the lights if it gets dark, warn you of moving objects at night, inform you
of a school in the vicinity, detect dangerous lane departures and raise alarm
if the driver is drowsy. You can control music volume by moving fingers in the
air and telling your car what song to play.

One of the primary objectives in this task was to make a design which is
compact and easy to integrate with a complex system such as a vehicle. Also,
we wanted to demonstrate how these relatively novel sensors can be integrated
with a microcontroller to develop an application. Modifications in the circuit
can be made with the objective of creating a system on-chip, which can be
easily plugged into existing vehicles. The sensor proposed in this model is
low cost and efficient to a great extent, however with the development of more
high quality and accurate sensors, much more desirable and reliable outputs
can be obtained. Another interesting area to explore into is controlling the
speed of the wiper to a more accurate sense. Currently, the wiper moves at two
different speeds. By modifying the code, we can have different speeds for a
different amount of rain. Also, we can use this automated car wiper along with
other automated features to make a Smart Car.

## References

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