$Z \to \pi^+\pi^-, K^+K^-$: A touchstone of the PQCD approach

We study two rare decays, $Z \to \pi^+\pi^-$ and $K^+K^-$, in the perturbative QCD approach up to the next-to-leading order of the strong coupling and the leading power of $1/m_Z$, $m_Z$ being the $Z$ boson mass. The branching ratios $\mathcal{B}(Z\to \pi^+\pi^-) = (0.83 \pm 0.02 \pm 0.02 \pm 0.04)\times 10^{-12}$ and $\mathcal{B}(Z\to K^+K^-) = (1.74^{+0.03}_{-0.05} \pm 0.04 \pm 0.02)\times 10^{-12}$ are obtained and can be measured at a tera-$Z$ factory. Because the subleading-power contributions to the branching ratios are negligible, and the leading one does not depend on any free parameter, the two channels can serve as a touchstone for the applicability of the perturbative QCD approach.


I. INTRODUCTION
Two-body non-leptonic B meson decays play an essential role in particle physics to help us understand the quantum chromodynamics (QCD) and the charge conjugation parity violation in the Standard Model, which have inspired development of many theoretical frameworks or approaches, including the collinear factorization in both the QCD [1] and the soft-collinear effective theory [2,3], the light-cone sum rules [4], the perturbative QCD (PQCD) approach [5] based on k T factorization [6][7][8][9], and the factorization-assisted topological-amplitude approach [10] proposed recently.
Among them, the PQCD approach is the most predictive one, since it factorizes each hadronic matrix element into universal distribution amplitudes of mesons and a perturbatively calculable hard kernel.However, it is also this unique feature of PQCD that has been questioned.An exercise of power counting of, for example, the B → π form factor or the timelike ππ form factor, as analysed in [1,11], tells us that the nonperturbative "small-x" (x is the momentum fraction of a constitute quark in π) region contributes from the leading power of 1/m B just as the perturbative x ∼ 1/2 region.On the other hand, calculation in PQCD shows that the small-x region is practically suppressed by a Sudakov factor after the k T resummation and thus the perturbative part is dominant [12][13][14].To test which argument is correct, here we propose the Z 0 → π + π − (or K + K − ) channel as a touchstone.In the PQCD calculation of the Z → π + π − decay rate, the power corrections of 1/m Z are so small that we can safely neglect them, which indicates that we only need to consider the twist-2 light-cone distribution amplitude (LCDA) of the pion and that the asymptotic form is good enough.As a result, the calculation does not have any arbitrariness, because even the nonperturbative pion LCDA is fixed [15][16][17].The two channels are expected to be measured or strictly constrained in the future at a Tera-Z factory like the Circular Electron-Positron Collider [18] and/or the FCC-ee, formerly known as TLEP [19], which can be used not only to precisely study the Higgs and Z properties (e.g.see [20]) and discover new particles (e.g.see [21]), but also to teach us more knowledge about the QCD as shown in this paper.
The Z → π + π − decay amplitude is proportional to the timelike pion form factor, which in principle can be studied in several different methods.One is the partial wave analysis to deal with elastic and inelastic scatterings, as well as resonances [22]; another one, the light-cone sum rules, is powerful for space-like form factors, while for the timelike region, dispersion relation as well as some resonance models are inevitable [23].Both these two approaches only work well in the low energy region and/or are model dependent, and to access the form factor with the dipion invariant mass at m Z , the PQCD is a good option [24][25][26].In this paper, we calculate the Z 0 → π + π − (K + K − ) decay rate up to the next-to-leading order (NLO) of α s and leading power of 1/m Z in the PQCD approach [5], and we find that the branching ratio is about 0.83 (1.74) ×10 −12 .It is quite hopeful to measure the two channels at a Tera-Z factory.Whichever of them is found, it will be the first observation of an exclusive hadronic Z decay, and serves as a touchstone to examine the PQCD approach.
The rest of the paper is organized as follows.In Section II the NLO PQCD calculation of the timelike pion form factor is performed, and the analytical formulas are given.In Section III we present the numerical results for the Z → π + π − , K + K − branching ratios.Section IV is the conclusion.

II. PERTURBATIVE CALCULATION
The Z qq interaction terms in the Lagrangian are given by L −J where g is the SU(2) gauge coupling, θ w is the weak mixing angle, and the hyperchages and electrocharges of the quarks are T u,d = ±1/2, Q u = 2/3 and Q d = -1/3.Then, we can write down the Z decaying amplitude, e.g. to π + π − , as with Z the polarization vector of the decaying Z boson.There are both vector and axial-vector currents in J (Z) µ , but an axial-vector current hadronic matrix element is forbidden by parity, and thus only the vector currents remain.
Therefore, the non-trivial timelike dipion vector form factors to calculate are with p 1,2 the momenta of π ± , q = p 1 + p 2 and Q 2 = q 2 .The above two definitions are equivalent to each other owing to isospin symmetry.Then, performing the phase space integral of |M| 2 and averaging the spins of Z, we obtain the decay width of the Z → π + π − as where g q V = g/(2 cos θ w ) × (T q − 2Q q sin 2 θ w ) [27,28], and the pion mass effect have been neglected.From the factor (g u V − g d V ) 2 , we can see that Z → π 0 π 0 or K 0 K0 is forbidden.Next, we will focus on the calculation of the

A. Kinematics and the LO form factor
As depicted in Fig. 1, the two upper and lower diagrams contribute to the timelike form factors from π + π − |ūγ µ u|0 and π + π − | dγ µ d|0 , respectively, at the leading order (LO) of QCD.In the calculation, we choose the following kinematics of the initial-and the final-state particles expressed in form of the light-cone coordinates, where p Z is the momentum of the Z boson.It can be read out that the Z boson is rest in frame, and two pion momenta are paralell with the two light-cone directions n and v, respectively, with neglecting the pion mass.The momenta of the constitute quarks and anti-quarks in Fig. 1 are written as We can equivalently get the timelike pion form factor at the leading power1 by calculating any of the diagrams in Fig. 1, where C F = 4/3, b i is the conjugate extend of the transverse momenta k iT , and φ π (x) is the twist-2 pion LCDA.The renormalization scale µ is chosen to be max The Sudakov factor obtained by the k T resummation for the non-convergent logarithms is where the exponent terms s(Q i , b i ) collect both the double and single logarithms in the vertex correction for an energetic light quark, as given in Eq (10) of [8], and the terms 1 log[2 log(t/Λ (5) )] + 1 log(t/Λ (5) ) − log[−2 log(bΛ (5) )] + 1 − log(bΛ (5) )) , resum the single logarithm in the quark self-energy correction [5].Here, we have accepted the two-loop expression for the strong coupling with β 1 = (33 − 2n f )/12, β 2 = (153 − 19n f )/24 and the flavor number n f = 5.We do not take into account the threshold resummation factor [12][13][14] for the hard kernel, which is only important in subleading contributions from higher-twist LCDAs.
The hard function h II (x 1 , b 1 , x 2 , b 2 , Q) in the form factor is the internal propagators conjugated in their coordinate space of the transverse momenta, which is further written in terms of the Bessel functions of the first kind J 0 and the Hankel functions of the first kind H (1) 0 , We find that Eq (11) oscillates violently as Q2 goes beyond 50 GeV 2 , which results from the large hierarchy between the two scales, Q 2 and k 2 T .Practically, the heavy oscillation causes difficulty in convergence of the multiple integral (7) numerically 2 .To overcome this difficulty, we take the hierarchy ansatz acquiesced in the PQCD, neglecting the transverse momentum in the quark propagator, while retaining the transverse momentum in the propagator of the hard gluon.As a result, the double-b hard function in (11) is reduced to a single-b one, with b = b 1 = b 2 read off the derivation of the above expression.This manipulation simplifies the computational task, and also extends the calculable region with Q 2 from dozens to thousands of GeV 2 .In this way, the form factor at LO is modified to with S I (x 1 , x 2 , b, µ) = S II (x 1 , b, x 2 , b, µ).Numerically, as will be observed in FIG. 2 where the double-b and single-b results are compared, the single-b approximation works very well in the high-Q 2 region.

B. Next-to-Leading-Order QCD correction
The NLO correction to the timelike pion form factor is derived in the PQCD approach with the single-b convolution configuration, and it reads [29] The explicit expression of the NLO correction function h(x 1 , x 2 , b, Q, µ) is given by Eq (18) in Ref. [24].For the second derivative of the Hankel function on the order parameter in practice we use the following fit function

Re[H
(1) 0 Im[H (1) 0 This NLO correction (14) has recently been applied to study the B c pair production at electron-positron colliders with using the B c distribution amplitudes obtained in NRQCD [30], where only the small argument limit for H

III. NUMERICS
In the calculation, the asymptotic form of the twist-2 pion LCDA is accepted, with the pion decay constant f π = 130.2 ± 1.4GeV [16].The other numerical inputs include [31] Γ Z = 2.4952 ± 0.0023 GeV and with the quantities consistently defined at the m Z scale under the modified minimal subtraction (MS) scheme.To reproduce the central value of α s (m 2 Z ) with the two-loop accuracy, the scale Λ MS = 0.2327 GeV is chosen.Using a Monte-Carlo integration strategy with the Vegas [32] algorithm from the GNU Scientific Library [33], we run the integration with 500,000,000 integration points for the real and imaginary parts of Eqs ( 13) and ( 14), which gives us a relative precision of better than permillage level.The central values of the LO pion form factor and the NLO correction at the m 2 Z point are from which we find that the NLO correction enhances the LO result reasonably by about 10%.The PQCD prediction up to the NLO for the Z → π + π − branching ratio is with the three uncertainties coming from the scale variation from µ/2 to 2µ, the strong coupling constant and the pion decay constant, respectively.Replacing the pion decay constant in the calculation with the kaon decay constant f K = 155.6 ± 0.4 GeV [16], we obtain the corresponding result for Z → K + K − , B(Z → K + K − ) = (1.74+0.03 −0.05 ± 0.04 ± 0.02) × 10 −12 .
Owing to almost 100% detection efficiencies of charged pions and kaons, the PQCD calculation predicts 1 and 2 events for Z → π + π − and K + K − at a Tera-Z factory.If the contributions from the "small-x" region are actually dominant in pion and kaon timelike form factors [1,11], more events are expected.
To confirm the validity of the single-b configuration, we numerically calculate the timelike pion form factor at the LO in the region Q 2 ∈ [1, 50] GeV 2 using both the double-b and single-b formulas ( 7) and ( 13), and depict them in Fig. 2. The discrepancy between the two results is visible in the low-Q 2 region, while starting from ∼ 40 GeV 2 , we can safely omit the transverse momentum effect in the internal quark propagator and accept the single-b approximation.
The magnitude and the strong phase of the pion form factor at the LO and NLO with Q 2 from 50 GeV 2 to m 2 Z are depicted in Fig. 3.We find that the NLO correction to the magnitude is around 11% in the whole Q 2 region considered here.For the strong phase, the LO prediction is about 180 • and the NLO correction brings an enhancement of not larger than 20 •3 .We suggest a parameterization formula to interpolate the form factor far away from the resonance region with Q in units of GeV, which is inspired from the parameterization with the reciprocal of the square polynomial [34].Here we add another Q 2 term in the numerator to relieve a sudden drop with Q 2 around several hundred GeV 2 .The equality of the constant terms in the numerator and denominator is guaranteed by the normalization condition of pion form factor G π (0) = 1 (for references see e.g.[35]).For the timelike pion form factor at the LO, we get A (0) = 0.0879, B (0) = 46.1,C (0) = 10.9; while after including the NLO correction, we have A = 0.0996, B = 48.2,C = 12.6.

IV. CONCLUSION
We study the Z → π + π − , K + K − decays whose branching ratios are determined by the timelike form factors of the corresponding mesons.With a high Q 2 = m 2 Z , we can safely neglect the power corrections in the calculation of the form factors in the PQCD approach, which then do not depend on any unknown nonperturbative parameters and serve as clean theoretical predictions.The PQCD predictions up to the NLO for the the branching ratios of the two channels are B(Z → π + π − ) = (0.83±0.02±0.02±0.04)×10−12 and B(Z → K + K − ) = (1.74+0.03 −0.05 ±0.04±0.02)×10−12 .
They can be touched by a future Tera-Z factory, and the measurements will act as a touchstone of the PQCD approach.
V. ACKNOWLEDGEMENTS

10 Q 2 2 FIG. 2 .
FIG. 2. The magnitude of the timelike pion form factor G(Q 2 ) at the LO obtained with the double-b and single-b configurations.

2 FIG. 3 .
FIG.3.The LO (G (0) ) and NLO (G) predictions for the magnitude (left) and the phase (right) of the pion form factor with Q 2 from 50 GeV 2 to m 2 Z .
1.35 log x +J 0 (x) −0.581 + 1.48 log x − 0.497 log 2 x 58 + 0.720x − 0.151x 2 + 0.00643x 3 + 2.57 log x +J 0 (x) 3.16 − 0.794x + 0.0179x 2 − 5.65 log x + 2.26 log 2 x +Y 0 (x) 4.10 − 2.03 log x − 0.00708 log 2 x , x 10, We thank Hsiang-nan Li, Xin Liu, Yue-Long Shen and Yan-Bing Wei for helpful discussion.S. C is supported by the National Science Foundation of China under the No. 11805060 and "the Fundamental Research Funds for the Central Universities" under No 020400/531107051171.Q. Q is supported by the DFG Research Unit FOR 1873 "Quark Flavour Physics and Effective Theories".S. C is grateful to Theoretical Division of Institute of High Energy Physics at Beijing for hospitality and for financial support where this work is finalized.