Predictions of $\Upsilon(4S) \to h_b(1P,2P) \pi^+\pi^-$ transitions

TL;DR

This paper investigates the mechanisms behind certain bottomonium transitions, predicting branching ratios influenced by intermediate states and loops, with implications for understanding heavy quark spin flip processes.

Contribution

It introduces a detailed analysis of bottom meson loops and $Z_b$-exchange contributions in $S$ transitions, providing new predictions for branching ratios and the relative importance of these mechanisms.

Findings

Bottom meson loops dominate over $Z_b$-exchange in $S o h_b(1P)\pi\pi$ transitions.

Predicted branching ratios vary significantly depending on interference effects.

Bottom meson loops are insufficient to explain decay rates of $S$ to $S$ states.

Abstract

In this work, we study the contributions of the intermediate bottomoniumlike $Z_b$ states and the bottom meson loops in the heavy quark spin flip transitions $\Upsilon(4S) \to h_b(1P,2P) \pi^+\pi^-$. Depending on the constructive or destructive interferences between the $Z_b$-exchange and the bottom meson loops mechanisms, we predict two possible branching ratios for each process: BR$_{\Upsilon(4S) \to h_b(1P)\pi^+\pi^-}\simeq\big(1.2^{+0.8}_{-0.4}\times10^{-6}\big)$ or $\big( 0.5^{+0.5}_{-0.2}\times10^{-6}\big)$, and BR$_{\Upsilon(4S) \to h_b(2P)\pi^+\pi^-}\simeq \big(7.1^{+1.7}_{-1.1}\times10^{-10}\big)$ or $\big( 2.4^{+0.2}_{-0.1}\times10^{-10}\big)$. The bottom meson loops contribution is found to be much larger than the $Z_b$-exchange contribution in the $\Upsilon(4S) \to h_b(1P) \pi\pi$ transitions, while it can not produce decay rates comparable to the heavy quark spin conserved $\Upsilon(4S) \to \Upsilon(1S,2S) \pi\pi$ processes. We also predict the branch fractions of $\psi(3S,4S) \to h_c(1P)\pi^+\pi^-$ contributed from the charm meson loops.

Equations

$${\mathcal{L}}_{Z_b{\rm Y}\pi }=\sum _{j=1,2}{C}_{Z_{bj}{\rm Y}(mS)\pi }{{\rm Y}}^i(mS)⟨{Z_{bj}^i}^{†}{u}_{\mu }⟩v^{\mu }+h.c.,$$

$${\mathcal{L}}_{Z_b{h}_b\pi }=\sum _{j=1,2}{g}_{Z_{bj}{h}_b(nP)\pi }{ϵ}_{ijk}⟨{Z_{bj}^i}^{†}{u}^j⟩h_b^k+h.c.,$$

$$Z^{i}_{bj}=\left({\begin{array}[]{*{2}c}\frac{1}{\sqrt{2}}Z^{0i}_{bj}&Z^{+i}_{bj}\\ Z^{-i}_{bj}&-\frac{1}{\sqrt{2}}Z^{0i}_{bj}\end{array}}\right)\,.$$

$$u_{\mu }$$

$$\Phi$$

$${\mathcal{L}}_{{\rm Y}HH}=\frac{i{g}_{JHH}}{2}⟨J^{†}{H}_a\mathbf{\sigma }\cdot \overleftrightarrow{\partial }{\bar{H}}_a⟩+h.c.,$$

$${\mathcal{L}}_{h_{b}HH}=\frac{i{g}_1}{2}⟨h_b^{†i}{H}_a{\sigma }^i{\bar{H}}_a⟩+h.c.,$$

$${\mathcal{L}}_{HH\Phi }=\frac{g_{\pi }}{2}⟨{\bar{H}}_a^{†}\mathbf{\sigma }\cdot {\mathbf{u}}_{ab}{\bar{H}}_b⟩-\frac{g_{\pi }}{2}⟨H_a^{†}{H}_b\mathbf{\sigma }\cdot {\mathbf{u}}_{ba}⟩,$$

$$\frac{q_{\pi }^3}{{\nu }^3}:\frac{m_B{q}_{\pi }^2}{\nu }:\frac{q_{\pi }^3}{\nu }:\frac{q_{\pi }^3}{\nu }:m_B{q}_{\pi }^2\nu$$

$$=$$

$${\rm Y}(mS)(p_a)\to h_b(nP)(p_b)\pi (p_c)\pi (p_d)$$

$$s$$

$$\displaystyle M_{Zb}=\frac{2\sqrt{m_{\Upsilon(mS)}m_{h_{b}(nP)}}}{F_{\pi}^{2}}\epsilon_{abj}\epsilon_{\Upsilon(mS)}^{a}\epsilon_{h_{b}(nP)}^{b}\sum_{i=1,2}m_{Z_{bi}}C_{Z_{bi}\Upsilon(mS)\pi}g_{Z_{bi}h_{b}(nP)\pi}\Big{\{}p_{c}^{0}p_{d}^{j}\frac{1}{t-m_{Z_{bi}}^{2}}+p_{d}^{0}p_{c}^{j}\frac{1}{u-m_{Z_{bi}}^{2}}\Big{\}}\,.$$

$$\displaystyle M_{\text{loop}}=\epsilon_{\Upsilon(mS)}^{a}\epsilon_{h_{b}(nP)}^{b}\Big{\{}\epsilon_{abi}q^{i}A_{1}+\epsilon_{abi}q_{\perp}^{i}A_{2}+\epsilon_{bij}q^{i}q_{\perp}^{j}q_{\perp}^{a}A_{3}+\epsilon_{bij}q^{i}q_{\perp}^{j}q^{a}A_{4}+\epsilon_{aij}q^{i}q_{\perp}^{j}q_{\perp}^{b}A_{5}+\epsilon_{aij}q^{i}q_{\perp}^{j}q^{b}A_{6}\Big{\}}\,.$$

$$\Gamma ={\int }_{s_{-}}^{s_{+}}{\int }_{t_{-}}^{t_{+}}\frac{|M_{Zb}+M_{\text{loop}}{|}^{2}dsdt}{768{\pi }^3{m}_{{\rm Y}(mS)}^3},$$

$$s_{-}$$

$$s_{+}$$

$$t_{\pm }$$

$$\lambda (a,b,c)$$

$$|g_{Z_{b}h_{b}\pi}|=\Bigg{\{}\frac{6\pi F_{\pi}^{2}m_{Z_{b}}\Gamma_{Z_{b}\to h_{b}\pi}}{|\mathbf{p}_{f}|^{3}m_{h_{b}}}\Bigg{\}}^{\frac{1}{2}}\,,$$

$$|g_{Z_{b1}{h}_b(1P)\pi }^{\text{naive}}|$$

$$|g_{Z_{b2}{h}_b(1P)\pi }^{\text{naive}}|$$

$$|g_{Z_{b1}{h}_b(2P)\pi }^{\text{naive}}|$$

$$|g_{Z_{b2}{h}_b(2P)\pi }^{\text{naive}}|$$

$$|g_{Z_b{h}_b(1P)\pi }|$$

$$|g_{Z_b{h}_b(2P)\pi }|$$

$${\text{BR}}_{{\rm Y}(4S)\to h_b(1P){\pi }^{+}{\pi }^{-}}$$

$${\text{BR}}_{{\rm Y}(4S)\to h_b(2P){\pi }^{+}{\pi }^{-}}$$

$${\text{BR}}_{{\rm Y}(4S)\to h_b(1P){\pi }^{+}{\pi }^{-}}^{Z_b}$$

$${\text{BR}}_{{\rm Y}(4S)\to h_b(1P){\pi }^{+}{\pi }^{-}}^{\text{Box}}$$

$${\text{BR}}_{\psi (3S)\to h_c(1P){\pi }^{+}{\pi }^{-}}^{\text{Box}}$$

$${\text{BR}}_{\psi (4S)\to h_c(1P){\pi }^{+}{\pi }^{-}}^{\text{Box}}$$

$$J_1^{(0)}$$

$$\simeq \frac{-i}{16m_1{m}_2{m}_3{m}_4}\int \frac{d^{4}l}{(2\pi {)}^4}\frac{1}{\left[l^0-\frac{{\mathbf{l}}^2}{2m_1}-m_1+iϵ\right]\left[l^0-M+\frac{{\mathbf{l}}^2}{2m_2}+m_2-iϵ\right]}$$

$$\times \frac{1}{\left[l^0-q_1^0-q_2^0-\frac{(\mathbf{l}+\mathbf{q}{)}^2}{2m_3}-m_3+iϵ\right]\left[l^0-q_1^0-\frac{(\mathbf{l}-{\mathbf{q}}_1{)}^2}{2m_4}-m_4+iϵ\right]}.$$

$$-\frac{{\mu }_{12}{\mu }_{23}{\mu }_{24}}{2m_1{m}_2{m}_3{m}_4}\int \frac{d^{3}l}{(2\pi {)}^3}\frac{1}{[{\mathbf{l}}^2+c_{12}-iϵ][{\mathbf{l}}^2+2\frac{{\mu }_{23}}{m_3}\mathbf{l}\cdot \mathbf{q}+c_{23}-iϵ][{\mathbf{l}}^2-2\frac{{\mu }_{24}}{m_4}\mathbf{l}\cdot {\mathbf{q}}_1+c_{24}-iϵ]},$$

$$c_{12}$$

Full text

I Introduction

The hadronic transitions $\Upsilon(mS)\to\Upsilon(lS)\pi\pi$ and $\Upsilon(mS)\to h_{b}(nP)\pi\pi$ are important processes for understanding the heavy-quarkonium dynamics and low-energy QCD. Because the bottomonia are expected to be compact and nonrelativistic, the method of the QCD multipole expansion (QCDME) Voloshin1980 ; Novikov1981 ; Kuang1981 ; Kuang2006 is often used in analysis of these transitions, where the pions emitted come from the hadronization of soft gluons. The decay rates of $\Upsilon(2S,3S)\to\Upsilon(1S,2S)\pi\pi$ can be well described by the method of QCDME ref:Kuang . Since the total spin of $b\bar{b}$ system in $\Upsilon(mS)$ and $h_{b}(nP)$ are 1 and 0, respectively, and thus in general the heavy quark spin flip $\Upsilon(mS)\to h_{b}(nP)\pi\pi$ processes are expected to be suppressed compared with the heavy quark spin conserved $\Upsilon(mS)\to\Upsilon(nS)\pi\pi$ processes. Within the framework of QCDME, references ref:standardmultipoleTuan ; ref:Kuang ; ref:kuangyan predict that the branching fraction of $\Upsilon(3S)\to\Upsilon(1P)\pi\pi$ is suppressed by 2 orders of magnitude relative to that of $\Upsilon(3S)\to\Upsilon(1S)\pi\pi$, while Ref. ref:Voloshin predicts an at least 3 orders of magnitude suppression. The prediction of Ref. ref:Voloshin is supported by the experimental data Lees:2011bv . While in the decay processes $\Upsilon(5S)\rightarrow\Upsilon(lS)\pi^{+}\pi^{-}$ ($l=1,2,3$) and $\Upsilon(5S)\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$ ($n=1,2$) where the two charged bottomoniumlike resonances $Z_{b}(10610)^{\pm}$ and $Z_{b}(10650)^{\pm}$ were observed, the $\Upsilon(5S)\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$ proceed at a rate comparable to the $\Upsilon(5S)\rightarrow\Upsilon(lS)\pi^{+}\pi^{-}$ processes Belle2011:1 ; Belle2012:1 . The mechanism that mitigates the expected suppression has remained controversial. In Refs. Chen:2011pv ; Swanson:2014tra , the $\Upsilon(5S)\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$ processes are interpreted via bottom meson loops mechanism, while genuine $S$-matrix $Z_{b}$ poles are required as in Refs. Guo:2014iya ; Huo:2015uka ; Guo:2016bjq ; Wang:2018jlv . Note that the meson loops mechanism has been explored by many previous works Chen:2011qx ; Chen:2011zv ; Meng:2007tk ; Meng:2008bq ; Wang:2016qmz ; Simonov:2008qy ; Wang:2015xsa to study the dipion and $\eta$ transitions of higher charmonia and bottomonia since the branch ratios and the dipion invariant mass spectra cannot be described by using QCDME.

In this work, we will study that whether the bottom meson loops mechanism can produce the $\Upsilon(4S)\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$ transitions at the decay ratios comparable with $\Upsilon(4S)\rightarrow\Upsilon(lS)\pi^{+}\pi^{-}$. Since in the dipion emission processes of the $\Upsilon(4S)$ the crossed-channel exchanged $Z_{b}$ can not be on-shell, one may expect that these transitons are good channels to study the bottom meson loops’ effect. In our previous works Chen:2016mjn ; Chen:2019gty , using the nonrelativistic effective field theory (NREFT) we calculated the effects of the bottom meson loops as well as the $Z_{b}$-exchange in the $\Upsilon(4S)\to\Upsilon(1S,2S)\pi\pi$ processes, and found that the experimental data can be described well. Here within the same theoretical scheme, we will calculate the contributions of the bottom meson loops and the $Z_{b}$-exchange in the $\Upsilon(4S)\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$ processes, and give the theoretical predictions of the decay branching ratios. We find that the contribution of the bottom meson loops is much larger than that of the $Z_{b}$-exchange in the $\Upsilon(4S)\rightarrow h_{b}(1P)\pi^{+}\pi^{-}$ process, while it can not produce a rate comparable with $\Upsilon(4S)\rightarrow\Upsilon(1S,2S)\pi^{+}\pi^{-}$.

This paper is organized as follows. In Sec. II, the theoretical framework is described in detail. In Sec. III, we give the theoretical predictions for the decay branching fractions of $\Upsilon(4S)\rightarrow h_{b}(1P,2P)\pi^{+}\pi^{-}$, and discuss the contributions of different mechanisms. A summary will be given in Sec. IV.

II Theoretical framework

II.1 Lagrangians

To calculate the contribution of the mechanism $\Upsilon(mS)\to Z_{b}\pi\to h_{b}(nP)\pi\pi$, we need the effective Lagrangians for the $Z_{b}\Upsilon\pi$ interaction and $Z_{b}h_{b}\pi$ interaction Guo2011 ,

[TABLE]

[TABLE]

where $Z_{b1}$ and $Z_{b2}$ denote $Z_{b}(10610)$ and $Z_{b}(10650)$, respectively, and $v^{\mu}=(1,\mathbf{0})$ is the velocity of the heavy quark. The $Z_{b}$ states are collected in the matrix as

[TABLE]

The pions as Goldstone bosons of the spontaneous breaking of the chiral symmetry can be parametrized as

[TABLE]

where $F_{\pi}=92.2\,\text{MeV}$ is the pion decay constant.

To calculate the box diagrams, we need the Lagrangian for the coupling of the $\Upsilon$ to the bottom mesons and the coupling of the $h_{b}$ to the bottom mesons Guo2009:PRL ; Guo2011 ,

[TABLE]

[TABLE]

where $J\equiv\mathbf{\Upsilon}\cdot\bm{\sigma}+\eta_{b}$ denotes the heavy quarkonia spin multiplet, $H_{a}=\mathbf{V}_{a}\cdot\bm{\sigma}+P_{a}$ with $P_{a}(V_{a})=(B^{(*)-},\bar{B}^{(*)0})$ collects the bottom mesons, and $A\overleftrightarrow{\partial}\!B\equiv A(\overrightarrow{\partial}B)-(\overrightarrow{\partial}A)B$. We also need the Lagrangian for the axial coupling of the pion fields to the bottom and antibottom mesons, which at the lowest order in heavy-flavor chiral perturbation theory is given by Burdman:1992gh ; Wise:1992hn ; Yan:1992gz ; Casalbuoni:1996pg ; Mehen2008

[TABLE]

where $u^{i}=-\sqrt{2}\partial^{i}\Phi/F+\mathcal{O}(\Phi^{3})$ denotes the three-vector components of $u_{\mu}$ as defined in Eq. (4). Here we will use $g_{\pi}=0.492\pm 0.029$ from a recent lattice QCD calculation Bernardoni:2014kla .

II.2 Power counting of the loops

Since the $\Upsilon(4S)$ meson is above the $B\bar{B}$ threshold and decays predominantly into $B\bar{B}$ pairs, the loop mechanism with intermediate bottom mesons may be important in the transitions $\Upsilon(4S)\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$. Following the formalism set-up based on NREFT Guo2009:PRL ; Guo2011:effect ; Martin2013 , we will analyze the power counting of different kinds of loops. In NREFT, the expansion parameter is the velocity of the intermediate heavy meson, namely $\nu_{X}=\sqrt{|m_{X}-m_{B^{(*)}}-m_{B^{(*)}}|/m_{B^{(*)}}}$, which is small since the bottomonia $X$ are close to the $B^{(*)}\bar{B}^{(*)}$ thresholds. In this power counting, each nonrelativistic propagator scale as $1/\nu^{2}$, and the measure of one-loop integration scales as $\int d^{4}l\sim\nu^{5}$.

There are five different kinds of loop contributions, namely the box diagrams displayed in Fig. 1 (b), (c), the triangle diagrams displayed in Fig. 2 (a)–(c) and the bubble loop in Fig. 2 (d). We analyze them one by one as follows:

First we analyze the power counting of the box diagrams, namely Fig. 1 (b), (c). As indicated in Eq. (7), the vertex of $B^{(*)}B^{(*)}\pi$ is proportional to the external momentum of the pion $q_{\pi}$. The $\Upsilon B^{(*)}\bar{B}^{(*)}$ vertex is in a $P$-wave, and the $h_{b}B^{(*)}\bar{B}^{(*)}$ vertex is in an $S$-wave, so the loop momentum must contract with the external pion momentum and hence the $P$-wave vertex scales as $\mathcal{O}(q_{\pi})$. Thus the box diagrams scales as $\nu^{5}q_{\pi}^{3}/\nu^{8}=q_{\pi}^{3}/\nu^{3}.$

As for the triangle diagram Fig. 2 (a), the leading $\Upsilon B^{(*)}\bar{B}^{(*)}\pi$ vertex given by $g_{JHH\pi}\langle J\bar{H}_{a}^{\dagger}H_{b}^{\dagger}\rangle u_{ab}^{0}$ Mehen2013 is proportional to the energy of the pion, $E_{\pi}\sim q_{\pi}$. Therefore, Fig. 2 (a) is counted as $m_{B}\nu^{5}q_{\pi}^{2}/\nu^{6}=m_{B}q_{\pi}^{2}/\nu$, where the factor $m_{B}$ has been introduced to match the dimension with the scaling for the box diagrams.

In Fig. 2 (b), the leading $h_{b}B^{(*)}\bar{B}^{(*)}\pi$ vertex given by $g_{h_{b}HH\pi}\langle h_{b}^{{\dagger}i}H_{a}\sigma^{j}\bar{H}_{b}\rangle\epsilon_{ijk}u_{ab}^{k}$ Mehen2008 is proportional to the momentum the pion $q_{\pi}$. The loop momentum due to the $\Upsilon B^{(*)}\bar{B}^{(*)}$ coupling has to contract with the external pion momentum. Thus, Fig. 2 (b) scales as $\nu^{5}q_{\pi}^{3}/\nu^{6}=q_{\pi}^{3}/\nu$.

The leading $B^{(*)}B^{(*)}\pi\pi$ vertex comes from the chiral derivative term $\langle H_{a}^{\dagger}(iD_{0})_{ba}H_{b}\rangle=\langle H_{a}^{\dagger}(i\partial_{0}-iV_{0})_{ba}H_{b}\rangle$ Stewart ; Mehen2006 , in which the pion pair produced by the vector current, $V^{\mu}=\frac{1}{2}(u^{\dagger}\partial^{\mu}u+u\partial^{\mu}u^{\dagger})$, cannot form a positive-parity and $C$-parity state, therefore this leading vertex does not contribute to the $\Upsilon(mS)\rightarrow h_{b}(nP)\pi\pi$ processes. Isoscalar, $PC=++$ pion pairs only enter in the next order $\mathcal{O}(q_{\pi}^{2})$ from point vertices. Therefore, Fig. 2 (c) scales as $\nu^{5}q_{\pi}^{3}/\nu^{6}=q_{\pi}^{3}/\nu$.

In Fig. 2 (d), both the initial vertex and the final vertex are proportional to $q_{\pi}$, so the bubble loop scales as $m_{B}\nu^{5}q_{\pi}^{2}/\nu^{4}=m_{B}q_{\pi}^{2}\nu$.

Therefore, we expect that the ratios of the contributions of the box diagrams, triangle diagram Fig. 2 (a)–(c), and the bubble loop Fig. 2 (d) are

[TABLE]

where $q_{\pi}\simeq(m_{\Upsilon(4S)}-m_{h_{b}(nP)})/2$ and $\nu=(\nu_{\Upsilon(4S)}+\nu_{h_{b}(nP)})/2$, with $\nu_{\Upsilon(4S)}\simeq 0.06$, $\nu_{h_{b}(1P)}\simeq 0.35$, and $\nu_{h_{b}(2P)}\simeq 0.24$. Thus for the $\Upsilon(4S)\to h_{b}(1P)\pi^{+}\pi^{-}$ transition, the ratios in Eq. (II.2) are $1:0.67:0.04:0.04:0.03$. For the $\Upsilon(4S)\to h_{b}(2P)\pi^{+}\pi^{-}$ transition, the ratios are $1:0.75:0.02:0.02:0.02$. Therefore according to the power counting the box diagrams and the triangle diagram in Fig. 2 (a) are dominant among the loop contributions, and they are of the same order. While the $\Upsilon(4S)$ is below the $B^{(*)}\bar{B}^{(*)}\pi$ threshold and the coupling $g_{JHH\pi}$ in the triangle diagram Fig. 2 (a) is unknown. Thus for a rough estimation of the loop contributions, we will only calculate the box diagrams in the present study. Note that all the box and triangle loop contributions discussed here are ultraviolet-finite, and do not require the additional introduction of counterterms.

II.3 Tree-level amplitudes and box diagram calculation

The decay amplitude for

[TABLE]

is described in terms of the Mandelstam variables

[TABLE]

Using the effective Lagrangians in Eqs. (1) and (2), the tree amplitude of $\Upsilon(mS)\to Z_{b}\pi\to h_{b}(nP)\pi\pi$ can be obtained

[TABLE]

Notice that the nonrelativistic normalization factor $\sqrt{m_{Y}}$ has been multiplied to the amplitude for every heavy particle, with $Y=\Upsilon(mS),h_{b}(nP),Z_{bi}$. The widths of the $Z_{b}$ states are neglected in the present study, since they are of the order of $10\,\text{MeV}$ and are much smaller than the difference between the $Z_{b}$ masses and the $\Upsilon(mS)\pi/h_{b}(nP)\pi$ threshold.

Now we discuss the calculation of the box diagrams. In the box diagrams Fig. 1 (b) and (c), we denote the top left intermediate bottom meson as $M1$, and the other intermediate bottom mesons as $M2$, $M3$, and $M4$, in counterclockwise order. Concerned with the pseudoscalar or vector content of $[M1,M2,M3,M4]$, there are twelve possible patterns and we number them in order: 1, $[PPPV]$; 2, $[PPVV]$; 3, $[PVPV]$; 4, $[PVVP]$; 5, $[VVPP]$; 6, $[VPVP]$; 7, $[VPPV]$; 8, $[PVVV]$; 9, $[VPVV]$; 10, $[VVPV]$; 11, $[VVVP]$; 12, $[VVVV]$. For each pattern, we also need to consider six possibilities of different flavor of the intermediate bottom mesons: $[B^{(*)+},B^{(*)-},B^{(*)+},B^{(*)0}]$, $[B^{(*)+},B^{(*)-},{\bar{B}}^{(*)0},B^{(*)0}]$, $[B^{(*)0},{\bar{B}}^{(*)0},B^{(*)0},B^{(*)+}]$, $[B^{(*)-},B^{(*)+},B^{(*)-},{\bar{B}}^{(*)0}]$, $[{\bar{B}}^{(*)0},B^{(*)0},B^{(*)+},B^{(*)-}]$, and $[{\bar{B}}^{(*)0},B^{(*)0},{\bar{B}}^{(*)0},B^{(*)-}]$. The full amplitude contains the sum of all possible ones.

For the tensor reduction of the loop integrals it is convenient to define $\mathbf{q}=-\mathbf{p}_{b}$ and the perpendicular momentum $\mathbf{q}_{\perp}=\mathbf{p}_{c}-\mathbf{q}(\mathbf{q}\cdot\mathbf{p}_{c})/\mathbf{q}^{2}$, which satisfy $\mathbf{q}\cdot\mathbf{q}_{\perp}=0$. The result of the amplitude of the box diagrams can be written as

[TABLE]

Details on the analytic calculation of the box diagrams and the explicit expressions of $A_{i}$ $(i=1,2,...,6)$ are given in Appendix A.

The decay width for $\Upsilon(mS)\to h_{b}(nP)\pi\pi$ is given by

[TABLE]

where the lower and upper limits are given as

[TABLE]

III Phenomenological discussion

To estimate the contribution of the $Z_{b}$-exchange mechanism we need to know the coupling strengths of $Z_{b}\Upsilon(4S)\pi$ and $Z_{b}h_{b}(nP)\pi$. The mass difference between $Z_{b}(10610)$ and $Z_{b}(10650)$ is much smaller than the difference between their masses and the $\Upsilon(mS)\pi/h_{b}(nP)\pi$ threshold, and they have the same quantum numbers and thus the same coupling structures as dictated by Eqs. (1) and (2). Therefore it is very difficult to distinguish their effects from each other in the dipion transitions of $\Upsilon(4S)$, so we only use one $Z_{b}$, the $Z_{b}(10610)$, which approximately combine both $Z_{b}$ states’ effects. In Ref. Chen:2019gty , we has studied the $\Upsilon(4S)\to\Upsilon(mS)\pi\pi$ processes to extract the coupling constant $|C_{Z_{b}\Upsilon(4S)\pi}|=(3.3\pm 0.1)\times 10^{-3}$, which containing effects from both $Z_{b}$ states. For the couplings of $Z_{b}h_{b}(nP)\pi$, in principle they can be extracted from the partial widths of the $Z_{b}$ states decay into $h_{b}(nP)\pi\,(n=1,2)$

[TABLE]

where |\mathbf{p}_{f}|\equiv\lambda^{1/2}\big{(}m_{Z_{b}}^{2},m_{h_{b}}^{2},m_{\pi}^{2}\big{)}/(2m_{Z_{b}}). The branching fractions of the decays of both $Z_{b}$ states into $h_{b}(nP)\pi\,(n=1,2)$ have been given in Garmash:2015rfd , where the $Z_{b}$ line shapes were described using Breit-Wigner forms. If we naively use these branching fractions, we would obtain

[TABLE]

Here all the $Z_{b}h_{b}\pi$ couplings are labeled by a superscript “naive” since this is not the appropriate way to extract the coupling strengths in this case; the $Z_{b}$ states are very close to the $B^{(*)}\bar{B}^{*}$ thresholds, and thus the Flatté parametrization for the $Z_{b}$ spectral functions should be used, which will lead to much larger partial widths into $(b\bar{b}\pi)$ channels, and thus the relevant coupling strengths. As analyzed in Ref. Chen2016 , in the the Flatté parametrization the sum of the partial widths of the $Z_{b}(10610)$ other than that for the $B\bar{B}^{*}$ channel should be larger than the nominal width, which is about $20\,\text{MeV}$. While summing over all the $\Upsilon(nS)\pi\,(n=1,2,3)$ and $h_{b}(mP)\pi\,(m=1,2)$ branching fractions in Ref. Garmash:2015rfd gives about 14% or $3\,\text{MeV}$ in terms of partial widths. Therefor for a rough estimation we will use three times the results from Eq. (III), namely

[TABLE]

We will find that even considering this enlarging factor of three for the couplings $|g_{Z_{b}h_{b}(nP)\pi}|$, the $Z_{b}$-exchange contribution is still much smaller than the bottom meson loops contributions.

In the calculation of the box diagrams, the coupling strength $g_{JHH(4S)}$ can be extracted from the measured open-bottom decay widths of the $\Upsilon(4S)$, and we have $g_{JHH(4S)}=1.43\pm 0.01\,\text{GeV}^{-3/2}$. For the $h_{b}B^{*}\bar{B}^{*}$ coupling $g_{1}$, we can use the results from Ref. Guo2011 . In Guo2011 , the $Z_{b}$-exchange mechanism in the $\Upsilon(5S)\to h_{b}(1P,2P)\pi\pi$ processes has been studied assuming the $Z_{b}$ states are $B^{(\ast)}\bar{B}^{\ast}$ bound states, and the physical coupling of the $Z_{b}$ states to the bottom and anti-bottom mesons, $z_{1}$, as well as the product $g_{1}z_{1}$ have been determined.111In Guo2011 , in order to reduce the number of free parameters, the couplings of $h_{b}(1P)B^{*}\bar{B}^{*}$ and $h_{b}(2P)B^{*}\bar{B}^{*}$ are assumed to be the same. Using their results $z_{1}=0.75^{+0.08}_{-0.11}\,\text{GeV}^{-1/2}$ and $g_{1}z_{1}=0.40\pm 0.06\,\text{GeV}^{-1}$, we can extract that $g_{1}=0.53^{+0.19}_{-0.13}\,\text{GeV}^{-1/2}$.

Using the coupling strengths above, we can predict the decay branching fractions of $\Upsilon(4S)\to h_{b}(1P,2P)\pi^{+}\pi^{-}$. Depending on the sign of the couplings in Eq. (III), the interferences can be constructive or destructive between the $Z_{b}$-exchange and box graph mechanisms, so there are two possible results for each process

[TABLE]

We find that the $\text{BR}_{\Upsilon(4S)\to h_{b}(1P)\pi^{+}\pi^{-}}$ is at least one order of magnitude smaller than the branching fractions $\text{BR}_{\Upsilon(4S)\to\Upsilon(1S,2S)\pi^{+}\pi^{-}}$, which are about $8\times 10^{-5}$ given in PDG PDG , and the $\text{BR}_{\Upsilon(4S)\to h_{b}(2P)\pi^{+}\pi^{-}}$ is tiny due to the very small phase space. We discuss the $\Upsilon(4S)\to h_{b}(1P)\pi^{+}\pi^{-}$ transition in more details. To illustrate the effects of the $Z_{b}$-exchange and box graph mechanisms in $\Upsilon(4S)\to h_{b}(1P)\pi\pi$, we give the predictions only including the $Z_{b}$-exchange terms or only including the box diagrams

[TABLE]

One observes that the bottom meson loops contribution is much larger than the $Z_{b}$-exchange contribution, while it is 2 orders of magnitude smaller than the $\Upsilon(4S)\to\Upsilon(1S,2S)\pi^{+}\pi^{-}$ transitions. Note that there is no calculation of the direct gluon hadronization mechanism contribution within QCDME for the $\Upsilon(4S)\to h_{b}(1P)\pi^{+}\pi^{-}$ process in the literature. In references ref:standardmultipoleTuan ; ref:Kuang ; ref:kuangyan ; ref:Voloshin , QCDME predict that the branching fraction of $\Upsilon(3S)\to h_{b}(1P)\pi\pi$ is 2-3 orders of magnitude suppressed relative to that of $\Upsilon(3S)\to\Upsilon(1S)\pi\pi$, and the 3 orders of magnitude suppression is supported by the experiment Lees:2011bv . Since the mass difference between the $\Upsilon(4S)$ and $h_{b}(1P)$ is about 0.68 GeV, the pions in the $\Upsilon(4S)\to h_{b}(1P)\pi^{+}\pi^{-}$ process also can be considered to be in the soft region. If one approximates that the gluon hadronization mechanism within QCDME in $\Upsilon(4S)\to h_{b}(1P)\pi^{+}\pi^{-}$ is also 2-3 orders of magnitudes suppressed relative to that in the $\Upsilon(4S)\to\Upsilon(1S)\pi^{+}\pi^{-}$ process, as in the $\Upsilon(3S)$ decay cases, then one obtains that the gluon hadronization mechanism contribution is at most at the same order of the bottom meson loops contribution. Due to the lack of exact information concerning the gluon hadronization within QCDME and the neglecting of triangle diagram Fig. 2 (a) as discussed in section II.2, it is important to keep in mind that the results presented in this paper are order-of-magnitude estimates.

In Fig. 3, we plot the distributions of the $\pi\pi$ and $h_{b}\pi$ invariant mass spectra, and the distribution of cos$\theta$, where $\theta$ is defined as the angle between the initial $\Upsilon(mS)$ and the $\pi^{+}$ in the rest frame of the $\pi\pi$ system. To illustrate the effects of different mechanisms, the contributions of the box diagrams, $Z_{b}$-exchange, the sum of them with constructive interference, and the sum with destructive interference are shown as the darker green dashed, magenta dotted, red solid, and blue dot-dashed lines, respectively. One observes that there is a broad bump around 0.5 GeV in the dipion invariant mass distribution. While the $\pi\pi$ invariant mass spectra with unknown normalization predicted within QCDME in Ref. ref:Kuang display a peak at low $\pi\pi$ masses. Thus the $\pi\pi$ invariant mass spectra can be useful to identify the effects of bottom meson loops and the gluon hadronization mechanism with future experimental data. Also one observes that the angular distribution is far from flat. In the $\Upsilon(mS)\to h_{b}(nP)\pi\pi$ process, isospin conservation combined with Bose symmetry requires the pions to have even relative angular momentum. Therefore it means that there is a large $D$-wave component from the box diagrams, if neglecting yet higher partial waves.

The $\Upsilon(mS)\to h_{b}(nP)\pi\pi$ are heavy quark spin flip processes and they are forbidden in the heavy quark limit. We have checked that in the heavy quark limit, i.e. $m_{B}=m_{B^{*}}$, all the box diagrams are cancelled with each other so the bottomed loops contribute nothing to the $\Upsilon(mS)\to h_{b}(nP)\pi\pi$ transitions. With the small mass splitting of $B$ and $B^{*}$ in the real world, in Eqs. (III) and (III) one observes that the bottomed meson loops contribution does not produce $\Upsilon(4S)\to h_{b}(1P)\pi\pi$ at a rate comparable to the heavy quark spin conserved $\Upsilon(4S)\to\Upsilon(1S,2S)\pi\pi$ transitions. Note that the datasets collected at $\Upsilon(4S)$ by BABAR and Belle II collaborations are $471\times 10^{6}$ and $772\times 10^{6}$  Kou:2018nap , respectively, and thus they should contain several hundreds of $\Upsilon(4S)\to h_{b}(1P)\pi\pi$ events according to our calculation. We hope future experimental analysis by BABAR and Belle can test our predictions. On the other hand, as stated in the introduction, the observed $\Upsilon(5S)\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$ proceed at a rate comparable to the $\Upsilon(5S)\rightarrow\Upsilon(lS)\pi^{+}\pi^{-}$ processes Belle2011:1 ; Belle2012:1 . The enhancements may come from the effects of the on-shell $Z_{b}$ exchange and the two-cut condition complexity of the bottom meson loops in the $\Upsilon(5S)$ decays. A detailed analysis of the $\Upsilon(5S)\rightarrow h_{b}(nP)\pi^{+}\pi^{-}$ processes is beyond the scope of this paper.

Due to the similarity between the bottomonium and charmonium families, we can extend the box diagrams calculation to give a rough estimation of the branch fractions of the $\psi(3S)(\psi(4040))\to h_{c}(1P)\pi^{+}\pi^{-}$ and $\psi(4S)(\psi(4415))\to h_{c}(1P)\pi^{+}\pi^{-}$ transitions. The relevant Feynman diagrams can be obtained by replacing the external $\Upsilon(mS)$ and $h_{b}(nP)$ by $\psi(mS)$ and $h_{c}(nP)$, respectively, and replacing the intermediate $B^{(*)}$ by $D^{(*)}$ in Fig. 1 (b) and (c). Note that the experimental decay widths of the $\psi(3S,4S)\to D^{(*)}\bar{D}^{(*)}$ transitions have not been given in PDG, and we will use the theoretical predictions of the decay widths in Ref. Barnes:2005pb to estimate the coupling strengths $g_{JHH(\psi(3S))}$ and $g_{JHH(\psi(4S))}$. Since among the different decay modes $D^{(*)}\bar{D}^{(*)}$, the $D\bar{D}^{*}$ and $D^{*}\bar{D}^{*}$ modes are dominant for $\psi(3S)$ and $\psi(4S)$, respectively, we will use corresponding coupling constants in the calculation, namely $g_{JHH(\psi(3S))}=g_{\psi(3S)D\bar{D}^{*}}=0.97\,\text{GeV}^{-3/2}$ and $g_{JHH(\psi(4S))}=g_{\psi(4S)D^{*}\bar{D}^{*}}=0.25\,\text{GeV}^{-3/2}$. For the $h_{c}D^{*}\bar{D}^{*}$ coupling, we use the result from Ref. Chen:2011xk , $g_{h_{c}D^{*}\bar{D}^{*}}=-(\sqrt{m_{\chi_{c0}}/3})/(f_{\chi_{c0}})=-(\sqrt{3.415/3})/(0.297)\,\text{GeV}^{-1/2}=-3.59\,\text{GeV}^{-1/2}$. The predictions of the box diagrams contributions to the branch fractions of $\psi(3S,4S)\to h_{c}(1P)\pi^{+}\pi^{-}$ are

[TABLE]

The prediction of $\text{BR}_{\psi(3S)\to h_{c}(1P)\pi^{+}\pi^{-}}^{\text{Box}}$ is below the upper limit given in PDG PDG . As expected the branch fractions $\text{BR}_{\psi(3S,4S)\to h_{c}(1P)\pi^{+}\pi^{-}}^{\text{Box}}$ are much larger than $\text{BR}_{\Upsilon(4S)\to h_{b}(1P)\pi^{+}\pi^{-}}^{\text{Box}}$, since the mass splitting of $D$ and $D^{*}$ is much larger than that of $B$ and $B^{*}$. It should be kept in mind that this is just a preliminary rough estimation, due to the lack of sufficient information concerning the $\psi(3S,4S)D^{(*)}\bar{D}^{(*)}$ coupling constants and the neglecting of the loop diagrams with intermediate $D_{1}$ state in the present calculation. A detailed theoretical study of the $\psi(3S,4S)\to h_{c}(1P)\pi^{+}\pi^{-}$ transitions will be pursued in the future.

IV Conclusions

In this paper, we study the effects of $Z_{b}$ exchange and bottom meson loops in the heavy quark spin flip transitions $\Upsilon(4S)\to h_{b}(nP)\pi\pi(n=1,2)$. The bottom meson loops are treated in the NREFT scheme, in which the dominant box diagrams are taken into account. We find that the bottom meson loops contribution is much larger than the $Z_{b}$-exchange contribution in the $\Upsilon(4S)\to h_{b}(1P)\pi\pi$ transition, while it can not produce decay rates comparable to the heavy quark spin conserved $\Upsilon(4S)\to\Upsilon(1S,2S)\pi\pi$ processes. The theoretical prediction of the decay rate and the dipion invariant mass spectra of $\Upsilon(4S)\to h_{b}(1P)\pi\pi$ in this work may be useful for identification of the effect of the bottom meson loops with future experimental analysis. We also predict the branch fractions of $\psi(3S,4S)\to h_{c}(1P)\pi\pi$ contributed from the charm meson loops.

Acknowledgments

We are grateful to referees’ useful suggestions and constructive remarks which are helpful in formulating the present version of this paper. We are grateful to Martin Cleven for the collaboration at the early stages of this study. We acknowledge Guo-Ying Chen, Meng-Lin Du, and Qian Wang for helpful discussions, and Feng-Kun Guo for a careful reading of the manuscript and valuable comments. This research is supported in part by the Fundamental Research Funds for the Central Universities under Grant No. FRF-BR-19-001A, and by the National Natural Science Foundation of China under Grants No. 11975028.

Appendix A Remarks on the box diagrams and four-point integrals

In this appendix, first we will discuss the parametrization and simplification of the scalar four-point integrals in the box diagrams. Then we introduce a tensor reduction scheme to deal with higher-rank loop integrals. Finally, we give the amplitude of the box diagrams for the $\Upsilon(mS)\to h_{b}(nP)\pi\pi$ process.

A.1 Scalar four-point integrals

For the first topology as shown in Fig. 4, the scalar integral evaluated for the initial bottomonium at rest ($p=(M,\mathbf{0})$) reads

[TABLE]

Performing the contour integration, we find

[TABLE]

where we defined

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The second topology in Fig. 4 is just the crossed diagram of the first topology with $q_{1}\leftrightarrow q_{2}$, so the scalar integral reads

[TABLE]

where

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For the third topology we have

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Performing the contour integration, we find

[TABLE]

where we defined

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In all the three cases the remaining three-dimensional momentum integration will be carried out numerically.

A.2 Tensor reduction

Since the $\Upsilon B^{(*)}\bar{B}^{(*)}$ vertex scales with the momentum of the bottom meson pair, for topology I we have to deal with

[TABLE]

where $f(l)=\{1,\,l^{i}\}$ for the fundamental scalar and vector integrals, respectively. A convenient parametrization of the tensor reduction reads

[TABLE]

where $\mathbf{q}_{1\perp}=\mathbf{q}_{1}-\mathbf{q}(\mathbf{q}\cdot\mathbf{q}_{1})/\mathbf{q}^{2}$. The expressions of the scalar integrals $J^{(r)}_{1}$ can easily be disentangled and have to be evaluated numerically. The corresponding expressions for topology II and III can be obtained by changing the denominators accordingly.

A.3 Amplitudes

We define the scalar integrals $J1(i,r,k)$ based on the $J_{1}^{(r)}$ in the tensor reduction of vector integral in Eq. (A.2), where $i=1,2,3$ denotes the three topologies of box diagrams as shown in Fig. 4, $r=1,2$ refers to the two components $J_{1}^{(r)}$, and $k=1,2,...,12$ represents the twelve patterns with different pseudoscalar or vector content of the intermediate bottom mesons in $[M1,M2,M3,M4]$ as displayed in Sec. II.3.

We give the amplitude of the box diagrams for the $\Upsilon(mS)\to h_{b}(nP)\pi\pi$ process, namely the $A_{l}(l=1,2,...,6)$ in the Eq. (12).

[TABLE]

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