Predictions of Υ ( 4 S ) → h b ( 1 P , 2 P ) π + π − transitions

In this work, we study the contributions of the intermediate bottomoniumlike Zb states and the bottom meson loops in the heavy quark spin flip transitions Υ(4S)→ hb(1P, 2P )π+π−. Depending on the constructive or destructive interferences between the Zb-exchange and the bottom meson loops mechanisms, we predict two possible branching ratios for each process: BRΥ(4S)→hb(1P )π+π− ' ( 1.3 −0.4 × 10−6 ) or ( 0.5 −0.2 × 10−6 ) , and BRΥ(4S)→hb(2P )π+π− ' ( 9.2 −1.2 × 10−10 ) or ( 4.4 −0.3 × 10−10 ) . The bottom meson loops are found to play the leading role in the Υ(4S) → hb(nP )ππ transitions, while they can not produce decay rates comparable to the heavy quark spin conserved Υ(4S)→ Υ(1S, 2S)ππ processes.

In this work, we will study that whether the bottom meson loops mechanism can produce the Υ(4S) → h b (nP )π + π − transitions at the decay ratios comparable with Υ(4S) → Υ(lS)π + π − . Since in the dipion emission processes of the Υ(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 [9,10], 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 Υ(4S) → Υ(1S, 2S)ππ 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 Υ(4S) → h b (nP )π + π − processes, and give the theoretical predictions of the decay branching ratios. We find that the effect of the bottom meson loops is dominant in the Υ(4S) → h b (nP )π + π − process, while it can not produce a rate comparable with Υ(4S) → Υ(1S, 2S)π + π − . 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 Υ(4S) → h b (1P, 2P )π + π − , and discuss the contributions of different mechanisms. A summary will be given in Sec. IV.

A. Lagrangians
To calculate the contribution of the mechanism Υ(mS) → Z b π → h b (nP )ππ, we need the effective Lagrangians for the Z b Υπ interaction and Z b h b π interaction [11], where Z b1 and Z b2 denote Z b (10610) and Z b (10650), respectively, and v µ = (1, 0) is the velocity of the heavy quark. The Z b states are collected in the matrix as The pions as Goldstone bosons of the spontaneous breaking of the chiral symmetry can be parametrized as where F π = 92.2 MeV is the pion decay constant.
To calculate the box diagrams, we need the Lagrangian for the coupling of the Υ to the bottom mesons and the coupling of the h b to the bottom mesons [11,12], where J ≡ Υ · σ + η b denotes the heavy quarkonia spin multiplet, 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 [13][14][15][16][17] where denotes the three-vector components of u µ as defined in Eq. (4).
Here we will use g π = 0.492 ± 0.029 from a recent lattice QCD calculation [18].
FIG. 1: Feynman diagrams considered for the Υ(mS) → h b (nP )ππ processes. The crossed diagrams of (a) and (b) are not shown explicitly. 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), (b), and the bubble loop in Fig. 2 (c).
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 ( * ) π is proportional to the external momentum of the pion q π . The ΥB ( * )B( * ) vertex is in a P -wave, and the h b B ( * )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 O(q π ).
As for the triangle diagram Fig. 2 (a), the leading ΥB ( * )B( * ) π vertex given by [21] is proportional to the energy of the pion, E π ∼ q π . Therefore, Fig. 2 (a) is counted as m B ν 5 q 2 π /ν 6 = m B q 2 π /ν, where the factor m B has been introduced to match the dimension with the scaling for the box diagrams.
In Fig. 2 [13] is proportional to the momentum the pion q π . The loop momentum due to the ΥB ( * )B( * ) coupling has to contract with the external pion momentum. Thus, Fig. 2 (b) scales as ν 5 q 3 π /ν 6 = q 3 π /ν. In Fig. 2 (c), both the initial vertex and the final vertex are proportional to q π , so the bubble loop scales as m B ν 5 q 2 π /ν 4 = m B q 2 π ν. Therefore, we expect that the ratios of the contributions of the box diagrams, triangle diagram where

C. Tree-level amplitudes and box diagram calculation
The decay amplitude for is described in terms of the Mandelstam variables Using the effective Lagrangians in Eqs. (1) and (2), the tree amplitude of Υ(mS) Notice that the nonrelativistic normalization factor √ m Y has been multiplied to the amplitude for  For the tensor reduction of the loop integrals it is convenient to define q = −p b and the perpendicular momentum q ⊥ = p c − q(q · p c )/q 2 , which satisfy q · q ⊥ = 0. The result of the amplitude of the box diagrams can be written as 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 Υ(mS) → h b (nP )ππ is given by where the lower and upper limits are given as

III. PHENOMENOLOGICAL DISCUSSION
To estimate the contribution of the Z b -exchange mechanism we need to know the coupling strengths of Z b Υ(4S)π and Z b h b (nP )π. The mass difference between Z b (10610) and Z b (10650) is much smaller than the difference between their masses and the Υ(mS)π/h b (nP )π 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 Υ(4S), so we only use one Z b , the Z b (10610), which approximately combine both Z b states' effects. In Ref. [10], we has studied the Υ(4S) → Υ(mS)ππ processes to extract the coupling constant |C Z b Υ(4S)π | = (3.3 ± 0.1) × 10 −3 , which containing effects from both Z b states.
For the couplings of Z b h b (nP )π, in principle they can be extracted from the partial widths of the Z b states decay into h b (nP )π(n = 1, 2) where . The preliminary results for the branching fractions of the decays of both Z b states into h b (nP )π(n = 1, 2) have been given in [22], where the Z b line shapes were described using Breit-Wigner forms. If we naively use these branching fractions, we would obtain Here all the Z b h b π 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 ( * )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 (bbπ) channels, and thus the relevant coupling strengths.
As analyzed in Ref. [23], we expect that the |g Z b h b π | 2 should be about 1 order of magnitude larger than the naive one, so for a rough estimation we will use three times the results from Eq. (16), In We find that the BR Υ(4S)→h b (1P )π + π − is at least one order of magnitude smaller than the branching fractions BR Υ(4S)→Υ(1S,2S)π + π − , which are about 8 × 10 −5 given in PDG [24], and the BR Υ(4S)→h b (2P )π + π − is tiny due to the very small phase space. To illustrate the effects of the Z b -exchange and box graph mechanisms in Υ(4S) → h b (1P )ππ, we give the predictions only including the Z b -exchange terms or only including the box diagrams The Υ(mS) → h b (nP )ππ are heavy quark spin flip processes and they are forbidden in the heavy quark limit. We have checked that

Scalar four-point integrals
For the first topology as shown in Fig. 3, the scalar integral evaluated for the initial bottomonium at rest (p = (M, 0)) reads Performing the contour integration, we find − µ 12 µ 23 µ 24 2m 1 m 2 m 3 m 4 where we defined The second topology in Fig. 3 is just the crossed diagram of the first topology with q 1 ↔ q 2 , so the scalar integral reads where Performing the contour integration, we find where we defined In all the three cases the remaining three-dimensional momentum integration will be carried out numerically.

Tensor reduction
Since the ΥB ( * )B( * ) vertex scales with the momentum of the bottom meson pair, for topology I we have to deal with −µ 12 µ 23 µ 24 where f (l) = {1, l i } for the fundamental scalar and vector integrals, respectively. A convenient parametrization of the tensor reduction reads where q 1⊥ = q 1 −q(q·q 1 )/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.

Amplitudes
We define the scalar integrals J1(i, r, k) based on the J (r) 1 in the tensor reduction of vector integral in Eq. (A10), where i = 1, 2, 3 denotes the three topologies of box diagrams as shown in Fig. 3, r = 1, 2 refers to the two components J We give the amplitude of the box diagrams for the Υ(mS) → h b (nP )ππ process, namely the A l (l = 1, 2, ..., 6) in the Eq. (12).