$$\eta ^{(\prime )}$$-meson twist-2 distribution amplitude within QCD sum rule approach and its application to the semi-leptonic decay $$ D_s^+ \rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell $$
Authors: Hu Dan-Dan , Fu Hai-Bing , Zhong Tao , Zeng Long , Cheng Wei , Wu Xing-Gang 
Published: January 2022

In this paper, we make a detailed discussion on the $$\eta $$
η
and $$\eta ^\prime $$

η
′

-meson leading-twist light-cone distribution amplitude $$\phi _{2;\eta ^{(\prime )}}(u,\mu )$$

ϕ

2
;

η

(
′
)

(
u
,
μ
)

by using the QCD sum rules approach under the background field theory. Taking both the non-perturbative condensates up to dimension-six and the next-to-leading order (NLO) QCD corrections to the perturbative part, its first three moments $$\langle \xi ^n_{2;\eta ^{(\prime )}}\rangle |_{\mu _0} $$

⟨

ξ

2
;

η

(
′
)

n

⟩

|

μ
0

with $$n = (2, 4, 6)$$

n
=
(
2
,
4
,
6
)

can be determined, where the initial scale $$\mu _0$$

μ
0

is set as the usual choice of 1 GeV. Numerically, we obtain $$\langle \xi _{2;\eta }^2\rangle |_{\mu _0} =0.231_{-0.013}^{+0.010}$$

⟨

ξ

2
;
η

2

⟩

|

μ
0

=
0
.

231

-
0.013

+
0.010

, $$\langle \xi _{2;\eta }^4 \rangle |_{\mu _0} =0.109_{ - 0.007}^{ + 0.007}$$

⟨

ξ

2
;
η

4

⟩

|

μ
0

=
0
.

109

-
0.007

+
0.007

, and $$\langle \xi _{2;\eta }^6 \rangle |_{\mu _0} =0.066_{-0.006}^{+0.006}$$

⟨

ξ

2
;
η

6

⟩

|

μ
0

=
0
.

066

-
0.006

+
0.006

for $$\eta $$
η
-meson, $$\langle \xi _{2;\eta '}^2\rangle |_{\mu _0} =0.211_{-0.017}^{+0.015}$$

⟨

ξ

2
;

η
′

2

⟩

|

μ
0

=
0
.

211

-
0.017

+
0.015

, $$\langle \xi _{2;\eta '}^4 \rangle |_{\mu _0} =0.093_{ - 0.009}^{ + 0.009}$$

⟨

ξ

2
;

η
′

4

⟩

|

μ
0

=
0
.

093

-
0.009

+
0.009

, and $$\langle \xi _{2;\eta '}^6 \rangle |_{\mu _0} =0.054_{-0.008}^{+0.008}$$

⟨

ξ

2
;

η
′

6

⟩

|

μ
0

=
0
.

054

-
0.008

+
0.008

for $$\eta '$$

η
′

-meson. Next, we calculate the $$D_s\rightarrow \eta ^{(\prime )}$$

D
s

→

η

(
′
)

transition form factors (TFFs) $$f^{\eta ^{(\prime )}}_{+}(q^2)$$

f
+

η

(
′
)

(

q
2

)

within QCD light-cone sum rules approach up to NLO level. The values at large recoil region are $$f^{\eta }_+(0) = 0.476_{-0.036}^{+0.040}$$

f
+
η

(
0
)

=
0
.

476

-
0.036

+
0.040

and $$f^{\eta '}_+(0) = 0.544_{-0.042}^{+0.046}$$

f
+

η
′

(
0
)

=
0
.

544

-
0.042

+
0.046

. After extrapolating TFFs to the allowable physical regions within the series expansion, we obtain the branching fractions of the semi-leptonic decay, i.e. $$D_s^+\rightarrow \eta ^{(\prime )}\ell ^+ \nu _\ell $$

D
s
+

→

η

(
′
)

ℓ
+

ν
ℓ

, i.e. $${{\mathcal {B}}}(D_s^+ \rightarrow \eta ^{(\prime )} e^+\nu _e)=2.346_{-0.331}^{+0.418}(0.792_{-0.118}^{+0.141})\times 10^{-2}$$

B

(

D
s
+

→

η

(
′
)

e
+

ν
e

)

=
2
.

346

-
0.331

+
0.418

(
0
.

792

-
0.118

+
0.141

)

×

10

-
2

and $${{\mathcal {B}}}(D_s^+ \rightarrow \eta ^{(\prime )} \mu ^+\nu _\mu )=2.320_{-0.327}^{+0.413}(0.773_{-0.115}^{+0.138})\times 10^{-2}$$

B

(

D
s
+

→

η

(
′
)

μ
+

ν
μ

)

=
2
.

320

-
0.327

+
0.413

(
0
.

773

-
0.115

+
0.138

)

×

10

-
2

for $$\ell = (e, \mu )$$

ℓ
=
(
e
,
μ
)

channels respectively. And in addition to that, the mixing angle for $$\eta -\eta '$$

η
-

η
′

with $$\varphi $$
φ
and ratio for the different decay channels $${{\mathcal {R}}}_{\eta '/\eta }^\ell $$

R

η
′

/
η

ℓ

are given, which show good agreement with the recent BESIII measurements.