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Physics Letters B www.elsevier.com/locate/physletb

To understand the rare decay B s → π + π − + − Wei Wang a,b , Rui-Lin Zhu a,b,∗ a

INPAC, Shanghai Key Laboratory for Particle Physics and Cosmology, Department of Physics and Astronomy, Shanghai Jiao-Tong University, Shanghai, 200240, China b State Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Beijing, 100190, China

a r t i c l e

i n f o

Article history: Received 3 January 2015 Received in revised form 4 March 2015 Accepted 6 March 2015 Available online 10 March 2015 Editor: J. Hisano

a b s t r a c t Motivated by the LHCb measurement, we analyze the B s → π + π − + − decay in the kinematics region where the pion pairs have invariant masses in the range 0.5–1.3 GeV and muon pairs do not originate from a resonance. The scalar π + π − form factor induced by the strange s¯ s current is predicted by the unitarized approach rooted in the chiral perturbation theory. Using the two-hadron light-cone distribution amplitude, we then can derive the B s → π + π − transition form factor in the light-cone sum rules approach. Merging these quantities, we present our results for differential decay width which can generally agree with the experimental data. More accurate measurements at the LHC and KEKB in future are helpful to validate our formalism and determine the inputs in this approach. © 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

Very recently, the LHCb has performed an analysis of rare B s decays into the π + π − μ+ μ− ﬁnal state [1] and the branching fraction is measured as

B( B s → π + π − μ+ μ− ) = (8.6 ± 1.5 ± 0.7 ± 0.7) × 10−8 ,

(1)

where the ﬁrst two errors are statistical, and systematic respectively. The third error is due to uncertainties on the normalization, i.e. the branching fraction of the B 0 → J /ψ(→ μ+ μ− ) K ∗ (→ K + π − ). The branching fraction for B s → f 0 (980)μ+ μ− [1] is determined as:

B( B s → f 0 (980)(→ π + π − )μ+ μ− ) = (8.3 ± 1.7) × 10−8 ,

(2)

which lies in the vicinity of the total branching fraction in Eq. (1). Despite the errors, the closeness of the two branching fractions and the differential distribution as shown later in Fig. 4(b) may indicate the dominance of the f 0 (980) contributions in the B s → π + π − μ+ μ− . The B s → π + π − μ+ μ− is a four-body process. Its decay amplitude shows two distinctive features. On the one side, the π + π − ﬁnal state interaction is constrained by unitarity and analyticity. On the other side, the b mass scale is much higher than the hadronic scale QCD , which allows an expansion of the hard-scattering kernels in terms of the strong coupling constant αs and the dimensionless power-scaling parameter QCD /mb . In Refs. [2–4], we

have developed a formalism that makes use of these two advantages. This approach was also pioneered in Refs. [6,7], and see also Refs. [8–11] for applications to charmless three-body B decays. In doing this, the new formalism can simultaneously merge the perturbation theory at the mb scale and the low-energy effective theory based on the chiral symmetry to describe the S-wave ππ scattering. The aim of this work is to further examine this formalism by confronting this theoretical framework with the recent data on B s → π + π − μ+ μ− . An independent analysis that is based on the perturbative QCD approach is also under progress [12]. We start with the differential decay width for B s → π + π − + − . The effective Hamiltonian for the transition b → s+ −

GF

∗ Heff = − √ V tb V ts

2

10

C i (μ) O i (μ)

i =1

involves various four-quark and the magnetic penguin operators O i . The C i (μ) are the corresponding Wilson coeﬃcients for these local operators O i . G F is the Fermi constant, and V tb = 0.99914 ± 0.011 0.00005 and V ts = −0.0405+ −0.012 [13] are the CKM matrix elements. The b and s quark masses are mb = (4.66 ± 0.03) GeV and ms = (0.095 ± 0.005) GeV [13]. The b → sl+l− transition has the decay amplitude

i M(b → s+ − )

*

Corresponding author. E-mail address: [email protected] (R.-L. Zhu).

¯ V +A = iN 1 × (C 9 + C 10 )[¯sb] V − A []

http://dx.doi.org/10.1016/j.physletb.2015.03.011 0370-2693/© 2015 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3 .

468

W. Wang, R.-L. Zhu / Physics Letters B 743 (2015) 467–471

¯ V −A + (C 9 − C 10 )[¯sb] V − A [] + 4C 7L mb [¯si σμν (1 + γ5 )b] + 4C 7R mb [¯si σμν (1 − γ5 )b]

The B s → ππ form factors used in Eq. (8) are deﬁned by

qμ

× [¯γ ν ]

q2

× [¯γ ν ] ,

qμ q2

(3)

αem ∗ N1 = √ V tb V ts . 4 2 π GF

(4)

The B → M 1 M 2 + − is a four-body decay mode, whose decay amplitude can be obtained by sandwiching Eq. (3) between the initial and ﬁnal hadronic states. The spinor product [¯sb] will be replaced by corresponding hadronic matrix elements. A general differential decay width for B → M 1 M 2 + − with various partial wave contributions has been derived using the helicity amplitude in Ref. [14]. In the B s → π + π − μ+ μ− case, the S-wave contribution will dominate and thus the angular distribution is derived as

3

=

dm2π π dq2 d cos θl

8

J 1c + J 2c cos(2θl ) ,

(5)

where θl is the polar angle between the μ− and the B s moving direction in the lepton pair rest frame. The angular coeﬃcients are given by

J 1c

0 ˆ l2 |A0L0 A0R0∗ | cos(δ L0 = |A0L0 |2 + |A0R0 |2 + 8m − δ 0R0 ) (6)

J 2c = −βl2 |A0L0 |2 + |A0R0 |2 . In the above equations, β =

(7)

N2i

ˆ = m / q2 . The 1 − 4m2 /q2 , and m

1

√ (C 9 ∓ C 10 )

mπ π

√ + 2(C 7L − C 7R ) A0L / R ,t =

N2i

N2 =

(C 9 ∓ C 10 )

mπ π

16π 2

Nπ π =

8

N1 Nπ π

√

q2

F1 (q )

m2B

=

du

= NF

u

exp −

F T (q ) ,

(8)

q2

π

uM 2 1

× − mb π π (u ) + umπ π πs π (u ) + mπ π σπ π (u ) mb2

+ q − u mπ π mπ π σπ π (u ) 2

2

uM 2

6

+ exp [−s0 / M ]

mπ π σπ π (u 0 ) mb2 − u 20 m2π π + q2

2

F− (mπ π , q ) = N F

du u

exp −

2

¯ 2π π − uq ¯ 2 mb2 + u um

(9)

= 2 N 2 C 10 i

1 mπ π

m2B s

− mπ π F0 (q2 ) . (10) 2

q2

,

(12)

uM 2

× mb π π (u ) + (2 − u )mπ π πs π (u )

−

1−u

mπ π σπ π (u ) 3u u (mb2 + q2 − u 2 m2π π ) + 2(mb2 − q2 + u 2 m2π π )

mπ π σπ π (u )

u2 M 2

6 u 0 (mb2

+ q2 − u 20m2f 0 ) + 2(mb2 − q2 + u 20m2π π ) u 0 (mb2 + u 20 m2π π − q2 ) mπ π σπ π (u 0 )

F0 (m2π π , q2 ) = F1 (m2π π , q2 ) +

q2 m2B s − m2π π

F T (m2π π , q2 ) = 2N F (m B s + mπ π ) 1 du

×

u

exp −

6

ππ ,

mb2 + u 20 m2π π − q2

6

1 2

3

2

/m2

.

u0

×

−m π π F0 (q2 ) ,

¯ 2π π − uq ¯ 2 mb2 + u um

× exp [−s0 / M ] 1 − 4m2

λq β

− A0L ,t

F1 (m2π π , q2 ) 1

2

2

(11)

As we have shown in Ref. [2], an explicit calculation of the B s → π + π − form factors requests the knowledge on generalized light-cone distribution amplitudes [16–20]. The expressions in the light-cone sum rules are given as [2],

2

3 256π 3m3B

A0R ,t

(π π ) S ( p π π )|¯sσμν qν γ5 b| B s ( p B s )

F T (m2π π , q2 ) 2 2 2 (m B − mπ π )qμ − q P μ . = mπ π (m B + mπ π )

−

λmb

Here the script t denotes the time-like component of a virtual state decays into a lepton pair. The function λ is related to the magnitude of the π + π − momentum in B s meson rest frame: λ ≡ λ(m2B s , m2π + π − , q2 ), and λ(a2 , b2 , c 2 ) = (a2 − b2 − c 2 )2 − 4b2 c 2 . The combination of the time-like decay amplitude is introduced in the differential distribution

At0

q2

qμ F0 (m2π π , q2 ) ,

−

+

2

q2 (m B + mπ π )

1

λ

where

1

2

u0

helicity amplitude is

=

q

− mπ π

m2B

2

ˆ l2 |At0 |2 , + 4m

A0L / R ,0

+

+

mπ π

+

where C 7L = C 7 and C 7R = C 7L ms /mb , and

d3

(π + π − ) S ( p π π )|¯sγμ γ5 b| B s ( p B s )

m2B − m2π π 1 Pμ − qμ F1 (m2π π , q2 ) = −i 2

(13)

,

F− (m2π π , q2 ),

¯ 2π π ) ¯ 2 + u um (mb2 − uq

uM 2

u0

π π ( u ) mπ π σπ π (u ) × − + mb 2 2

+ mb

6uM

mπ π σπ π (u 0 )

exp[−s0 / M 2 ]

6

mb2 − q2 + u 20 m2π π

,

(14)

W. Wang, R.-L. Zhu / Physics Letters B 743 (2015) 467–471

Table 1 The B s → f 0 (980) form factors in the light-cone sum rules at LO and NLO in

469

αs [15].

LO

F (0)

aF

bF

NLO

F (0)

aF

F1

0.185 ± 0.029

0.07 0.59+ −0.05

F1

0.238 ± 0.036

F0

0.238 ± 0.036

0.13 1.50+ −0.09

F0

0.185 ± 0.029

0.13 1.44+ −0.09

FT

0.228 ± 0.036

FT

0.308 ± 0.049

0.12 0.47+ −0.09 0.13 1.42+ −0.10

where

N F = B 0 F π π (m2π π )

u0 =

mb + m s 2m2B s f B

m2π π + q2 − s0 +

0.08 0.01+ −0.09

exp

m2B s M2

0.06 0.60+ −0.05

(m2π π + q2 − s0 )2 + 4m2π π (mb2 − q2 )

.

ππ form factor is deﬁned as

0|¯s s|π + π − = B 0 F π π (m2π π ),

(16)

and the B 0 is the QCD condensate parameter:

0|¯qq|0 ≡ − f π2 B 0 ,

(17)

with f π MeV being the pion decay constant at LO. For the numerics, we use f π = 91.4 MeV and 0|¯qq|0 = −(0.24 ± 0.01) GeV3 (for a review see Ref. [21]), which corresponds to B 0 = (1.7 ± 0.2) GeV. The M is a Borel parameter introduced to suppress higher twist contributions. Our formulae can be compared to the results for the B s → f 0 (980) transition [15], with the correspondence

if 0 (u ) ↔ πi π (u ),

f f 0 ↔ B 0 F π π (m2π π ), (18)

where f f 0 is the decay constant of f 0 (980) deﬁned by the scalar s σ current. The twist-3 distribution amplitudes, π π (u ) and π π (u ), for the scalar ππ state have the same asymptotic forms with the ones for a scalar resonance [22], while the twist ones can be similarly expanded in terms of the Gegenbauer moments. Inspired by this similarity, we can plausibly introduce an intuitive matching: B s →π π

Fi

(m2π π , q2 ) =

1 f f0

Bs→ f0

B 0 F π π (m2π π ) F i

(q2 ).

F i (0) 1 − ai

q2 /m

B 2s

+ bi (q2 /m2B s )

.

Fig. 1. Feynman diagrams for the scalar form factor at tree-level and one-loop level in CHPT. The wave function renormalization diagrams are not shown here.

1 1 |ππ I=0 = √ π + π − + √ π 0 π 0 ,

(20)

Numerical results for these quantities where f f 0 = (0.18 ± 0.015) GeV [33] are taken from Ref. [15] and are collected in Table 1. Using a different value for f f 0 for instance in Refs. [22,34] will not induce any difference to the generalized form factor, since such effects will cancel as demonstrated in Eq. (19). In the following calculation, we will use the NLO results for the B s → f 0 transition. Using the LO results can reduce the differential decay width by about 40%. The scalar ππ form factor F π π (m2π π ) has been calculated within a variety of approaches using (unitarized) chiral perturbation theory (CHPT) [35–42] and dispersion relations [43]. In terms of the isoscalar S-wave states

(21)

3 6 1 + − 1 0 0 | K K¯ I=0 = √ K K + √ K K¯ , 2 2

(22)

the scalar form factors are deﬁned as

√

2B 0 F 1s (s) = 0|¯s s|ππ I=0 ,

(23)

= 0|¯s s| K K¯ I=0 ,

(24)

√

2B 0 F 2s (s)

where the notation (π = 1, K = 2) has been √ introduced for simplicity, and the convention F π π (m2π π ) = 2/ 3F 1s (m2π π ). In the CHPT, expressions have already been derived by calculating the diagrams in Fig. 1 up to NLO [36,40–42]:

√ F 1CHPT (s)

8s 3 16m2π r s r 2L 6 − L r4 + 2 L r4 + J (s) 2 2 f f 2 f 2 KK

=

+

2 m2π 9 f2

F 2CHPT (s) = 1 +

(19)

Here we have assumed the dominance of the f 0 (980) which is justiﬁed in the B s → π + π − μ+ μ− as shown in the data in Eq. (2) and Eq. (1). The B s → f 0 (980) form factors have been calculated in the light-cone sum rules at leading order (LO) and next-to-leading order (NLO) in αs [15,23–25], and in the perturbative QCD approach [26–31] in Ref. [32]. The momentum distribution in the form factors has been parametrized in the form:

F i (q2 ) =

0.14 1.46+ −0.10

,

(15)

m f 0 ↔ mπ π ,

0.09 0.58+ −0.07 0.09 −0.36+ −0.08 0.09 0.58+ −0.07

0.14 0.53+ −0.10

2m2π π

In the above the scalar

bF

+

r J ηη (s) ,

8L r4 f2

16L r6

+

f2

(25)

s − m2π − 4m2K +

4m2K + m2π +

9s − 8m2K 18 f 2

4L r5 f2

32L r8 f2

r J ηη (s) +

3s 4f2

s − 4m2K

m2K +

2 3

μη

J rK K (s).

(26)

With the increase of the invariant mass of the ππ system, higher order contributions become more important. It has been proposed that the unitarized approach can sum higher order corrections and extend the applicability to the scale around 1 GeV [44]. A sketch of the resummation scheme is shown in Fig. 2. In this ﬁgure, the K (s) is the S-wave projected kernel of meson– meson scattering amplitudes [40,41]:

K (s) = K 11 =

K 11 K 21

2s − m2π 2f2

K 12 K 22

,

(27)

, √ K 12 = K 21 =

3s

4f2

,

K 22 =

3s 4f2

,

(28)

where the subscripts 1, 2 denote the ππ and K K¯ state, respectively. The function g (s) is the loop integral which can be calculated in the cutoff-regularization scheme with qmax ∼ 1 GeV being the cutoff [cf. Erratum of Ref. [44]] or in dimensional regularization. In the latter scheme, the meson loop function g ii (s) is given by

470

W. Wang, R.-L. Zhu / Physics Letters B 743 (2015) 467–471

Fig. 2. The s-channel diagrams to the scalar π π form factors in CHPT. With the increase of the unitarized approach [44], these diagrams can be summed.

Fig. 3. The π π scalar form factor obtained in the unitarized chiral perturbation theory. The modulus, real part and imaginary part are shown in solid, dashed and dotted curves, respectively.

J iir (s)

≡

1

1 − log

16π 2

m2i

μ2

σi ( s ) + 1 − σi (s) log σi ( s ) − 1

σi ( s ) =

B( B s → f 0 (980)(→ π + π − )μ+ μ− ) = (4.1 ± 1.6) × 10−8 , (33)

1 − 4m2i /s. Imposing the unitarity constraints, the

which deviates from the data by about 2σ . However, one expects the experimental result in Eq. (2) would get somewhat reduced. This can be witnessed by the B − → J /ψ K − and B − → K − μ+ μ− [13]

scalar form factor can be expressed in terms of the algebraic coupled-channel equation [36,38]

F (s) = R (s)[ I + g (s) K (s)]−1 6

= R (s)[ I − g (s) K (s)] + O( p ),

(30)

where the above equation has been expanded up to NLO in the chiral expansion. The R (s) = ( R 1 (s), R 2 (s)) includes both tree-level contributions, and other higher order corrections that have not been summed. Thus this function has no right-hand cut, and can be obtained by matching onto the CHPT results in Eqs. (25)–(26) [38,45]:

√

R 1 (s) =

3

16m2π f2

2

−

m2π

+ +

8L r4 f2

16L r6 f2 36π

−

L r4

1 + log

+

m2η

Lr f2 4

f2

1 + log

4L r5 f2

32L r8 f2

μ2

s − 4m2K

m2K +

2

mη

(31)

,

μ2

4m2K + m2π +

=

(4.49 ± 0.23) × 10−7 ∼ 4.4 × 10−4 . (1.027 ± 0.031) × 10−3

(34)

If this ratio were not sensitive the light meson in the ﬁnal state which is true in most cases, the branching fraction for the B s → J /ψ f 0 (980) [13]

would indicate

s − 4m2K − m2π +

B( B − → K − μ+ μ− ) B( B − → J /ψ K − )

B( B s → J /ψ f 0 ) = (1.39 ± 0.14) × 10−4 ,

8s

m2K 2

2L r6

72π 2 f 2

R 2 (s) = 1 +

modulus, real part and imaginary part are shown as solid, dashed and dotted curves. Equipped with the results for scalar form factor and heavy to light transition, we can explore the differential branching fraction for the B s → π + π − μ+ μ− . Our theoretical results for dB /dmπ π are given in the left panel of Fig. 4. This clearly shows the peak corresponding to the f 0 (980). In order to compare with the experimental data, we also give the binned results on the right panel in Fig. 4 from 0.5 GeV to 1.3 GeV. Theoretical errors shown in this panel arise from the ones in the form factors. The experimental data (with triangle markers) has been normalized to the central value given in Eq. (1). The comparison in this panel shows a general agreement between our theoretical prediction and the experimental data except in a few bins. This agreement is very encouraging. In spite of the agreement, there exist some differences in our results and data. For instance our theoretical result does not show the enhancement at mπ π (800, 1100, 1250) MeV as given in the data. The excess at 800 MeV may come from the tail of the B s → η(→ π + π − π 0 , π + π − γ )μ+ μ− , while in the range above 1 GeV, the contribution from the f 0 (1370) may not be negligible. Integrating out the mπ π , we have the branching fraction:

(29)

= − g ii (s), with

π π invariant mass, higher order contributions may become important. In the

.

2 3

μη (32)

With the above formulae and the ﬁtted results for the lowenergy constants L ri in Ref. [38] (evolved from mρ to the scale √ μ = 2qmax / e), we show the strange ππ form factor in Fig. 3. The

B( B s → f 0 (980)μ+ μ− ) ∼ 6.1 × 10−8 .

(35)

This value is smaller by about 30% than the central value given in Eq. (1), and is more consistent with our theoretical result. The future measurement with more data at the experimental facilities like LHC and KEKB will be able to clarify this point, and thus to examine our theoretical formalism more precisely. We strongly encourage our experimental colleagues to conduct such measurements. In summary, in this work we have analyzed the B s → π + π − + − that has focused on the region where the pion pairs have invariant masses in the range 0.5–1.3 GeV and muon pairs do not originate from a resonance. We have adopted the approach proposed in our previous work [2–4] (see also Ref. [5] for an overview) which makes uses of the two-hadron light-cone

W. Wang, R.-L. Zhu / Physics Letters B 743 (2015) 467–471

471

Fig. 4. The differential branching ratio for the B s → π + π − + − . The experimental data (with triangle markers) has been normalized to the central value of the branching fraction: B( B 0s → π + π − μ+ μ− ) = (8.6 ± 1.5 ± 0.7 ± 0.7) × 10−8 . Theoretical predictions (with square markers) are based on the result for the time-like scalar form factors derived in the unitarized CHPT.

distribution amplitude. The scalar π + π − form factor induced by the strange s¯ s current is predicted by the unitarized chiral perturbation theory. The heavy to light transition can then be handled by the light-cone sum rules approach. Merging these quantities, we have presented our theoretical results for differential decay width and compared with the experimental data. Except in a few bins, our theoretical results are in alignment with the data. We have also discussed the disagreement and given our expectation. More accurate measurements at the LHC and KEKB in future are helpful to validate/falsify our formalism and determine the inputs in this approach. Acknowledgements The authors thank Michael Döring, Feng-Kun Guo, Bastian Kubis, Hsiang-Nan Li, Cai-Dian Lü, Ulf-G. Meissner, Eulogio Oset, WenFei Wang for enlightening discussions. This work was supported in part by a key laboratory grant from the Oﬃce of Science and Technology, Shanghai Municipal Government (No. 11DZ2260700), and by Shanghai Natural Science Foundation under Grant No. 15ZR1423100. References [1] R. Aaij, et al., LHCb Collaboration, arXiv:1412.6433 [hep-ex]. [2] U.G. Meißner, W. Wang, Phys. Lett. B 730 (2014) 336, arXiv:1312.3087 [hep-ph]. [3] U.G. Meißner, W. Wang, J. High Energy Phys. 1401 (2014) 107, arXiv:1311.5420 [hep-ph]. [4] M. Döring, U.G. Meißner, W. Wang, J. High Energy Phys. 1310 (2013) 011, arXiv:1307.0947 [hep-ph]. [5] W. Wang, Int. J. Mod. Phys. A 29 (2014) 1430040, arXiv:1407.6868 [hep-ph]. [6] S. Gardner, U.G. Meißner, Phys. Rev. D 65 (2002) 094004, arXiv:hep-ph/ 0112281. [7] M. Maul, Eur. Phys. J. C 21 (2001) 115, arXiv:hep-ph/0104078. [8] C.H. Chen, H.n. Li, Phys. Lett. B 561 (2003) 258, arXiv:hep-ph/0209043. [9] C.H. Chen, H.n. Li, Phys. Rev. D 70 (2004) 054006, arXiv:hep-ph/0404097. [10] W.F. Wang, H.C. Hu, H.n. Li, C.D. Lü, Phys. Rev. D 89 (2014) 074031, arXiv: 1402.5280 [hep-ph]. [11] H.s. Wang, S.m. Liu, J. Cao, X. Liu, Z.j. Xiao, Nucl. Phys. A 930 (2014) 117. [12] W.F. Wang, H.n. Li, W. Wang, C.D. Lü, arXiv:1502.05483 [hep-ph]. [13] K.A. Olive, et al., Particle Data Group Collaboration, Chin. Phys. C 38 (2014) 090001. [14] C.D. Lu, W. Wang, Phys. Rev. D 85 (2012) 034014, arXiv:1111.1513 [hep-ph]. [15] P. Colangelo, F. De Fazio, W. Wang, Phys. Rev. D 81 (2010) 074001, arXiv: 1002.2880 [hep-ph].

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