ψ(4040) and ψ(4160) Decays into the Dd± Ds∓$ D_{d}^{\pm } D_{s}^{\mp }$ Pair and the S- D Mixing Effects

Junfeng Sun; Yueling Yang; Na Wang; Qin Chang; Gongru Lu

doi:10.1007/s10773-017-3335-5

International Journal of Theoretical Physics

June 2017, Volume 56, Issue 6, pp 1892–1902 | Cite as

ψ(4040) and ψ(4160) Decays into the $D_{d}^{\pm }D_{s}^{\mp }$ Pair and the S-D Mixing Effects

Authors
Authors and affiliations

Junfeng Sun
Yueling YangEmail author
Na Wang
Qin Chang
Gongru Lu

Article

First Online: 25 March 2017

Received: 04 January 2017

Accepted: 13 March 2017

90 Downloads

Abstract

The ψ(4040) and ψ(4160) →$D_{d}^{\pm }D_{s}^{\mp }$ decays are studied with the perturbative QCD approach phenomenologically. It is found that branching ratios for these decays are insensitive to the S-D mixing angle 𝜃 ∈ [−30^∘,30^∘], and too tiny to be measured in the near future experiments.

Keywords

ψ(4040) ψ(4160) S-D mixing Weak decay Perturbative QCD

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Notes

Acknowledgments

The work is supported by the National Natural Science Foundation of China (Grant Nos. U1632109, 11547014 and 11475055).

Appendix A: Wave Functions for the 3S and 2D Charmonium States

It is usually assumed that the heavy quarkonium system, such as the charmonium system, is nonrelativistic. The fact that the charmonium system has a hydrogen atom like spectrum evinces that wave functions for the charmonium system could be obtained from the solutions of a Schrödinger type equation. Here, we will take the conventional notation to specify the ψ(n L) states, where n = 1, 2, 3, ⋯ is the radial quantum quantum number, and the orbital angular momentum L = 0, 1, 2 ⋯ correspond to S, P, D ⋯ waves, respectively. The wave functions for the 3S and 2D states corresponding to a spherically symmetric linear-harmonic-oscillator potential are [2]

$$ {\phi}_{3S}(\vec{k})\ {\propto}\ e^{-\frac{ \vec{k}^{2} }{ 2\,{\beta}^{2} } }\, \{ 15\,{\beta}^{4}-20\,\vec{k}^{2}\,{\beta}^{2}+4\,\vec{k}^{4} \} , $$

(22)

$$ {\phi}_{2D}(\vec{k})\ {\propto}\ e^{-\frac{ \vec{k}^{2} }{ 2\,{\beta}^{2} } }\, \vec{k}^{2}\, \{ 7\,{\beta}^{2} -2\,\vec{k}^{2} \} , $$

(23)

where the parameter β determines the average transverse momentum, i.e., ${\left \langle 1S{\vert }\vec {k}^{2}_{\perp }{\vert }1S\right \rangle }$ = β ². Based on the power counting rules of nonrelativistic QCD effective theory [48, 49, 50], the characteristic velocity v of a valence quark with a mass m in the heavy quarkonium is about v ∼ α _s(m), so the value of β ≃ m _c α _s(m _c) is taken for the ψ meson in our calculation, where α _s is the QCD coupling constant. To construct wave function model, we adopt the light-cone momentum k = $(x,\vec {k}_{\perp })$, and employ the substitution ansatz [51, 52],

$$ \vec{k}^{2}\ {\to}\ \frac{(x-\bar{x})^{2}\,{m_{c}^{2}}+\vec{k}_{\perp}^{2}}{4\,x\,\bar{x}} , $$

(24)

where $\bar {x}$ ≡ 1 −x; x and $\vec {k}_{\perp }$ are the longitudinal parton momentum fraction and transverse momentum of valence quarks, respectively. By integrating out $\vec {k}_{\perp }$ and combining with their asymptotic forms [42], one can obtain the distribution amplitudes of (12)–(15).

Appendix B: Building Blocks for ψ(3S,2D) →$D_{d}^{+}D_{s}^{-}$ Decays

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{a} &=& {\int}_{0}^{1}dx_{2} {\int}_{0}^{1}dx_{3} {\int}_{0}^{\infty}b_{2}db_{2} {\int}_{0}^{\infty}b_{3}db_{3}\, {\phi}_{D_{d}}(x_{2})\,{\phi}_{D_{s}}(x_{3})\, E_{f}(t_{a})\, a_{2}(t_{a}) \\ &&{\times}\,H_{f}({\alpha},{\beta}_{a},b_{2},b_{3})\, {\alpha}_{s}(t_{a})\, \left\{ 2\,m_{D_{d}}\,m_{D_{s}}\,x_{2} -m_{\psi}^{2}\,\bar{x}_{2}-m_{D_{s}}^{2}\,x_{2} \right\} , \end{array} $$

(25)

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{b} &=& {\int}_{0}^{1}dx_{2} {\int}_{0}^{1}dx_{3} {\int}_{0}^{\infty}b_{2}db_{2} {\int}_{0}^{\infty}b_{3}db_{3}\, {\phi}_{D_{d}}(x_{2})\,{\phi}_{D_{s}}(x_{3})\, E_{f}(t_{b})\, a_{2}(t_{b}) \\ &&{\times}\, H_{f}({\alpha},{\beta}_{b},b_{3},b_{2})\, {\alpha}_{s}(t_{b})\, \left\{ 2\,m_{D_{d}}\,m_{D_{s}}\,x_{3} -m_{\psi}^{2}\,\bar{x}_{3}-m_{D_{d}}^{2}\,x_{3} \right\} , \end{array} $$

(26)

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{c} &=& \frac{1}{N_{c}}\, {\int}_{0}^{1}dx_{1} {\int}_{0}^{1}dx_{2} {\int}_{0}^{1}dx_{3} {\int}_{0}^{\infty}b_{1}db_{1} {\int}_{0}^{\infty}b_{2}db_{2} {\int}_{0}^{\infty}db_{3}\, {\delta}(b_{2}-b_{3}) \\ &&{\times} {\phi}_{D_{d}}(x_{2})\, {\phi}_{D_{s}}(x_{3})\, E_{n}(t_{c})\, {\alpha}_{s}(t_{c})\, \left\{ {\phi}_{\psi}^{v}(x_{1})\, \left[ s\,\bar{x}_{3}+2\,m_{D_{d}}^{2}\,\bar{x}_{2}-t\,x_{1} \right.\right. \\ &&\left.\left.+\, m_{D_{d}}\,m_{D_{s}}\,(\bar{x}_{3}-\bar{x}_{2}) \right] +{\phi}_{\psi}^{t}(x_{1})\,m_{\psi}\,m_{c} \right\}\, C_{1}(t_{c})\, H_{n}({\alpha},{\beta}_{c},b_{1},b_{2}) , \end{array} $$

(27)

$$\begin{array}{@{}rcl@{}} \mathcal{A}_{d} &=& \frac{1}{N_{c}}\, {\int}_{0}^{1}dx_{1} {\int}_{0}^{1}dx_{2} {\int}_{0}^{1}dx_{3} {\int}_{0}^{\infty}b_{1}db_{1} {\int}_{0}^{\infty}b_{2}db_{2} {\int}_{0}^{\infty}db_{3}\, {\delta}(b_{2}-b_{3}) \\ &&{\times} {\phi}_{D_{d}}(x_{2})\, {\phi}_{D_{s}}(x_{3})\, E_{n}(t_{d})\, {\alpha}_{s}(t_{d})\, \left\{ {\phi}_{\psi}^{v}(x_{1})\, \left[ s\,\bar{x}_{2}+2\,m_{D_{s}}^{2}\,\bar{x}_{3}-u\,\bar{x}_{1} \right.\right. \\ &&\left.\left.+ m_{D_{d}}\,m_{D_{s}}\,(\bar{x}_{2}-\bar{x}_{3}) \right] +{\phi}_{\psi}^{t}(x_{1})\,m_{\psi}\,m_{c} \right\}\, C_{1}(t_{d})\, H_{n}({\alpha},{\beta}_{d},b_{1},b_{2}) , \end{array} $$

(28)

where the subscript i of $\mathcal {A}_{i}$, β _i and t _i corresponds to the indices of Fig. 2; x _i and b _i are the longitudinal parton momentum fraction and the conjugate variable of transverse momentum of the relatively light valence quarks; a ₂ = C ₂ + C ₁/N _c; C _1,2 are the Wilson coefficients.

The function H _f,n and Sudakov factor E _f,n are defined as

$$\begin{array}{@{}rcl@{}} H_{f}({\alpha},{\beta},b_{i},b_{j}) &=& \{ {\theta}(b_{i}-b_{j}) [ i\,J_{0}(b_{i}\sqrt{{\beta}}) - Y_{0}(b_{i}\sqrt{{\beta}}) ] J_{0}(b_{j}\sqrt{{\beta}}) \\ &&\left.+\, (b_{i} {\leftrightarrow} b_{j}) \right\}\, \left\{ i\,J_{0}(b_{j}\sqrt{{\alpha}}) - Y_{0}(b_{j}\sqrt{{\alpha}}) \right\}\, \frac{{\pi}^{2}}{4} , \end{array} $$

(29)

$$\begin{array}{@{}rcl@{}} H_{n}({\alpha},{\beta},b_{1},b_{2}) &=& \left\{ {\theta}(-{\beta}) K_{0}(b_{1}\sqrt{-{\beta}}) + \frac{\pi}{2}{\theta}({\beta}) [ i\,J_{0}(b_{1}\sqrt{{\beta}}) - Y_{0}(b_{1}\sqrt{{\beta}}) ] \right\} \\ & & {\times}\, \frac{\pi}{2} \left\{ {\theta}(b_{1}\,-\,b_{2}) \left[ iJ_{0}(b_{1}\sqrt{{\alpha}}) \,-\, Y_{0}(b_{1}\sqrt{{\alpha}}) \right] J_{0}(b_{2}\sqrt{{\alpha}}) \,+\, (b_{1} {\leftrightarrow} b_{2}) \right\} , \end{array} $$

(30)

$$ E_{f}(t)\ =\ {\exp}\{ -S_{D_{d}}(t)-S_{D_{s}}(t) \} , $$

(31)

$$ E_{n}(t)\ =\ {\exp}\{ -S_{\psi}(t)-S_{D_{d}}(t)-S_{D_{s}}(t) \} , $$

(32)

$$ S_{i}(t)\, =\, s\left( x_{i},b_{i},p_{i}^{+}\right) + s\left( \bar{x}_{i},b_{i},p_{i}^{+}\right) +2\,{\int}_{1/b_{i}}^{t}\frac{d{\mu}}{\mu}{\gamma}_{q} , $$

(33)

where I ₀, J ₀, K ₀ and Y ₀ are Bessel functions; γ _q = − α _s/π is the quark anomalous dimension; the expression of s(x,b,Q) can be found in Ref. [20]; α and β _i are virtualities of gluon and quarks,

$$ {\alpha}\ =\ \bar{x}_{2}^{2}\, m_{D_{d}}^{2}+\bar{x}_{3}^{2}\,m_{D_{s}}^{2}+\bar{x}_{2}\,\bar{x}_{3}\,s , $$

(34)

$$ {\beta}_{a}\ =\ m_{D_{s}}^{2}+\bar{x}_{2}^{2}\,m_{D_{d}}^{2}+\bar{x}_{2}\,s , $$

(35)

$$ {\beta}_{b}\ =\ m_{D_{d}}^{2}+\bar{x}_{3}^{2}\,m_{D_{s}}^{2}+\bar{x}_{3}\,s , $$

(36)

$$ {\beta}_{c}\ =\ {\alpha}+{x_{1}^{2}}\,m_{\psi}^{2} - x_{1}\,\bar{x}_{2}\,t - x_{1}\,\bar{x}_{3}\,u -{m_{c}^{2}} , $$

(37)

$$ {\beta}_{d}\ =\ {\alpha}+\bar{x}_{1}^{2}\,m_{\psi}^{2} - \bar{x}_{1}\,\bar{x}_{2}\,t - \bar{x}_{1}\,\bar{x}_{3}\,u -{m_{c}^{2}} , $$

(38)

$$ t_{a,b}\ =\ {\max}\{ \sqrt{{\alpha}},\sqrt{{\beta}_{a,b}},1/b_{2},1/b_{3} \} , $$

(39)

$$ t_{c,d}\ =\ {\max}\{ \sqrt{{\alpha}},\sqrt{{\vert}{\beta}_{c,d}{\vert}},1/b_{1},1/b_{2} \} , $$

(40)

$$ s\ =\ 2\,p_{D_{d}}{\cdot}p_{D_{s}}\, =\ m_{\psi}^{2}-m_{D_{d}}^{2}-m_{D_{s}}^{2} , $$

(41)

$$ t\ =\ 2\,p_{\psi}{\cdot}p_{D_{d}}\, =\ m_{\psi}^{2}+m_{D_{d}}^{2}-m_{D_{s}}^{2} , $$

(42)

$$ u\ =\ 2\,p_{\psi}{\cdot}p_{D_{s}}\, =\ m_{\psi}^{2}-m_{D_{d}}^{2}+m_{D_{s}}^{2} . $$

(43)

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Authors and Affiliations

Junfeng Sun
- 1
Yueling Yang
- 1
Email author
Na Wang
- 1
Qin Chang
- 1
Gongru Lu
- 1

1.Institute of Particle and Nuclear PhysicsHenan Normal UniversityXinxiangChina

ψ(4040) and ψ(4160) Decays into the \(D_{d}^{\pm }D_{s}^{\mp }\) Pair and the S-D Mixing Effects