Anatomy of $B_s \to PP $ decays and effects of the next-to-leading order contributions in the perturbative QCD approach

TL;DR

This paper systematically investigates $B_s o PP$ decays using perturbative QCD, highlighting the significant impact of next-to-leading order contributions on decay rates and CP asymmetries, and compares results with experimental data and other theoretical approaches.

Contribution

It provides a comprehensive analysis of NLO effects in $B_s o PP$ decays within PQCD, improving agreement with experimental results and contrasting with other theoretical frameworks.

Findings

NLO contributions enhance or reduce decay rates by 20-150%.

Predictions for CP asymmetries align well with measurements after NLO corrections.

Ratios of branching fractions match experimental data after NLO adjustments.

Abstract

By employing the perturbative QCD (PQCD) factorization approach, we made a systematic investigation for the CP-averaged branching ratios and the CP-violating asymmetries of the thirteen $\bar{B}^0_s \to PP $ decays ( here $P=(\pi,K, \eta, \eta^{\prime})$) with the inclusion of all currently known next-to-leading order (NLO) contributions, and compared our results with the measured values or the theoretical predictions from other different approaches. We focused on the examination of the effects of the NLO contributions and found the following points: (a) for $\bar{B}_s^0 \to (K^0 \bar{K}^0, K^+ K^-, \eta^\prime \eta^\prime,\pi^- K^+)$ decays, the NLO contributions can provide a $(40-150) \%$ enhancement or a $20\%$ reduction to the corresponding leading order (LO) PQCD predictions for their decay rates and result in a much better agreement between the PQCD predictions and the experimental measurements; (b) for the pure annihilation decay $\bar{B}_s^0 \to \pi^+\pi^-$, the PQCD prediction still remain consistent with the data after the inclusion of the small NLO reduction; (c) the PQCD predictions for the ratio $(f_s/f_d)\cdot {\cal B}(B_s^0\to \pi^+\pi^-)/{\cal B}(B^0\to K^+\pi^-)$ and $(f_s/f_d)\cdot {\cal B}(B_s^0\to \pi^+ K^-)/{\cal B}(B^0\to K^+\pi^-)$ become agree very well with the measured ones after the inclusion of a $40\%$ NLO reduction; (d) for $\bar{B}_s^0 \to K^+ K^-$ and $\bar{B}_s^0 \to \pi^- K^+$ decays, the NLO PQCD predictions for their CP-violating asymmetries do agree very well with the measured values in both the sign and the magnitude; and (e) for all $\bar{B}_s^0 \to P P$ decays, we also compared our results with those obtained in the QCD factorization approach and soft-collinear effective theory and discussed their similarities and differences.

Tables

Table 1. Table 1: The measured values of the branching ratios (in units of 10 − 6 superscript 10 6 10^{-6} ) of the five considered decay modes, as reported by the Belle bellepipi ; bellekk , CDF CDFpipi ; CDFpik ; CDFkk , LHCb Collaboration LHCbpipi ; LHCbepep ; LHCbpik , and the world averages as given in Refs.  hfag2018 ; pdg2018 .

Mode $\left({\overline{B}}_s^0\to \overline{f}\right)$	Bellebellepipi ; bellekk	CDFCDFpipi ; CDFpik ; CDFkk	LHCbLHCbpipi ; LHCbepep ; LHCbpik	HFLAVhfag2018	PDGpdg2018
${\overline{B}}_s^0\to {\pi }^{-}{K}^{+}$		$5.3\pm 0.9\pm 0.3$	$5.6\pm 0.6\pm 0.3$	$5.3\pm 0.5$	$5.7\pm 0.6$
${\overline{B}}_s^0\to K^{+}{K}^{-}$	${35}_{-9}^{+10}\pm 7$	$25.9\pm 2.2\pm 1.7$	$23.7\pm 1.6\pm 1.5$	$24.8\pm 1.7$	$25.9\pm 1.7$
${\overline{B}}_s^0\to K^0{\overline{K}}^0$	${19.6}_{-5.1}^{+5.8}\pm 2.0$	$-$	$-$	${19.6}_{-5.6}^{+6.2}$	$20\pm 6$
${\overline{B}}_s^0\to {\eta }^{\prime }{\eta }^{\prime }$	$-$	$-$	$33.1\pm 7.0\pm 1.2$	$33.1\pm 7.1$	$33\pm 7$
${\overline{B}}_s^0\to {\pi }^{+}{\pi }^{-}$		$0.60\pm 0.17\pm 0.04$	$0.691\pm 0.083\pm 0.044$	$0.67\pm 0.08$	$0.70\pm 0.08$

Table 2. Table 2: The measured values of A CP ​ ( B s 0 → π + ​ K − ) subscript 𝐴 CP → subscript superscript 𝐵 0 𝑠 superscript 𝜋 superscript 𝐾 A_{\rm CP}(B^{0}_{s}\to\pi^{+}K^{-}) and CP-violating asymmetries C K ​ K subscript 𝐶 𝐾 𝐾 C_{KK} , S K ​ K subscript 𝑆 𝐾 𝐾 S_{KK} and A K ​ K Δ ​ Γ subscript superscript 𝐴 Δ Γ 𝐾 𝐾 A^{\Delta\Gamma}_{KK} of B s 0 → K + ​ K − → subscript superscript 𝐵 0 𝑠 superscript 𝐾 superscript 𝐾 B^{0}_{s}\to K^{+}K^{-} decay, as reported by the CDF CDFpikcp and LHCb Collaboration LHCbpikcp1 ; LHCbpikcp2 ; lhcb-18a ; LHCbkkcp , and the world averages as given in Refs.  hfag2018 ; pdg2018 .

Mode	CDFCDFpikcp	LHCbLHCbpikcp1 ; LHCbpikcp2 ; lhcb-18a ; LHCbkkcp	HFLAVhfag2018	PDGpdg2018
$A_{\mathrm{CP}}\left(B_s^0\to K^{-}{\pi }^{+}\right)$	$0.22\pm 0.07$	$0.27\pm 0.08\pm 0.02$ LHCbpikcp1	$0.213\pm 0.017$	$0.221\pm 0.015$
		$0.27\pm 0.04\pm 0.01$ LHCbpikcp2
		$0.213\pm 0.015\pm 0.007$ lhcb-18a
$C_{KK}\left(B_s^0\to K^{+}{K}^{-}\right)$		$0.14\pm 0.11\pm 0.03$ LHCbkkcp		$0.14\pm 0.11$
		$0.20\pm 0.06\pm 0.02$ lhcb-18a
$S_{KK}\left(B_s^0\to K^{+}{K}^{-}\right)$		$0.30\pm 0.12\pm 0.04$ LHCbkkcp		$0.30\pm 0.13$
		$0.18\pm 0.06\pm 0.02$ lhcb-18a
$A_{KK}^{\Delta \Gamma }\left(B_s^0\to K^{+}{K}^{-}\right)$		$-0.79\pm 0.07\pm 0.10$ lhcb-18a

Table 3. Table 3: The LO and NLO PQCD predictions for the CP-averaged branching ratios (in units of 10 − 6 superscript 10 6 10^{-6} ) of the considered B ¯ s 0 → P ​ P → superscript subscript ¯ 𝐵 𝑠 0 𝑃 𝑃 \bar{B}_{s}^{0}\to PP decays. The theoretical predictions as given in Refs.  ali07 ; xiao2012 ; xiao14a ; xiao14b ; bs08 ; npb675 ; chengbs09 ; scet06 , and the world average of the measured values as given in PDG 2018 pdg2018 .

Mode	Class	$\mathrm{LO}$	$\mathrm{NLO}$	$\mathrm{PQCD}$	$\mathrm{PQCD}$	QCDF	SCET	PDG2018
(${\overline{B}}_s^0\to \overline{f}$)				ali07	bs08 ; xiao2012 ; xiao14a ; xiao14b	npb675 ; chengbs09	scet06	pdg2018
${\overline{B}}_s^0\to {\pi }^{-}{K}^{+}$	T	$6.9$	${5.4}_{-1.5}^{+2.4}$	$7.6$	${5.7}_{-1.9}^{+2.3}$ xiao14a	$10.2$	$4.9$	$5.6\pm 0.6$
${\overline{B}}_s^0\to {\pi }^0{K}^0$	C	$0.18$	${0.27}_{-0.08}^{+0.10}$	$0.16$	${0.28}_{-0.07}^{+0.11}$xiao14a	$0.49$	$0.76$	$-$
${\overline{B}}_s^0\to K^0\eta$	C	$0.08$	${0.14}_{-0.04}^{+0.06}$	$0.11$	${0.19}_{-0.07}^{+0.04}$bs08	$0.34$	$0.80$	$-$
${\overline{B}}_s^0\to K^0{\eta }^{\prime }$	C	$0.60$	${1.36}_{-0.28}^{+0.43}$	$0.72$	${1.87}_{-0.56}^{+0.45}$bs08	$2.0$	$4.5$	$-$
${\overline{B}}_s^0\to K^{+}{K}^{-}$	P	$13.4$	${18.6}_{-5.3}^{+6.4}$	$13.6$	${19.7}_{-5.7}^{+6.6}$xiao14a	$22.7$	$18.2$	$25.4\pm 1.7$
${\overline{B}}_s^0\to K^0{\overline{K}}^0$	P	$14.4$	${19.7}_{-3.8}^{+5.9}$	$15.6$	${20.2}_{-5.8}^{+7.3}$xiao14a	$24.7$	$17.7$	$20\pm 6$
${\overline{B}}_s^0\to \eta \eta$	P	$6.7$	${10.4}_{-3.4}^{+4.9}$	$8.0$	${10.6}_{-2.7}^{+3.8}$xiao14b	$15.6$	$7.1$	$-$
${\overline{B}}_s^0\to \eta {\eta }^{\prime }$	P	$17.2$	${36.2}_{-11.1}^{+14.7}$	$21$	${41.4}_{-12.0}^{+16.4}$xiao14b	$54.0$	$24.0$	$-$
${\overline{B}}_s^0\to {\eta }^{\prime }{\eta }^{\prime }$	P	$12.3$	${30.8}_{-8.6}^{+11.2}$	$14$	${41.0}_{-13.4}^{+17.5}$xiao14b	$41.7$	$44.3$	$33\pm 7$
${\overline{B}}_s^0\to {\pi }^0\eta$	$P_{\mathrm{EW}}$	$0.04$	$0.04\pm 0.02$	$0.05$	$0.06\pm 0.03$xiao14b	$0.075$	$0.014$	$-$
${\overline{B}}_s^0\to {\pi }^0{\eta }^{\prime }$	$P_{\mathrm{EW}}$	$0.06$	$0.07\pm 0.03$	$0.11$	$0.13\pm 0.06$xiao14b	$0.11$	$0.006$	$-$
${\overline{B}}_s^0\to {\pi }^{+}{\pi }^{-}$	ann	$0.62$	${0.52}_{-0.18}^{+0.21}$	$0.57$	${0.57}_{-0.22}^{+0.24}$bs08	$0.02$	$-$	$0.68\pm 0.08$
					${0.51}_{-0.19}^{+0.23}$xiao2012	$0.26$chengbs09
${\overline{B}}_s^0\to {\pi }^0{\pi }^0$	ann	$0.25$	${0.21}_{-0.09}^{+0.10}$	$0.28$	$0.29\pm 0.12$ bs08	$0.01$	$-$	$-$

Table 4. Table 4: The LO and NLO PQCD predictions for some ratios of the branching ratios for several B s / B → P ​ P → subscript 𝐵 𝑠 𝐵 𝑃 𝑃 B_{s}/B\to PP decays. The measured values of these ratios as reported by CDF CDFpipi ; CDFpik ; CDFkk and LHCb Collaboration LHCbpipi ; LHCbpik , as well as the averages from HFLAV hfag2018 , are listed as comparison.

Mode	LO	NLO	CDFCDFpipi ; CDFpik ; CDFkk	LHCb LHCbpipi ; LHCbpik	HFLAV hfag2018
$R_1$	${0.013}_{-0.001}^{+0.002}$	$0.008\pm 0.001$	$0.008\pm 0.002$	$0.009\pm 0.001$	$0.009\pm 0.001$
$R_2$	$0.026\pm 0.003$	$0.027\pm 0.003$	$-$	${0.050}_{-0.011}^{+0.012}$	${0.050}_{-0.010}^{+0.012}$
$R_3$	${0.149}_{-0.014}^{+0.013}$	${0.087}_{-0.009}^{+0.011}$	$0.071\pm 0.012$	$0.074\pm 0.008$	$0.073\pm 0.007$
$R_4$	${0.290}_{-0.028}^{+0.029}$	$0.301\pm 0.022$	$0.35\pm 0.03$	$0.32\pm 0.02$	$0.327\pm 0.017$

Table 5. Table 5: The LO and NLO PQCD predictions for A C ​ P ​ ( B ¯ s 0 → π − ​ K + ) subscript 𝐴 𝐶 𝑃 → superscript subscript ¯ 𝐵 𝑠 0 superscript 𝜋 superscript 𝐾 A_{CP}(\bar{B}_{s}^{0}\to\pi^{-}K^{+}) and 𝒜 f subscript 𝒜 𝑓 {\cal A}_{f} ( in unit of 10 − 2 superscript 10 2 10^{-2} ) for other twelve B ¯ s 0 → P ​ P → superscript subscript ¯ 𝐵 𝑠 0 𝑃 𝑃 \bar{B}_{s}^{0}\to PP decays. As a comparison, we also listed the theoretical predictions as given in Refs.  ali07 ; bs08 ; xiao14a ; xiao14b ; npb675 ; scet06 and the data as given in Ref.  pdg2018 .

Mode	LO	NLO	$\mathrm{PQCD}$	$\mathrm{PQCD}$	QCDF	SCET	PDG
$\left({\overline{B}}_s^0\to \overline{f}\right)$			ali07	bs08 ; xiao14a ; xiao14b	npb675	scet06	pdg2018
${\overline{B}}_s^0\to {\pi }^{-}{K}^{+}$	$22.9$	${23.8}_{-6.5}^{+6.4}$	$24.1$	$38.7$ xiao14a	$-6.7$	$20$	$22.1\pm 1.5$
${\overline{B}}_s^0\to {\pi }^0{K}^0$	$53.4$	${87.5}_{-8.3}^{+6.4}$	$59.4$	$83$ xiao14a	$41.6$	$76$
${\overline{B}}_s^0\to K^0\eta$	$47.1$	${95.6}_{-7.6}^{+3.7}$	$56.4$	$96.7$ bs08	$46.8$	$-56$
${\overline{B}}_s^0\to K^0{\eta }^{\prime }$	$-20.6$	$-{42.6}_{-2.0}^{+2.2}$	$-19.9$	$-35.4$ bs08	$-36.6$	$-14$
${\overline{B}}_s^0\to K^{+}{K}^{-}$	$-24.2$	$-{13.2}_{-1.6}^{+2.3}$	$-23.3$	$-16.4$ xiao14a	$4.0$	$-6$	$-14\pm 11$
${\overline{B}}_s^0\to K^0{\overline{K}}^0$	$0$	$0.5\pm 0.1$	$0$	$-0.7$ xiao14a	$0.9$	$-$
${\overline{B}}_s^0\to \eta \eta$	$-0.3$	$-{2.7}_{-0.3}^{+0.4}$	$-0.6$	$-2.3$ xiao14b	$-1.6$	$7.9$
${\overline{B}}_s^0\to \eta {\eta }^{\prime }$	$-0.8$	$-{0.6}_{-0.1}^{+0.2}$	$-1.3$	$-0.2$ xiao14b	$0.4$	$0.04$
${\overline{B}}_s^0\to {\eta }^{\prime }{\eta }^{\prime }$	$1.3$	${2.7}_{-0.3}^{+0.4}$	$1.9$	$2.8$ xiao14b	$2.1$	$0.9$
${\overline{B}}_s^0\to {\pi }^0\eta$	$-4.5$	${32.2}_{-5.2}^{+5.3}$	$-0.4$	$40.3$ xiao14b	$-$	$-$
${\overline{B}}_s^0\to {\pi }^0{\eta }^{\prime }$	$28.9$	${59.2}_{-1.5}^{+2.1}$	$20.6$	$51.9$ xiao14b	$27.8$	$-$
${\overline{B}}_s^0\to {\pi }^{+}{\pi }^{-}$	$-1.3$	$-{0.6}_{-1.3}^{+1.0}$	$-1.2$	$0.2$ bs08	$-$	$-$
${\overline{B}}_s^0\to {\pi }^0{\pi }^0$	$-1.3$	$-{0.6}_{-1.3}^{+1.0}$	$-1.2$	$0.2$ bs08	$-$	$-$

Table 6. Table 6: The PQCD predictions for the mixing-induced CP asymmetries (in unit of 10 − 2 superscript 10 2 10^{-2} ) 𝒮 f subscript 𝒮 𝑓 {\cal S}_{f} (the first row) and ℋ f subscript ℋ 𝑓 {\cal H}_{f} (the second row) for the twelve B ¯ s 0 → P ​ P → superscript subscript ¯ 𝐵 𝑠 0 𝑃 𝑃 \bar{B}_{s}^{0}\to PP decays. In last four columns, we also listed the theoretical predictions as given in Refs.  ali07 ; xiao2012 ; xiao14a ; xiao14b ; chengbs09 ; scet06 .

Mode	LO	NLO	$\mathrm{PQCD}$	$\mathrm{PQCD}$	QCDF	SCET	LHCb
			ali07	bs08 ; xiao14a ; xiao14b	chengbs09	scet06	lhcb-18a
${\overline{B}}_s^0\to {\pi }^0{K}^0$	$-57.1$	$-{45.3}_{-10}^{+8}$	$-61$	$-52.9$ xiao14a	$8$	$-16$
	$-62.4$	$-{19.2}_{-4.2}^{+6.3}$	$-52$	$-17.4$ xiao14a	$-$	$80$
${\overline{B}}_s^0\to K^0\eta$	$-52.6$	$-{20.2}_{-15.4}^{+10.7}$	$-43$	$-18$ bs08	$26$	$82$
	$-64.2$	$-{20.1}_{-11.5}^{+12.6}$	$-70$	$-18$ bs08	$-$	$7$
${\overline{B}}_s^0\to K^0{\eta }^{\prime }$	$-65.6$	$-{38.9}_{-13.5}^{+14.2}$	$-68$	$-46$ bs08	$8$	$38$
	$-72.1$	$-{81.6}_{-5.4}^{+7.2}$	$-70$	$-82$ bs08	$-$	$-92$
${\overline{B}}_s^0\to K^{+}{K}^{-}$	$22.2$	${23.8}_{-4.2}^{+5.4}$	$28$	$20.6$ xiao14a	$22$	$19$	$18\pm 6\pm 2$
	$94.4$	${96.2}_{-1.2}^{+0.3}$	$93$	$96.5$ xiao14a	$-$	$97.9$	$79\pm 7\pm 10$
${\overline{B}}_s^0\to K^0{\overline{K}}^0$	$4.0$	$-3.3$	$4$	$-0.2$ xiao14a	$0.4$	$-$
	$\sim 100$	$\sim 100$	$\sim 100$	$\sim 100$ xiao14a	$-$	$-$
${\overline{B}}_s^0\to \eta \eta$	$3.0$	$-{1.2}_{-0.3}^{+0.4}$	$3$	$-2.2$ xiao14b	$-7$	$-2.6$
	$99.9$	$99.9$	$\sim 100$	$99.9$ xiao14b	$-$	$99.6$
${\overline{B}}_s^0\to \eta {\eta }^{\prime }$	$3.0$	${4.1}_{-1.5}^{+1.0}$	$4$	$0.1$ xiao14b	$-1$	$4.1$
	$99.9$	$99.9$	$\sim 100$	$\sim 100$ xiao14b	$-$	$99.2$
${\overline{B}}_s^0\to {\eta }^{\prime }{\eta }^{\prime }$	$3.0$	${2.0}_{-0.3}^{+0.5}$	$4$	$2.5$ xiao14b	$4$	$4.9$
	$99.9$	$99.9$	$\sim 100$	$99.9$ xiao14b	$-$	$99.9$
${\overline{B}}_s^0\to {\pi }^0\eta$	$11.8$	${8.7}_{-2.5}^{+2.6}$	$17$	$8$ xiao14b	$26$	$45$
	$98.8$	${94.2}_{-2.2}^{+1.5}$	$99$	$91.2$ xiao14b	$-$	$-89$
${\overline{B}}_s^0\to {\pi }^0{\eta }^{\prime }$	$-11.6$	$-{16.9}_{-9.0}^{+5.2}$	$-17$	$-24.9$ xiao14b	$88$	$45$
	$95.1$	${78.8}_{-4.3}^{+1.0}$	$96$	$81.8$ xiao14b	$-$	$-89$
${\overline{B}}_s^0\to {\pi }^{+}{\pi }^{-}$	$11.2$	${10.6}_{-1.0}^{+1.2}$	$14$	$9$ bs08	$15$	$-$
	$99.9$	$99.8$	$99$	$99$ bs08	$-$	$-$
${\overline{B}}_s^0\to {\pi }^0{\pi }^0$	$11.2$	${10.6}_{-1.0}^{+1.2}$	$14$	$8.1$ bs08	$15$	$-$
	$99.9$	$99.9$	$99$	$99$ bs08	$-$	$-$

Equations

$$P_1$$

$$k_1$$

$$\mathcal{A}$$

$${\Phi }_{B_s}=\frac{1}{\sqrt{6}}(P{/}_{B_s}+m_{B_s}){\gamma }_5{\varphi }_{B_s}({\mathbf{k}}_1),$$

$${\varphi }_{B_s}(x,b)$$

$$\displaystyle\left(\begin{array}[]{c}\eta\\ \eta^{\prime}\end{array}\right)=\left(\begin{array}[]{cc}\cos\phi&-\sin\phi\\ \sin\phi&\cos\phi\\ \end{array}\right)\left(\begin{array}[]{c}\eta_{q}\\ \eta_{s}\end{array}\right),$$

$$f_q=(1.07\pm 0.02)f_{\pi },f_s=(1.34\pm 0.06)f_{\pi },\varphi =39.3^{\circ }\pm 1.0^{\circ }.$$

$${\Phi }_{M_i}(P_i,x_i)\equiv \frac{1}{\sqrt{6}}{\gamma }_5\left[P{/}_i{\varphi }_{M_i}^A(x_i)+m_{0i}{\varphi }_{M_i}^P(x_i)+\zeta m_{0i}(n/v/-1){\varphi }_{M_i}^T(x_i)\right],$$

$${\varphi }_M^A(x)$$

$${\varphi }_M^P(x)$$

$${\varphi }_M^T(x)$$

$$m_{qq}^2$$

$$m_{ss}^2$$

$$a_1^{\pi ,{\eta }_q,{\eta }_s}$$

$$a_2^{{\eta }_q,{\eta }_s}$$

$${\mathcal{H}}_{eff}$$

$$a_1$$

$$a_i$$

$$H_{eff}^{ql}$$

$$H_{eff}^{mp}$$

$${\mathcal{A}}_{{\pi }^{-}{K}^{+}}^{(u)}$$

$${\mathcal{A}}_{{\pi }^0{K}^0}^{(u)}$$

$${\mathcal{A}}_{K^{+}{K}^{-}}^{(u)}$$

$${\mathcal{A}}_{K^0{\bar{K}}^0}^{(u)}$$

$${\mathcal{A}}_{K^0{\eta }_{n(s)}}^{(u)}$$

$${\mathcal{A}}_{{\eta }_s{\eta }_s}^{(u)}$$

$$f^{+}(q^2){|}_{\mathrm{NLO}}^{B_s\to K}$$

$$F_{T2}^{(1)}$$

$$F_{T3}^{(1)}$$

$$t^a=\max (\sqrt{x_3\eta }{m}_{B_s},1/b_1,1/b_3),\mathrm{or}{t}^b=\max (\sqrt{x_1\eta }{m}_{B_s},1/b_1,1/b_3),$$

$$\mu =t_s({\mu }_{\mathrm{f}})={\left\{\mathrm{Exp}\left[c_1+\left(\ln \frac{m_{B_s}^2}{{\xi }_1^2}+\frac{5}{4}\right)\ln \frac{{\mu }_{\mathrm{f}}^2}{m_{B_s}^2}\right]x_1^{c_2}{x}_3^{c_3}\right\}}^{2/21}{\mu }_{\mathrm{f}},$$

$$c_1$$

$$c_2$$

$$c_3$$

$${\Lambda }_{\overline{\mathrm{MS}}}^{(f=5)}$$

$$m_{B_s}$$

$$\lambda =0.2254\pm 0.0006,A=0.814_{-0.024}^{+0.023},\bar{\rho }=0.117\pm 0.021,\bar{\eta }=0.353\pm \mathrm{0.013.}$$

$$\mathcal{B}$$

Full text

Key Words:The charmless two-body $B_{s}$ meson decays; the PQCD factorization approach; branching ratios; the CP-violating asymmetries

I Introduction

During the past three decades, the two-body charmless hadronic $B_{s}\to PP$ decays ( $P$ refers to the light pseudoscalar mesons $\pi,K,\eta^{(\prime)}$) have been studied intensively by many authors for example in Refs. npb675 ; sun2003 ; chengbs09 ; scet06 ; ali07 ; bspipi ; bs08 ; xiao2012 ; xiao14a ; xiao14b ; Cheng:2011qh ; Chang:2014yma ; Cheng:2014rfa and measured by CDF, Belle and LHCb Collaborations  bellepipi ; bellekk ; CDFpipi ; CDFpik ; CDFpikcp ; CDFkk ; LHCbpipi ; LHCbepep ; LHCbpik ; LHCbpikcp1 ; LHCbpikcp2 ; lhcb-18a ; LHCbkkcp . The studies for these decays can offer us good opportunities to test the Standard Model (SM) and to search for the new physics (NP) beyond the SM.

Among the thirteen $B_{s}^{0}\to PP$ decays, only five of them have been observed by CDF CDFpipi ; CDFpik ; CDFkk ; CDFpikcp , Belle bellepipi ; bellekk and LHCb Collaboration LHCbpipi ; LHCbepep ; LHCbpik ; LHCbpikcp1 ; LHCbpikcp2 ; lhcb-18a ; LHCbkkcp . The measured values of the branching ratios and/or the CP-violating asymmetries are collected in Tables 1 and 2 . Of course, more measurements with higher precision for these decays in the LHCb and Belle-II experiments are expected in the following years  belle-2 ; lhcb-2 ; bfac ; hfag2018 ; pdg2018 . On the theory side, $B_{s}^{0}\to PP$ decays have been studied by employing rather different kinds of theoretical approaches: such as the generalized factorization approach chenbs99 ; xiaobs01 ; aag1 , the QCD factorization ( QCDF) approach  prl99 ; npb591 , the soft-collinear effective theory (SCET) scet01 ; scet02 and the perturbative QCD (PQCD) factorization approach  pqcd1 ; pqcd2 ; li2003 . Although there exist many clear differences between the theoretical predictions from rather different approaches, specifically for the pattern and magnitudes of the CP-violating asymmetries, they are generally consistent with each other for the branching ratios within still large theoretical errors.

In the framework of the PQCD factorization approach, the $B_{s}\to PV$ and $B_{s}\to VV$ decays have been calculated very recently with the inclusion of all currently known next-to-leading order (NLO) contributions xiao18a ; xiao18b . For $B_{s}\to PP$ decays, the situation is a little complicated:

1. (1)

In 2004, the $B_{s}\to\pi\pi$ decay was firstly studied by using the PQCD approach at the leading order (LO)  bspipi . In 2007, all thirteen $B_{s}\to PP$ decays were studied in the PQCD approach at leading order in Ref. ali07 . The large branching ratio for $B_{s}\to\pi^{+}\pi^{-}$ decay as predicted in Refs. bspipi ; ali07 are confirmed several years later by both CDF CDFpipi and LHCb measurements LHCbpipi ; LHCbpik . 2. (2)

In 2008, all $B_{s}\to PP$ decays were studied in the PQCD approach in Ref. bs08 with the inclusion of the NLO contributions from different sources known at that time: (a) the NLO Wilson coefficients $C_{i}(\mu)$ with other relevant functions at the NLO level  buras96 ; (b) the NLO vertex and quark-loop corrections  npb675 ; nlo05 ; joint and (c) the NLO contribution from the operator $O_{8g}$ nlo05 ; o8g2003 . 3. (3)

After 2012, the NLO twist-2 and twist-3 contributions to $B/B_{s}\to P$ form factors are calculated in Refs. prd85-074004 ; cheng14a . The $B_{s}\to(K\pi,KK)$ and $B_{s}\to(\pi\eta^{(\prime)},\eta^{(\prime)}\eta^{(\prime)})$ decays are studied very soon in Refs. xiao14a ; xiao14b with the inclusion of newly known NLO contributions to the relevant form factors. Although the pure annihilation decays $B_{s}\to\pi\pi$ do not receive the NLO contributions to $B/B_{s}\to P$ form factors xiao2012 , the studies for $B_{s}\to K\eta^{(\prime)}$ at the same NLO level as that for $B_{s}\to(K\pi,KK,\pi\eta^{(\prime)},\eta^{(\prime)}\eta^{(\prime)})$ decays  xiao14a ; xiao14b are very interesting and worth of being done now.

From the above mentioned works xiao2012 ; xiao14a ; xiao14b ; xiao18a ; xiao18b , we get to know that (a) the NLO contributions can interfere with the LO part constructively or destructively for different decay modes and can therefore result in large variations to the LO predictions; and (b) the agreement between the PQCD predictions for the decay rates and CP violating asymmetries and those currently available experimental measurements can be improved effectively after the inclusion of the NLO contributions.

In this paper , we will calculate the $B_{s}\to K\eta^{(\prime)}$ decays with the inclusion of all currently known NLO contributions, reexamine other $B_{s}\to PP$ decays simultaneously by using the same set of wave functions and input parameters, compare our PQCD predictions with those obtained based on other different approaches, as well as currently available measured values for five decay modes, and finally check the effects of the NLO contributions.

This paper is organized as follows. In Sec. II, we give a brief review about the PQCD factorization approach and we calculate analytically the relevant Feynman diagrams and present the various decay amplitudes for the considered decay modes at the LO and NLO level. We show the numerical PQCD predictions for the branching ratios and CP violating asymmetries of all thirteen $B_{s}\to PP$ decays in Sec III and make phenomenological analysis. The summary will be given in Sec. IV.

II Decay amplitudes at LO and NLO level

As usual, we consider the $B_{s}$ meson at rest and treat it as a heavy-light system. Using the light-cone coordinates, we define the $B_{s}^{0}$ meson with momentum $P_{1}$, the emitted meson $M_{2}$ and the recoiled meson $M_{3}$ with momentum $P_{2}$ and $P_{3}$ respectively. We also use $x_{i}$ to denote the momentum fraction of anti-quark in each meson and set the momentum $P_{i}$ and $k_{i}$ ( the momentum carried by the light anti-quark in $B_{s}$ and $M_{2,3}$ meson) in the following forms:

[TABLE]

The integration over $k_{1,2}^{-}$ and $k_{3}^{+}$ will lead conceptually to the decay amplitudes

[TABLE]

where $b_{i}$ is the conjugate space coordinate of $k_{\rm iT}$, $C(t)$ denotes the Wilson coefficients evaluated at the scale $t$, and $\Phi_{B_{s}}$ and $\Phi_{M_{i}}$ are wave functions of the $B_{s}$ meson and the final state mesons. The hard kernel $H(x_{i},b_{i},t)$ describes the four-quark operator and the spectator quark connected by a hard gluon. The Sudakov factors $e^{-S(t)}$ and $S_{t}(x_{i})$ together suppress the soft dynamics effectively li2003 .

II.1 Wave functions

Without the endpoint singularities in the evaluations, the hadron wave functions are the only input in the PQCD approach. These nonperturbative quantities are process independent and could be obtained with the techniques of QCD sum rule and/or Lattice QCD, or be fitted to the measurements for some relevant decay processes with good precision.

For $B_{s}^{0}$ meson, we consider only the contribution of Lorentz structure ali07

[TABLE]

and adopt the distribution amplitude $\phi_{B_{s}}$ as in Refs. bspipi ; ali07 ; xiao14a .

[TABLE]

We also take $\omega_{B_{s}}=0.50\pm 0.05$ GeV in numerical calculations. The normalization factor $N_{B_{s}}$ will be determined through the normalization condition: $\int_{0}^{1}dx\;\phi_{B_{s}}(x,b=0)=f_{B_{s}}/(2\sqrt{6})$.

For $\eta$-$\eta^{\prime}$ mixing, we also use the quark-flavor basis: $\eta_{q}=(u\bar{u}+d\bar{d})/\sqrt{2}$ and $\eta_{s}=s\bar{s}$ fks98 ; fks99 ; xiao08b ; fan2013 . The physical $\eta$ and $\eta^{\prime}$ can then be written in the form of

[TABLE]

where $\phi$ is the mixing angle. The relation between the decay constants $(f_{\eta}^{q},f_{\eta}^{s},f_{\eta^{\prime}}^{q},f_{\eta^{\prime}}^{s})$ and $(f_{q},f_{s})$ can be found for example in Ref. xiao08b . The chiral masses $m_{0}^{\eta_{q}}$ and $m_{0}^{\eta_{s}}$ have been defined in Ref. ckl06 by assuming the exact isospin symmetry $m_{q}=m_{u}=m_{d}$. The three input parameters $f_{q},f_{s},$ and $\phi$ in Eq. (11) have been extracted from the data fks98 ; fks99

[TABLE]

With $f_{\pi}=0.13$ GeV, the chiral masses $m_{0}^{\eta_{q}}$ and $m_{0}^{\eta_{s}}$ consequently take the values of $m_{0}^{\eta_{q}}=1.07$ GeV and $m_{0}^{\eta_{s}}=1.92$ GeV ckl06 .

For the final state pseudo-scalar mesons $M=(\pi,K,\eta_{q},\eta_{s})$, their wave functions are the same ones as those in Refs. pball-90 ; pball-98 ; pball-99 ; BL-04 ; KMM-04 ; pball-05 ; pball-06 ; csbwf1 ; csbwf2 :

[TABLE]

where $m_{0i}$ is the chiral mass of the meson $M_{i}$, $P_{i}$ and $x_{i}$ are the momentum and the fraction of the momentum of $M_{i}$s. The parameter $\zeta=1$ or $-1$ when the momentum fraction of the quark (anti-quark) of the meson is set to be $x$. The distribution amplitudes (DA’s) of the pseudo-scalar meson $M$ can be found easily in Refs. fan2013 ; xiao08b ; xiao18a :

[TABLE]

where $t=2x-1$, $f_{M}$ and $\rho_{M}$ are the decay constant and the mass ratio with the definition of $\rho_{M}=(m_{\pi}/m_{0}^{\pi},m_{K}/m_{0}^{K}$, $m_{qq}/m_{0}^{\eta_{q}},m_{ss}/m_{0}^{\eta_{s}})$. The parameters $m_{qq}$ and $m_{ss}$ have been defined in Ref. ckl06 :

[TABLE]

with the assumption of exact isospin symmetry $m_{q}=m_{u}=m_{d}$. The explicit expressions of those Gegenbauer polynomials $C_{1}^{3/2}(t)$ and $C_{2,4}^{1/2,3/2}(t)$ can be found for example in Eq. (20) of Ref. xiao08b The Gegenbauer moments $a_{i}^{M}$ and other input parameters are similar with those as being used in Refs. xiao2012 ; xiao14a ; xiao14b

[TABLE]

with the chiral masses $m_{0}^{\pi}=1.4\pm 0.1$ GeV, $m_{0}^{K}=1.9\pm 0.2$ GeV xiao2012 .

II.2 Example of the LO decay amplitudes

In the SM, for the considered $\bar{B}^{0}_{s}\to PP$ decays induced by the $b\to q$ transition with $q=(d,s)$, the weak effective Hamiltonian $H_{eff}$ can be written asburas96

[TABLE]

where $G_{F}=1.16639\times 10^{-5}$ GeV*-2* is the Fermi constant, and $V_{ij}$ is the Cabbibo-Kobayashi-Maskawa (CKM) matrix element, $C_{i}(\mu)$ are the Wilson coefficients and $O_{i}(\mu)$ are the four-fermion operators. For convenience, the combinations $a_{i}$ of the Wilson coefficients are defined as usual ali07 :

[TABLE]

At leading order, as illustrated in Fig. 1, there are eight types of Feynman diagrams contributing to the $B_{s}\to PP$ decays, which can be classified into three types: the factorizable emission diagrams ( Fig. 1(a) and 1(b)); the nonfactorizable emission diagrams (Fig. 1(c) and 1(d)); and the annihilation diagrams (Fig. 1(e)-1(h)). As mentioned in the Introduction, the thirteen $B_{s}\to PP$ modes have been studied at LO or partial NLO in the PQCD approach in Refs. bspipi ; ali07 ; bs08 ; xiao14a ; xiao14b . The factorization formulas of the LO decay amplitudes with various topologies have been presented explicitly for example in Ref. ali07 . Therefore, after the confirmation by our independent recalculations, we shall not collect those analytic expressions here for simplicity. In this work, we try to examine the effects of all currently known NLO contributions to all thirteen $B_{s}\to PP$ decay modes in the PQCD approach by using the same set of the input parameters, and compare the PQCD predictions with those measured values becoming known recently.

II.3 The NLO contributions

During the past two decades, many authors have made great efforts to calculate the NLO contributions to the two-body charmless $B/B_{s}\to M_{2}M_{3}$ in the framework of the PQCD factorization approach. At present, almost all such NLO contributions become available now:

• (1)

The NLO Wilson coefficients $C_{i}(m_{W})$(NLO-WC), the renormalization group running matrix $U(m_{1},m_{2},\alpha)$ at NLO level and the strong coupling constant $\alpha_{s}(\mu)$ at two-loop level as presented in Ref. buras96 ;

• (2)

The NLO contributions from the vertex corrections(VC) nlo05 ; npb675 , the quark-loops(QL) nlo05 and the chromo-magnetic penguin (MP) operator $O_{8g}$ o8g2003 ; nlo05 , as illustrated in Figs. 2(a)-2(h).

• (3)

The NLO corrections to the $B_{s}\to P$ transition form factors, as shown in Fig. 2(i)-2(l).

In two previous works prd85-074004 ; cheng14a , we calculated the NLO twist-2 and twist-3 contributions to the form factors of $B\to\pi$ transitions. Based on the $SU(3)$ flavor symmetry, we could extend directly the formulas for the NLO contributions to the form factor $F_{0,1}^{B\to\pi}(0)$ to the cases for $B_{s}\to(K,\eta_{s})$ transitions after making some proper modifications for the relevant masses or decay constants of the mesons involved, as being done in Ref. xiao18a for the decays of $B_{s}\to PV$.

In this paper, we adopt directly the formulas for all currently known NLO contributions from Refs. npb675 ; nlo05 ; o8g2003 ; fan2013 ; xiao14a ; xiao14b ; prd85-074004 ; cheng14a without further discussions about the details. For the unknown NLO corrections to the nonfactorizable emission and annihilation decay amplitudes, however, some essential comments should be given qualitatively as follows:

• (1)

For the nonfactorizable emission diagrams as shown in Fig. 1(c,d), since the hard gluons are emitted from the upper quark line of Fig. 1(c) and the upper anti-quark line of Fig. 1(d) respectively, the LO contribution from these two figures will be largely cancelled each other. The remaining contribution after the cancellation will become very small in magnitude. At NLO level, with the insertion of second gluon propagator between two quark lines, another suppression factor $\alpha_{s}(t)$ will appear. Of course, it is worth of mentioning that the ”Color-suppressed tree” dominated decay modes involving $\pi^{0}$ and/or $K^{0}$ meson, such as $B_{s}^{0}\to\pi^{0}K^{0}$ decay where the glauber effects should be considered Li:2009wba ; Li:2014haa ; Liu:2015sra ; Liu:2016upa , may be exceptional and need more investigations in depth. Due to the strong cancellation and the second suppression factor, in general, the possible NLO contribution from the spectator diagrams should be much smaller than the dominant one from the tree emission diagrams (Fig. 1(a,b)).

• (2)

For the annihilation diagrams as presented in Fig. 1(e)-1(h), the possible NLO contributions are in fact doubly suppressed by the factors $1/m_{B_{s}}$ and $\alpha_{s}(t)$, and consequently must be much smaller than those dominant LO contribution from Fig. 1(a) and 1(b).

Therefore, these two kinds of still unknown NLO contributions in the PQCD approach are in fact the higher order corrections to the already small LO pieces, and should be much smaller than the dominant contribution for the considered decays.

According to Refs. npb675 ; nlo05 , the vertex corrections can be absorbed into the redefinition of the Wilson coefficients $a_{i}(\mu)$ by adding a vertex-function $V_{i}(P)$ to them. The expressions of the vertex-functions $V_{i}(P)$ can be found easily in Ref. nlo05 . The NLO “Quark-Loop” and “Magnetic-Penguin” contributions are in fact a kind of penguin corrections with the insertion of the four-quark operators and the chromo-magnetic operator $O_{8g}$ respectively, as shown in Figs. 2(e,f) and 2(g,h). For the $b\to s$ transition, for example, the corresponding effective Hamiltonian $H_{eff}^{ql}$ and $H_{eff}^{mp}$ can be written in the following form:

[TABLE]

where $l^{2}$ is the invariant mass of the gluon which attaches the quark loops in Figs. 2(e,f), and the functions $C^{q}(\mu,l^{2})$ can be found in Ref. nlo05 ; xiao08b . The $C_{8g}^{eff}$ in Eq. (24) is the effective Wilson coefficient with the definition of $C_{8g}^{eff}=C_{8g}+C_{5}$ buras96 .

For the thirteen $B_{s}\to PP$ decays, the analytical evaluations lead to the following three*(two?)* points:

1. (1)

For the $B_{s}^{0}\to\pi^{0}(\eta,\eta^{\prime})$ decays, only the Feynman diagrams Fig. 1(a)-1(d) with the $B_{s}\to\eta_{s}$ transition will contribute at leading order. The relevant NLO contributions are those from the vertex corrections to the emitted $\pi$ meson and the one to the $B_{s}\to\eta_{s}$ transition form factor. 2. (2)

For the remaining decay modes, besides the LO decay amplitudes, all currently known NLO contributions will contribute in different ways:

[TABLE]

where the terms ${\cal A}_{M_{2}M_{3}}^{(u,t)}$ refer to the LO amplitudes, while ${\cal M}_{M_{2}M_{3}}^{(u,c,t)}$ and ${\cal M}_{M_{2}M_{3}}^{(g)}$ are the NLO amplitudes, which describe the NLO contributions from the quark-loops, the QCD-penguin-loops and the magnetic-penguin diagrams, respectively. The explicit expressions and more details about these NLO amplitudes can be found easily for example in Refs. nlo05 ; xiao14a ; xiao14b .

As mentioned in previous section, we will extend the formulaes of the NLO contributions for $B\to\pi$ transition form factors as given in Refs. prd85-074004 ; cheng14a to the cases for $B_{s}\to(K,\eta_{s})$ transition form factors. The NLO form factor $f^{+}(q^{2})$ for $B_{s}\to K$ transition, for example, can be written in the following form:

[TABLE]

where r_{k}={\color[rgb]{0.8,0.1,0.1}{\it m_{0}^{K}}}/m_{B_{s}}, $\eta=1-q^{2}/m_{B_{s}}^{2}$ with $q^{2}=(P_{B_{s}}-P_{3})^{2}$ and $P_{3}$ is the momentum of the meson $M_{3}$ which absorbed the spectator $\bar{s}$ quark of the $\bar{B}_{s}^{0}$ meson, $\mu$ ($\mu_{f}$) is the renormalization (factorization ) scale, the hard scale $t_{1,2}$ are chosen as the largest scale of the propagators in the hard $b$-quark decay diagrams prd85-074004 ; cheng14a . The explicit expressions of the threshold Sudakov function $S_{t}(x)$ and the hard function $h(x_{i},b_{j})$ can be found easily in Refs. prd85-074004 ; cheng14a . The NLO factors $F^{(1)}_{\rm T2}(x_{i},\mu,\mu_{f},q^{2})$ and $F^{(1)}_{\rm T3}(x_{i},\mu,\mu_{f},q^{2})$ appeared in Eq. (26) describe the NLO twist-2 and twist-3 contributions to the form factor $f^{+,0}(q^{2})$ of the $B_{s}\to K$ transition respectively, and can be written in the following form prd85-074004 ; cheng14a :

[TABLE]

where $r_{1,2}=m^{2}_{B_{s}}/\xi_{1,2}^{2}$ with the choice of $\xi_{1}=25m_{B_{s}}$ and $\xi_{2}=m_{B_{s}}$. For $B_{s}\to PP$ decays considered in this paper, the large recoil region corresponds to the energy fraction $\eta\sim\textit{O}(1)$. The factorization scale $\mu_{f}$ is set to be the hard scales

[TABLE]

corresponding to the largest energy scales in Figs. 1(a) and 1(b), respectively. The renormalization scale $\mu$ is defined as prd85-074004 ; cheng14a ; fan2013

[TABLE]

with the coefficients

[TABLE]

III Numerical results

In the numerical calculations, the following input parameters will be used implicitly. The masses, decay constants and QCD scales are in units of GeV pdg2018 :

[TABLE]

For the CKM matrix elements, we adopt the Wolfenstein parametrization up to ${\cal O}(\lambda^{5})$ with the updated parameters as pdg2018

[TABLE]

For the thirteen $B_{s}^{0}\to PP$ decays, their CP-averaged branching ratios are defined as

[TABLE]

where $\tau_{B_{s}}$ is the lifetime of the $B_{s}$ meson.

Since the final state $\bar{f}=K^{+}\pi^{-}$ is not a CP eigenstate, the CP asymmetry ${\cal A}_{CP}$ for $\bar{B}_{s}^{0}\to K^{+}\pi^{-}$ decay is defined as lhcb-18a

[TABLE]

where $A_{f}$ ($\bar{A}_{\bar{f}}$) is the decay amplitude of $B_{s}^{0}\to f$ ( $\bar{B}_{s}^{0}\to\bar{f}$) decay.

When the final states are CP eigenstates, i.e. $\bar{f}=\eta_{f}\;f$ with $\eta_{f}=\pm 1$ for a CP-even or a CP-odd final state $f$, the direct CP violation ${\cal A}_{f}$, the CP-violating asymmetry ${\cal S}_{f}$ and ${\cal H}_{f}$ can be defined in the same way as in Refs. ali07 ; xiao18a ; xiao18b :

[TABLE]

with the CP-violating parameter $\lambda$

[TABLE]

where $\beta_{s}=\arg[-V_{ts}V_{tb}^{*}]$ is small in size for $B_{s}^{0}$ meson, and the three CP violations satisfy the normalization relation $|{\cal A}_{f}|^{2}+|{\cal S}_{f}|^{2}+|{\cal H}_{f}|^{2}=1$. It is worth of mentioning that the parameter ${\cal A}_{f}$ and ${\cal H}_{f}$ as defined in Eq. (36) have an opposite sign with ${\cal C}_{f}$ and $A_{f}^{\Delta\Gamma}$ as given in Ref. lhcb-18a : i.e: ${\cal A}_{f}=-{\cal C}_{f}$ and ${\cal H}_{f}=-A_{f}^{\Delta\Gamma}$ .

III.1 The branching ratios

In Table 3, we present our numerical results for the CP-averaged branching ratios of the thirteen $\bar{B}_{s}^{0}\to PP$ decays. In the second column of Table 3, we classify the LO dominant contribution to each decay mode with the symbol “${\rm T}$” (the color-allowed tree), “${\rm C}$” (the color-suppressed tree), “${\rm P}$” ( the QCD penguin), “${\rm P_{EW}}$” ( the electroweak penguin) and “${\rm ann}$” (the annihilation). The label “LO” and “NLO” denote the PQCD predictions at the leading order only, or with the inclusion of all currently known NLO contributions, including the NLO twist-2 and twist-3 contributions to the form factors of $B_{s}\to(K,\eta_{s})$ transitions. The theoretical errors mainly come from the uncertainties of various input parameters, in particular, the dominant ones come from the shape parameter $\omega_{B_{s}}=0.50\pm 0.05$, the decay constant $f_{B_{s}}=0.23\pm 0.02$ GeV and the Gegenbauer moments in the DAs of the relevant mesons. The total errors of the NLO PQCD predictions are given in the Tables by adding the individual uncertainties in quadrature. For comparison, we also show in the fifth to eighth column of Table 3 the LO PQCD predictions as given in Ref. ali07 , the previous PQCD predictions with the inclusion of the partial NLO contributions known at that time as given in Refs.  xiao2012 ; xiao14a ; xiao14b , the NLO QCDF predictions as given in Ref. npb675 and the SCET results as given in Ref. scet06 . In last column, we show the currently available measured values for five decay modes as presented in PDG 2018 pdg2018 ( one can see Table 1 for more details ).

From the theoretical predictions for the branching ratios of the considered thirteen $B_{s}\to PP$ decays and those currently available experimental measurements for the five $B_{s}\to PP$ decay modes, as listed in Table 1 and 3, we have the following observations:

• (1)

For all considered decay channels, the previous LO PQCD predictions of the branching ratios as given in Ref. ali07 are well confirmed by our independent calculations within the errors. The small differences between the LO PQCD predictions as given in Ref. ali07 and in Table 3 are induced by the update of some input parameters. For most considered decay channels, our new NLO PQCD predictions as listed in the fourth column of Table 3 also agree well with those as given in Refs. bs08 ; xiao2012 ; xiao14a ; xiao14b . For ${\cal B}(\bar{B}_{s}^{0}\to\eta^{\prime}\eta^{\prime})$ decay, the new PQCD prediction ${\cal B}(\bar{B}_{s}^{0}\to\eta^{\prime}\eta^{\prime})=(30.8^{+11.2}_{-8.6})\times 10^{-6}$ is smaller than the previous one, but become agree well with the measured value: $(33.1\pm 7.0)\times 10^{-6}$ LHCbepep . The reason is that we here used $a^{\eta_{q},\eta_{s}}_{2}=0.115$ as input instead of $a^{\eta_{q},\eta_{s}}_{2}=0.44$ as being employed in Ref. xiao14b .

• (2)

For the “tree” dominated decay $\bar{B}_{s}^{0}\to\pi^{-}K^{+}$, the NLO contribution will result in a $\sim 20\%$ reduction to the LO PQCD prediction for its branching ratio, and leads to a better agreement with the data. The QCDF prediction ${\cal B}(\bar{B}_{s}^{0}\to\pi^{-}K^{+})=(10.2^{+6.0}_{-5.2})\times 10^{-6}$ as given in Ref. npb675 is far above the measured value. In Ref. chengbs09 , however, the authors presented their QCDF result ${\cal B}(\bar{B}_{s}^{0}\to\pi^{-}K^{+})=(5.3^{+0.4+0.4}_{-0.8-0.5})\times 10^{-6}$ by using a smaller form factor $F^{B_{s}K}_{0}(0)=0.24$ instead of the large one $F^{B_{s}K}_{0}(0)=0.31$ in Ref. npb675 ,

• (3)

Among the five “QCD-Penguin” decays, three decay modes $\bar{B}_{s}^{0}\to(K^{-}K^{+},K^{0}\bar{K}^{0},\eta^{\prime}\eta^{\prime})$ have been measured. The NLO contributions can provide $\sim 40\%$ to $\sim 150\%$ enhancements to the LO PQCD predictions of the branching ratios and help us effectively to obtain a much better agreement between the theory and the data for these three decays. Of course, the QCDF and SCET predictions for the branching ratios of these three decays as listed in Table 3 are also consistent with the experimental measurements within the still large errors.

• (4)

For the three “Colour-suppressed” decays, $\bar{B}_{s}^{0}\to(K^{0}\eta,K^{0}\eta^{\prime},\pi^{0}K^{0})$, the theoretical predictions for their branching ratios are at the level of $10^{-7}-10^{-6}$, and have not been observed by experiments. In PQCD factorization approach, the NLO contributions can provide a factor of two enhancement to their decay rates. The difference between different factorization approaches will be examined by the future LHCb measurements.

• (5)

The two “Electroweak-Penguin” decays $\bar{B}_{s}^{0}\to\pi^{0}\eta^{(\prime)}$ are very rare decay modes, the theoretical predictions for their decay rates are at the range of $10^{-8}-10^{-7}$, and hardly be observed in near future. In the PQCD approach, the NLO contributions are coming from the so-called “Vertex corrections” only and lead to a small enhancement no more than $20\%$. The substantial cancelations between the contributions arising from the $u\bar{u}$ and $d\bar{d}$ components of the $\pi^{0}$ meson is one of the major reasons for so small branching fractions of these two decays.

• (6)

For the two pure annihilation decays $\bar{B}^{0}_{s}\to(\pi^{+}\pi^{-},\pi^{0}\pi^{0})$, the NLO correction comes only from the usage of the Wilson coefficients and their renormalization group evolution at the NLO level, which results in a $16\%$ reduction to the corresponding LO PQCD predictions for their branching ratios. It is easy to see that although the central value of the PQCD predictions for ${\cal B}(\bar{B}^{0}_{s}\to\pi^{+}\pi^{-})$ is a little smaller than the measured one, but it still agree well with the measured values within errors. Although we believe that the still unknown NLO contributions from the annihilation Feynman diagrams is a higher order corrections to a small LO quantity, but it may help us to cover the remaining difference between the PQCD prediction and the data. This is the major motivation for us to complete the calculation for those still unknown NLO pieces. As is well-known, both the QCDF approach and the SCET can not provide reliable predictions for these pure annihilation decay modes. In Ref. chengbs09 , the authors studied $\bar{B}^{0}_{s}\to\pi\pi$decays by including the subleading power corrections to the penguin annihilation topology, and gave their prediction ${\cal B}(\bar{B}^{0}_{s}\to\pi^{+}\pi^{-})=(0.26\pm 0.10)\times 10^{-6}$, which is much larger than the one given in Ref. npb675 , but it is still much smaller than the measured value.

Since the theoretical and experimental errors of the ratios of the branching ratios are generally much smaller than those for the branching ratios themselves, people tend to define and measure such kinds of ratios. CDF and LHCb Collaboration , for example, also defined and measured some ratios of the branching ratios for several $B/B_{s}\to PP$ decays CDFpipi ; CDFpik ; CDFkk ; LHCbpipi ; LHCbpik based on some considerations of flavor symmetries , as listed in Table 4.

[TABLE]

By employing the PQCD approach, we also calculate the above four ratios at the LO and NLO level and present our results in the second and third column of Table 4. In the numerical calculations, $f_{s}/f_{d}=0.267^{+0.021}_{-0.020}$ as given in Ref. LHCbpik is used. From the PQCD predictions and the measured values as listed in Table 4, we find the following points:

1. (1)

For the ratio $R_{1}$ and $R_{3}$, the NLO contributions lead to a significant reduction to the LO results, and such reduction can help us effectively to explain the measured values. The NLO PQCD results agree well with the corresponding data. 2. (2)

For the ratio $R_{2}$, the NLO contribution is very small in size. The PQCD predictions for $R_{2}$ is about half of the measured result, but still consistent with it within $2\sigma$ error. 3. (3)

For the ratio $R_{4}$, the NLO contribution is also very small in size. But the PQCD predictions for $R_{4}$ agree very well with the measured one within $1\sigma$ error.

It is easy to see that the measured values of $R_{1,2,3,4}$ as listed in Table 4 can be understood properly in the framework of the PQCD factorization approach at the NLO level.

III.2 The CP-violating asymmetries

By using the formulaes as given in Eqs. (35,36,37), we calculate the direct and mixing-induced CP asymmetries of the thirteen $\bar{B}_{s}^{0}\to PP$ decays, show the numerical results in Table 5 for $A_{\rm CP}(\bar{B}_{s}^{0}\to\pi^{-}K^{+})$ and ${\cal A}_{f}$ for remaining twelve decays, and show the PQCD predictions for ${\cal S}_{f}$ and ${\cal H}_{f}$ in Table 6 for twelve $\bar{B}_{s}^{0}\to PP$ decays. As comparison, we also list the theoretical predictions as given in Refs. npb675 ; scet06 and the data as given in Refs. lhcb-18a ; pdg2018 . From these numerical results we find the following points:

• (1)

Our LO and NLO PQCD predictions for the direct and mixing-induced CP asymmetries of the considered $\bar{B}_{s}^{0}\to PP$ decays do agree well with those as given in Refs. ali07 ; bs08 ; xiao14a ; xiao14b . Some small differences between the central values are induced by the different choices or upgrade of some input parameters, such as the Gagenbauer moments and the CKM matrix elements.

• (2)

For most $\bar{B}_{s}^{0}\to PP$ decays, the effects of the NLO contributions to the CP asymmetries are small in magnitude. For $\bar{B}_{s}^{0}\to\pi^{0}(\eta^{\prime},K^{0})$ and $\bar{B}_{s}^{0}\to K^{0}\eta^{(\prime)}$ decays, however, the NLO enhancements can be as large as $(60-100)\%$.

• (3)

Among the thirteen $\bar{B}_{s}^{0}\to PP$ decays, only the CP asymmetries of the $\bar{B}^{0}_{s}\to\pi^{-}K^{+}$ and $\bar{B}^{0}_{s}\to K^{-}K^{+}$ decays have been measured by CDF and LHCb Collaboration  CDFpikcp ; LHCbpikcp1 ; LHCbpikcp2 ; lhcb-18a ; pdg2018 as listed in last column of Table 5 and 6 . For $\bar{B}^{0}_{s}\to(\pi^{-}K^{+},K^{-}K^{+})$ decays, fortunately, the NLO PQCD predictions do agree very well with those currently available measured values in both the sign and the magnitude within one standard deviation.

• (4)

For $\bar{B}^{0}_{s}\to(\bar{K}^{0}K^{0},\eta^{(\prime)}\eta^{(\prime)},\pi\pi)$ decays, the CP asymmetries ${\cal A}_{f}$ and ${\cal S}_{f}$ are all small in size and hardly be observed in future experiments. For $\bar{B}^{0}_{s}\to(K^{0}\pi^{0},K^{0}\eta^{\prime},\pi^{0}\eta^{(\prime)})$ decays, on the other hand, although their ${\cal A}_{f}$ and/or ${\cal S}_{f}$ may be large in size, but it is still very difficult to measure them due to their very small decay rates.

IV SUMMARY

In this paper, we studied the two-body charmless hadronic decays $\bar{B}^{0}_{s}\to PP$ ( here $P=(\pi,K,\eta,\eta^{\prime})$) by employing the PQCD factorization approach with the inclusion of all currently known NLO contributions: such as the NLO vertex corrections, the quark loop effects, the chromo-magnetic penguin diagrams and the NLO twist-2 and twist-3 contributions to the relevant form factors $F^{B_{s}\to K}_{0}(0)$ and $F^{B_{s}\to\eta_{s}}_{0}(0)$. In particular, we used the updated Gegenbauer moments for the distribution amplitudes of the final state mesons. We also compared our predictions for the branching ratios and CP violating asymmetries with those currently available experimental measurements, as well as the theoretical predictions obtained by using the QCDF approach and SCET method.

By the numerical evaluations and the phenomenological analyses, we found the following interesting points:

• (1)

For the three $\bar{B}_{s}^{0}\to(K^{0}\bar{K}^{0},K^{+}K^{-},\eta^{\prime}\eta^{\prime})$ decays, the NLO contributions can provide about $(40-150)\%$ enhancements to the LO PQCD predictions for their decay rates. For ${\cal B}(\bar{B}_{s}^{0}\to\pi^{-}K^{+})$ decay, however, the NLO contribution will result in a $20\%$ reduction to the LO PQCD prediction for its branching ratio. The agreement between the PQCD predictions and the measured values for these three decay modes, fortunately, are all improved effectively after the inclusion of the NLO contributions.

• (2)

For the pure annihilation $\bar{B}_{s}^{0}\to\pi^{+}\pi^{-}$ decay, the NLO contribution will lead to a $16\%$ reduction to the central value of the LO PQCD prediction. But the NLO PQCD prediction ${\cal B}(\bar{B}_{s}^{0}\to\pi^{+}\pi^{-})=(0.52^{+0.21}_{-0.18})\times 10^{-6}$ still agree well with the measured value $(0.68\pm 0.08)\times 10^{-6}$ pdg2018 within one standard deviation.

• (3)

Among the four ratios of the branching ratios $R_{1,2,3,4}$ defined and measured by CDF CDFpipi ; CDFpik ; CDFkk and LHCb Collaborations LHCbpipi ; LHCbpik , as illustrated in Table 4, the NLO PQCD predictions for $R_{1,3}$ become agree very well with the measured ones after the inclusion of a $40\%$ reduction from the NLO contributions. The NLO enhancements to ratio $R_{2,4}$ are very small (less than $4\%$ in size), the PQCD prediction for $R_{4}$ agrees very well with the measured value, while the PQCD prediction for $R_{2}$ is smaller than the measured one but still consistent with each other within $3\sigma$ errors .

• (4)

For both $\bar{B}_{s}^{0}\to K^{+}K^{-}$ and $\bar{B}_{s}^{0}\to\pi^{-}K^{+}$ decays, the NLO PQCD predictions for the CP-violating asymmetries do agree very well with the measured values pdg2018 in both the sign and the magnitude. For the direct CP violation ${\cal A}_{f}(\bar{B}_{s}^{0}\to K^{+}K^{-})$, the NLO contribution can help us to interpret the measured value.

• (5)

For all thirteen $\bar{B}_{s}^{0}\to PP$ decays, we also compared our results with those obtained in the QCDF and SCET approaches npb675 ; sun2003 ; chengbs09 ; scet06 and made some comments on the similarities and the differences between the theoretical predictions from different approaches. For most $\bar{B}_{s}^{0}\to PP$ decays, in fact, the experimental measurements are still absent now. The forthcoming precision measurements at LHCb and Belle-II could help us to test the theoretical predictions.

Acknowledgements.

This work is supported by the National Natural Science Foundation of China under Grants No. 11775117, 11875033 and No. 11765012 , by the Qing Lan Project of Jiangsu Province under Grant No. 9212218405, and by the Research Fund of Jiangsu Normal University under Grant No. HB2016004.