Decay widths and energy shifts of pi pi and pi K atoms

We calculate the S-wave decay widths and energy shifts for pi pi and pi K atoms in the framework of QCD+QED. The evaluation - valid at next-to-leading order in isospin symmetry breaking - is performed within a non-relativistic effective field theory. The results are of interest for future hadronic atom experiments.


Introduction
Nearly fifty years ago, Deser et al. [1] derived the formulae for the decay width and strong energy shift of pionic hydrogen at leading order in isospin symmetry breaking. Similar relations also hold for π + π − [2] and π − K + atoms, which decay predominantly into 2π 0 and π 0 K 0 , respectively. These Deser-type relations allow to extract the scattering lengths from measurements of the decay width and the strong energy shift. The DIRAC collaboration [3] at CERN intends to measure the lifetime of pionium in its ground state at the 10% level, which will allow to extract the scattering length difference |a 0 0 − a 2 0 | at 5% accuracy. The experimental result can then be compared with theoretical predictions for the S-wave scattering lengths [4][5][6] and with the results from other experiments [7]. Particularly interesting is the fact that one may determine in this manner the nature of the SU(2)×SU(2) spontaneous chiral symmetry breaking experimentally [8]. New experiments are proposed for CERN PS and J-PARC in Japan [9]. In order to determine the scattering lengths from such experiments, the theoretical expressions for the decay width and the strong energy shift must be known to an accuracy that matches the experimental Preprint submitted to Elsevier Science precision. For this reason, the ground state decay width of pionium has been evaluated at next-to-leading order [10][11][12][13][14][15] in the isospin symmetry breaking parameter δ, where both the fine-structure constant α and (m u − m d ) 2 count as O(δ). The aim of the present article is to provide the corresponding formulae for the S-wave decay widths and strong energy shifts of pionium and the π ± K ∓ atom at next-to-leading order in isospin symmetry breaking. A detailed derivation of the results will be provided elsewhere [16]. The strong energy shift of the π ± K ∓ atom is proportional to the sum of the isospin even and odd S-wave πK scattering lengths a + 0 + a − 0 . This sum [18][19][20][21][22] is sensitive to the combination of low-energy constants 2L r 6 + L r 8 [23]. The consequences of this observation for the SU(3)×SU(3) quark condensate [24] remain to be worked out.

Non-relativistic framework
The non-relativistic effective Lagrangian framework has proven to be a very efficient method to investigate bound state characteristics [12,15,25,26]. The non-relativistic Lagrangian is exclusively determined by symmetries, which are rotational invariance, parity and time reversal. It provides a systematic expansion in powers of the isospin breaking parameter δ. What concerns the π − K + atom, we count both α and m u − m d as order δ. The different power counting for the π + π − and π − K + atoms are due to the fact that in QCD, the chiral expansion of the pion mass difference ∆ π = M 2 π + −M 2 π 0 is of second order in m u − m d , while the kaon mass difference ∆ K = M 2 K + − M 2 K 0 starts at first order in m u − m d . In the sector with one or two mesons, the non-relativistic πK Lagrangian is L NR = L 1 + L 2 . The first term contains the one-pion and one-kaon sectors, where E = −∇A 0 −Ȧ, B = ∇ × A and the quantity h = π, K stands for the non-relativistic pion and kaon fields. We work in the Coulomb gauge and eliminate the A 0 component of the photon field by the use of the equations of motion. The covariant derivatives are given by D t h ± = ∂ t h ± ∓ ieA 0 h ± and Dh ± = ∇h ± ± ieAh ± , where e denotes the electromagnetic coupling. What concerns the one-pion-one-kaon sector, we only list the terms needed to evaluate the decay width and the energy shift of the π − K + atom at order δ 9/2 and δ 4 , respectively, The ellipsis stands for higher order terms 1 . We work in the center of mass system and thus omit terms proportional to the total 3-momentum. The total and reduced masses read The coupling constant C ′ 1 contains contributions coming from the electromagnetic form factors of the pion and kaon, where r 2 π + and r 2 K + denote the charge radii of the charged pion and kaon, respectively. The low energy constants C 1 , . . . , C 3 may be determined through matching the πK amplitude at threshold for various channels, see section 3.
To evaluate the energy shift and decay width of the π − K + atom at next-toleading order in isospin symmetry breaking, we make use of resolvents. For a detailed discussion of the technique, we refer to Ref. [15]. Here, we simply list the results. We use dimensional regularization, to treat both ultraviolet and infrared singularities. Up to and including order δ 9/2 , the decay into π 0 K 0 is the only decay channel contributing, and we get for the total S-wave decay width where 2n 2 ))] 1/2 is of order δ 1/2 . The function ξ n develops an ultraviolet singularity as d → 3, with ψ(n) = Γ ′ (n)/Γ(n) and the running scale µ. At order δ 4 , the total energy shift may be split into a strong part and an electromagnetic part, according to ∆E n = ∆E h n + ∆E em n .
(7) For the discussion of the electromagnetic energy shift, we refer to section 4. The strong S-wave energy shift reads The results for the decay width (5) and energy shift (8) are valid at next-toleading order in isospin symmetry breaking.

Matching the low-energy constants
The coupling constants C i can be determined through matching the nonrelativistic and the relativistic amplitudes at threshold. The coupling C 3 is needed at order δ 0 only. However, we have to determine both C 1 and C 2 at next-to-leading order in isospin symmetry breaking. The relativistic amplitudes are related to the non-relativistic ones through with ω i (p) = (M 2 i + p 2 ) 1/2 . The 3-momentum p denotes the center of mass momentum of the incoming particles, q the one of the outgoing particles. The effective Lagrangian in Eqs. (1) and (2), allows us to evaluate the nonrelativistic π − K + → π 0 K 0 and π − K + → π − K + scattering amplitudes at threshold at order δ. In the isospin symmetry limit, the effective couplings C 1 , C 2 and C 3 are where the S-wave scattering lengths 2 a + 0 = 1/3(a By substituting these relations into the expression for the decay width (5) and the strong energy shift (8), one obtains the Deser-type formulae [1,2]. We demonstrate the matching at next-to-leading order in δ by means of the π − K + → π − K + amplitude. In the presence of virtual photons, we first have to subtract the one-photon exchange diagram from the full amplitude, as displayed in Fig. 1. The coupling constant C 1 is determined by the truncated partT ±;± NR , which contains an infrared singular Coulomb phase θ c as d → 3, 2 a + 0 and a − 0 are normalized as in Ref. [18].
The blob describes the vector form factor of the pion and kaon.T ±;± NR denotes the truncated amplitude.
At order δ, the remainderT ±;± NR is free of infrared singularities at threshold. The real part ofT ±;± NR is given by Here, the ultraviolet pole term Λ(µ) is removed by renormalizing the coupling C 1 . The renormalization of C 1 eliminates at the same time the ultraviolet divergence contained in the expression for the energy shift (8). The calculation of the relativistic π − K + → π − K + scattering amplitude was performed at O(p 4 , e 2 p 2 ) in Refs. [20,21]. Both the Coulomb phase and the logarithmic singularity in Eq. (12) are absent in the real part of the relativistic amplitude at this order of accuracy, they first occur at order e 2 p 4 . The quantity Re A ±;± thr denotes the constant term occurring in the real part of the truncated relativistic threshold amplitude. The coupling constant C 2 may be determined analogously by matching the non-relativistic π − K + → π 0 K 0 amplitude to the relativistic one at order δ.
4 Results for the π − K + atom The result for the decay width and strong energy shift are valid at nextto-leading order in isospin symmetry breaking, and to all orders in the chiral expansion. We get for the decay width at order δ 9/2 , in terms of the relativistic π − K + → π 0 K 0 threshold amplitude, where The outgoing relative 3-momentum with λ(x, y, z) = x 2 + y 2 + z 2 − 2xy − 2xz − 2yz, is chosen such that the total final state energy corresponds to E n = Σ + − α 2 µ + /(2n 2 ). The quantity Re A 00;± thr is calculated as follows. One evaluates the relativistic π − K + → π 0 K 0 amplitude near threshold and removes the divergent Coulomb phase. The real part contains singularities ∼ 1/|p| and ∼ ln|p|/µ + . The constant term in this expansion corresponds to Re A 00;± thr . The normalization is chosen such that The isospin breaking corrections ǫ have been evaluated at O(p 4 , e 2 p 2 ) in Refs. [21,27]. See also the comments in section 6.
We now discuss the various energy shift contributions. According to Eq. (7), the energy shift at order δ 4 is split into an electromagnetic part ∆E em n and the strong part ∆E h n in Eq. (8). The electromagnetic energy shift contains both pure QED corrections as well as finite size effects due to the charge radii of the pion and kaon, contained in λ. The pure electromagnetic corrections have been evaluated in Ref. [28] for arbitrary angular momentum l. We checked 3 that the electromagnetic energy shift at order α 4 indeed amounts to 3 We thank A. Rusetsky for a very useful communication concerning technical aspects of the calculation.
Here, the first term is generated by the mass insertions, the second contains the finite size effects and the last stems from the one-photon exchange contribution. The strong S-wave energy shift reads at order δ 4 , with In the isospin limit, the normalized relativistic amplitude reduces to the sum of the isospin even and odd scattering lengths. The corrections ǫ ′ have been obtained at O(p 4 , e 2 p 2 ) in Refs. [20,21]. See also the comments in section 6. The result for ∆E h 1 in Eq. (19) agrees with the one obtained for the strong energy shift of the ground state in pionic hydrogen [26], if we replace µ + with the reduced mass of the π − p atom and Re A ±;± thr with the constant term in the threshold expansion for the real part of the truncated π − p → π − p amplitude.
What remains to be added are the vacuum polarization contributions [14,29], which are formally of higher order in α, however numerically not negligible. The vacuum polarization leads to an energy level shift ∆E vac nl as well as to a change in the Coulomb wave function of the π − K + atom at the origin δψ K,n (0). For the first two energy levels, ∆E vac nl [14,29] is given numerically in table 2, section 6. Formally of order α 2l+5 , this contribution is enhanced due to its large coefficient containing (µ + /m e ) 2l+2 . The modified Coulomb wave function affects both, the decay width and the strong energy shift, see section 6.
As discussed in section 6, the electromagnetic contributions (18) are known to a high precision. Further, the strong shift in the nP state is very much suppressed (order α 5 ). A future measurement of the energy splitting between the nS and nP states will therefore allow to extract the strong S-wave energy shift in Eq. (19), and to determine the combination a + 0 + a − 0 of the πK scattering lengths. The energy splitting between the 2S and 2P states is given by The uncertainty displayed is the one in ∆E h 2 only. For the numerical values of the various energy shift contributions, see table 2 in section 6.
Again the uncertainty displayed is the one in ∆E h π,2 only. The numerical values for the various energy shifts are listed in table 3, section 6.
We obtain for the decay width of the ground state, where the corrections δ h,1 , h = π, K are given in table 1. The strong energy shift reads For the first two energy levels, the corrections δ ′ h,n are specified in table 1. As mentioned in section 4, these corrections to the Deser-type formulae are modified by vacuum polarization,  Table 2 Numerical values for the energy shift and the lifetime of the π ± K ∓ atom.  Table 3 Numerical values for the energy shift and the lifetime of the π + π − atom. where Formally, the contribution δ vac h,n is of order α 2 , but enhanced because of the large coefficient containing µ + /m e . For the ground state, the corrections [14] yield δ vac K,1 = 0.45 · 10 −2 and δ vac π,1 = 0.31 · 10 −2 . The changes in δ π,1 and δ ′ π,1 due to δ vac π,1 are about 5%. For δ ′ K,1 however, the correction amounts to 27%. Here, we omit the contributions from δ vac h,n , because the uncertainties in δ h,n and δ ′ h,n are much larger than δ vac h,n .
The numerical values for the lifetime τ 1 . = Γ −1 1 , (τ π,1 . = Γ −1 π,1 ) and the energy shifts at next-to-leading order in isospin symmetry breaking are given in table 2 and 3. The energy shifts due to vacuum polarization ∆E vac nl are taken from Ref. [14,29]. In the evaluation of the uncertainties, the correlations between the S-wave scattering lengths have been taken into account. For the decay width and the strong energy shift of the π ± K ∓ atom, the dominant source of uncertainty is due to the uncertainties in the scattering lengths a + 0 and a − 0 . We do not display the error bars for the electromagnetic energy shifts, which stem at order α 4 from the uncertainties in r 2 π + and r 2 K + only. For pionium, the uncertainties of ∆E em π,10 at order α 4 amount to about 0.7%, while for the π ± K ∓ atom ∆E em 10 is known at the 5% level. To estimate the order of magnitude of the electromagnetic corrections at higher order, we may compare with positronium. Here, the α 5 and α 5 lnα corrections [34] amount to about 2% with respect to the α 4 contributions.

Summary and Conclusions
We provided the formulae for the energy shifts and decay widths of the π + π − and π ± K ∓ atoms at next-to-leading order in isospin symmetry breaking. To confront these predictions with data presents a challenge for future hadronic atom experiments. Should it turn out that these predictions are in conflict with experiment, one would have to revise our present understanding of the low-energy structure of QCD.