QCD视锥和规则的B′0→π+π0ℓν¯ℓ中的形状因子资源-CSDN文库

Open

Access

117 浏览量 2020-04-03 08:50:00 上传评论收藏 435KB PDF 举报

资源推荐

资源详情

资源评论

Available online at www.sciencedirect.com

ScienceDirect

Nuclear Physics B 905 (2016) 373–390

www.elsevier.com/locate/nuclphysb

Form factors in

→ π

¯ν



from QCD light-cone

sum rules

Christian Hambrock, Alexander Khodjamirian

∗

Theoretische

Physik 1, Naturwissenschaftlich-Technische Fakultät, Universität Siegen, D-57068 Siegen, Germany

Received 16

November 2015; received in revised form 25 February 2016; accepted 28 February 2016

Available

online 2 March 2016

Editor: Tommy

Ohlsson

Abstract

The

form factors of the semileptonic B → ππ¯ν decay are calculated from QCD light-cone sum rules

with the distribution amplitudes of dipion states. This method is valid in the kinematical region, where the

hadronic dipion state has a small invariant mass and simultaneously a large recoil. The derivation of the sum

rules is complicated by the presence of an additional variable related to the angle between the two pions.

In particular, we realize that not all invariant amplitudes in the underlying correlation function can be used,

some of them generating kinematical singularities in the dispersion relation. The two sum rules that are free

from these ambiguities are obtained in the leading twist-2 approximation, predicting the

→ π

form

factors F

⊥

and F



of the vector and axial b → u current, respectively. We calculate these form factors at the

momentum transfers 0 <q

 12 GeV

and at the dipion mass close to the threshold 4m

. The sum rule

results indicate that the contributions of the higher partial waves to the form factors are suppressed with

respect to the lowest P -wave contribution and that the latter is not completely saturated by the ρ-meson

term.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

Corresponding author.

E-mail

address: khodjamirian@physik.uni-siegen.de (A. Khodjamirian).

http://dx.doi.org/10.1016/j.nuclphysb.2016.02.035

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 SCOAP

374 C. Hambrock, A. Khodjamirian / Nuclear Physics B 905 (2016) 373–390

1. Introduction

The current tendency in the studies of the ﬂavour-changing decays of heavy hadrons is to

enlarge the set of exclusive processes used for the determination of the fundamental CKM

parameters. Probing different exclusive b → u processes may, in particular, help in the |V

determination. The interval of this CKM parameter obtained from the measurements of the

B → π¯ν



decay, combined with the B → π form factors from lattice QCD or from the QCD

light-cone sum rules (LCSR), deviates from the results obtained in the inclusive B → X

¯ν



decay studies (see, e.g., the review [1] and references therein).

Alternative e

xclusive b → u processes are being actively investigated, among them the

B → ππ¯ν



decay, where the ρ-meson contribution is prominent. The semileptonic B-decay

mode with the two-pion (dipion) ﬁnal state is not only important for the |V

| determination,

but also has a rich set of observables (see e.g., Ref. [2]) which can be used for nontrivial tests

of Standard Model. The B → ππ¯ν



decay has already been measured, but mainly its reso-

nant, B → ρ¯ν



part (see e.g., the BaBar [3] and Belle [4] Collaborations data). Signiﬁcantly

more detailed data on the B → ππ¯ν



observables are expected from the Belle-2 experiment in

future.

The dynamics of the B → ππ¯ν



decay is governed by general B →2π form factors, hence

the calculation of these form factors is becoming the next big task for the practitioners of QCD-

based methods. As discussed in Ref. [2] in detail, various non-lattice methods, from heavy-meson

chiral perturbation theory to the soft-collinear effective theory are applicable, depending on the

gion of the Dalitz plot formed by the invariant masses of the lepton pair and dipion.

In this paper

, we use the method of LCSRs [5] to calculate the B →2π form factors relevant

for the

→ π



−

¯ν



decay. We shall conﬁne ourselves with the charged dipion (isovector)

ﬁnal state, and postpone the case of the neutral (isoscalar) state with related scalar resonances

for the future work. The approach we use is applicable in the region of small and intermediate

lepton-pair masses, restricting simultaneously the dipion invariant mass by the  1

GeV region,

so that a large hadronic recoil takes place with two energetic and almost collinear pions in the

B-meson rest frame.

The technique we use has man

y similarities with the LCSRs obtained for B → π form factors,

but employs a different and more complicated nonperturbative input: the light-cone distribution

amplitudes (DAs) of the dipion state. These universal objects have been introduced in Refs. [6,

7] to encode the hadronization of the quark-pair in the γγ

∗

→ 2π process at large momentum

transfer. The properties of dipion DAs were worked out in details in Refs. [8,9]. In a differ-

ent context, two-meson wave functions in hard exclusive processes were discussed earlier in

Ref. [10].

In this paper we aim at the follo

wing goals. First, we demonstrate how the method works,

deriving the LCSRs for the two of the B → ππ form factors in the leading twist-2 approxi-

mation. The sum rules predict these form factors at large recoil and small mass of the dipion

state. Second, based on this calculation, we in

vestigate the role of higher partial waves in the

B → ππ form factors and assess the impact of the contributions beyond the ρ-meson in the low-

est P -wave. In what follows, the derivation of LCSRs for B → ππ form factors is presented in

Sect. 2. In Sect. 3 we compare our predictions with the B → ρ form f

actors. In Sect. 4 using the

available information on the chiral-odd dipion DA, we calculate the form factors numerically.

Our conclusions are presented in Sect. 5. The Appendices contain some details (A) on the decay

kinematics and (B) on the dipion DAs.

C. Hambrock, A. Khodjamirian / Nuclear Physics B 905 (2016) 373–390 375

2. Light-cone sum rules with dipion distribution amplitudes

The LCSR derivation starts from deﬁning an appropriate correlation function. We consider

the T -product of the b → u weak current j

V −A

(x) =¯u(x)γ

(1 − γ

)b(x) with the B-meson

interpolating current j

(B)

(0) = im

b(0)γ

d(0). Since we are interested in the ﬁnal state with

two pions, this T -product is then sandwiched between the vacuum and the on-shell dipion state:



(q, k

) = i



iqx

π

)π

)|T {j

V −A

(x), j

(0)}|0. (1)

The above correlation function has a more complicated kinematics than in the case of the one-

pion ﬁnal state and depends on three independent 4-momenta q, k

, k

. We denote by k = k

the total dipion four-momentum and by p = q + k the external four-momentum of the B-meson

interpolating current. At ﬁxed k

1,2

= m

these momenta form four independent invariant vari-

ables, as such we choose p

= (q + k)

, q

, k

and q · k, where k = k

− k

. Further details on

the kinematics are given in the Appendix A.

The correlation function (1) is decomposed in four independent Lorentz-vectors



(q, k

) = i

μαβρ



(V )



(A,q)



(A,k)



(A,k)

, (2)

where the ﬁrst term (the rest) corresponds to the contribution of the vector (axial) part of the

b → u weak current and the invariant amplitudes 

(V ),(A,q),...

depend on the four invariant vari-

ables: p

, q

, k

, q · k.

To guarantee the validity of the operator–product expansion (OPE) for the correlation function

(1) near the light-cone (x

∼ 0), we consider the region p

 m

and q

 m

, so that the

b-quark mass provides the large scale. In this respect, the conditions for the light-cone dominance

are practically the same as in the case of the vacuum-to-pion correlation functions used to obtain

the LCSRs for B → π form factors (for a detailed derivation of the latter sum rules see, e.g.,

Ref. [11]). An additional constraint concerns the in

variant mass of dipion which is also kept

small, k

 1GeV

 m

. In this region the two-pion system with isospin one is dominated

by the ρ(770) resonance, accompanied by a nonresonant background. In this paper, we only

consider the charged dipion state, so that only odd angular momenta contribute in the isospin

symmetry limit. This limitation simpliﬁes our analysis, whereas the case of neutral dipion state

where also the scalar/isoscalar f

resonances contribute, will be considered elsewhere.

Turning to the calculation of the correlation function (1), in the leading-order (LO) approxi-

mation (α

= 0), after inserting the free b-quark propagator, we obtain:



(q, k

) = i



(2π)

i(q−f)x

−f

×π

)π

)|¯u(x)γ

(1 −γ

)(



f + m

)γ

d(0)|0. (3)

This expression consists of the hard-scattering amplitude – the virtual b-quark propagator – con-

voluted with the vacuum → dipion matrix elements of bilocal quark–antiquark operators. These

matrix elements absorb long-distance dynamics and are expressed via universal dipion DAs, de-

ﬁned following Ref. [8]. The LO diagram of OPE for the correlation function (1) is sho

wn in

Fig. 1.

Here we use the convention 

0123

=−1.

剩余17页未读，继续阅读

评论收藏

内容反馈

weixin_38622962

粉丝: 3
资源: 903

QCD视锥和规则的B′0→π+π0ℓν¯ℓ中的形状因子

带有视锥QCD和规则的B-S过渡形状因数

QCD视锥和规则中的B→ρ螺旋形状因子

极化顶夸克衰变中的T-奇数相关性在顺序衰变t（↑）→Xb + W +（→ℓ++νℓ）和准三体衰变t（↑）→Xb +ℓ++νℓ中

晶格QCD的K +→π+νν¯衰减幅度

近似物理介子质量的稀有钾衰变K +→π+νν¯的格子QCD研究

B→Dℓν¯ℓ衰变中形状因子的新参数化

B s→Kℓνℓ和B（s）→π（K）ℓ+ℓ-在大后坐力和CKM矩阵元素下衰减

QCD和规则法框架下有限温度下的BcBcJ /ψ顶点形状因子

从QCD圆锥总和规则对Λb→Λ形状因数进行微扰校正

满足PCAC关系的晶格QCD的轴向矢量形状因子

QCD方法中的Pion和Kaon形状因子

B→π，K形状因子的QCD计算，带有较高扭转校正

基于QCD的方法中D→π−π+和D→K − K +中的直接CP不对称

B→ππℓν衰减的QCD分解在大二倍质量下

光锥QCD和规则中罕见的Λb（Λb*）→Nℓ+ℓ-衰变的形状因子

Dipion光锥分布幅度和B→ππ形状因子

在整个运动学上可及的区域重新检查了B→πℓ+ ℓ−形状因子

两味晶格QCD的核子广义形状因子

QCD中的平面四环形状因子和尖端异常尺寸

光铁蛋白对QCD中精确的希格斯-胶子形状因子的贡献

全息QCD对J /ψ和ϒ的衍射光产生：质子中胶子的引力形状因子和GPD

完全QCD中的半轻质Λb→Λcℓν́ℓ跃迁

扰动QCD方法中的准两体衰变B（s）→PD0 *（2400）→PDπ

QCD中的半轻体Bs→Ds2 ∗（2573）ℓν¯ℓ跃迁

Z→π+ π−，K + K−：微扰QCD方法的试金石

在简化的新物理模型中，K→πvν¯$$ K \ to \ pi \ nu \ overline {\ nu} $$和ε′/ε

最新资源