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Decay Widths and Energy Shifts of Ππ and Πk Atoms

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Decay Widths and Energy Shifts of Ππ and Πk Atoms Physics Letters B 587 (2004) 33–40 www.elsevier.com/locate/physletb Decay widths and energy shifts of ππ and πK atoms J. Schweizer Institute for Theoretical Physics, University of Bern, Sidlerstrasse 5, CH-3012 Bern, Switzerland Received 20 January 2004; accepted 2 March 2004 Editor: W.-D. Schlatter Abstract + − ± ∓ We calculate the S-wave decay widths and energy shifts for π π and π 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. 2004 Published by Elsevier B.V. PACS: 03.65.Ge; 03.65.Nk; 11.10.St; 12.39.Fe; 13.40.Ks Keywords: Hadronic atoms; Chiral perturbation theory; Non-relativistic effective Lagrangians; Isospin symmetry breaking; Electromagnetic corrections 1. Introduction [4–6] and with the results from other experiments [7]. Particularly interesting is the fact that one may deter- × Nearly fifty years ago, Deser et al. [1] derived the mine in this manner the nature of the SU(2) SU(2) formulae for the decay width and strong energy shift spontaneous chiral symmetry breaking experimentally of pionic hydrogen at leading order in isospin symme- [8]. New experiments are proposed for CERN PS and try breaking. Similar relations also hold for π+π− [2] J-PARC in Japan [9]. In order to determine the scat- and π−K+ atoms, which decay predominantly into tering lengths from such experiments, the theoretical 2π0 and π0K0, respectively. These Deser-type rela- expressions for the decay width and the strong en- tions allow to extract the scattering lengths from mea- ergy shift must be known to an accuracy that matches surements of the decay width and the strong energy the experimental precision. For this reason, the ground shift. The DIRAC Collaboration [3] at CERN intends state decay width of pionium has been evaluated at to measure the lifetime of pionium in its ground state next-to-leading order [10–15] in the isospin symme- at the 10% level, which will allow to extract the scat- try breaking parameter δ, where both the fine-structure 2 | 0 − 2| constant α and (mu − md ) count as O(δ).Theaim tering length difference a0 a0 at 5% accuracy. The experimental result can then be compared with the- of the present Letter is to provide the corresponding oretical predictions for the S-wave scattering lengths formulae for the S-wave decay widths and strong en- ergy shifts of pionium and the π±K∓ atom at next- to-leading order in isospin symmetry breaking. A de- E-mail address: [email protected] (J. Schweizer). tailed derivation of the results will be provided else- 0370-2693/$ – see front matter 2004 Published by Elsevier B.V. doi:10.1016/j.physletb.2004.03.007 34 J. Schweizer / Physics Letters B 587 (2004) 33–40 where [16]. The strong energy shift of the π±K∓ atom δ9/2 and δ4, respectively, is proportional to the sum of the isospin even and + − † † † † + L = C π K π−K+ + C π K π K + h.c. odd S-wave πK scattering lengths a0 a0 . This sum 2 1 − + 2 − + 0 0 [18–22] is sensitive to the combination of low-energy + † † +··· r + r C3π0 K0 π0K0 . (2) constants 2L6 L8 [23]. The consequences of this ob- servation for the SU(3) × SU(3) quark condensate [24] The ellipsis stands for higher order terms.1 We work in remain to be worked out. the center of mass system and thus omit terms propor- tional to the total 3-momentum. The total and reduced masses read 2. Non-relativistic framework = + Σi Mπi MKi , The non-relativistic effective Lagrangian frame- Mπi MKi work has proven to be a very efficient method to inves- µi = ,i=+, 0. (3) M i + M i tigate bound state characteristics [12,15,25,26]. The π K non-relativistic Lagrangian is exclusively determined The coupling constant C1 contains contributions com- by symmetries, which are rotational invariance, par- ing from the electromagnetic form factors of the pion ity and time reversal. It provides a systematic expan- and kaon, sion in powers of the isospin breaking parameter δ. − + What concerns the π K atom, we count both α and = − 2 = 1 2 + 2 C1 C1 e λ, λ rπ+ rK+ , (4) mu − md as order δ. The different power counting for 6 + − − + the π π and π K atoms are due to the fact that in 2 2 where r + and r + denote the charge radii of QCD, the chiral expansion of the pion mass difference π K 2 2 the charged pion and kaon, respectively. The low en- ∆π = M + − M is of second order in mu − md , π π0 ergy constants C1,...,C3 may be determined through 2 2 while the kaon mass difference ∆ = M + − M K K K0 matching the πK amplitude at threshold for various starts at first order in mu − md . In the sector with one channels, see Section 3. or two mesons, the non-relativistic πK Lagrangian is To evaluate the energy shift and decay width of − + LNR = L1 + L2. The first term contains the one-pion the π K atom at next-to-leading order in isospin and one-kaon sectors, symmetry breaking, we make use of resolvents. For a detailed discussion of the technique, we refer to L = 1 2 − 2 + † − + ∆ Ref. [15]. Here, we simply list the results. We use di- 1 E B h0 i∂t Mh0 2 2Mh0 mensional regularization, to treat both ultraviolet and 9/2 2 infrared singularities. Up to and including order δ , ∆ † 0 0 + +··· h0 + h± iDt − Mh+ the decay into π K is the only decay channel con- M3 8 h0 ± tributing, and we get for the total S-wave decay width 2 4 D D α3µ3 µ2k2C2 αµ2 C ξ + + +··· h±, (1) = + 2 − 0 0 3 − + 1 n 2M + 3 Γn µ0k0C2 1 h 8Mh+ n3π2 4π2 π 2 3 + 3 =−∇ − ˙ =∇× 5µ0k M 0 M 0 where E A0 A, B A and the quan- + 0 π K + O δ5 , (5) tity h = π,K stands for the non-relativistic pion and 8 M3 M3 π0 K0 kaon fields. We work in the Coulomb gauge and elim- 0 2 2 1/2 inate the A component of the photon field by the where k0 =[2µ0(Σ+ − Σ0 − α µ+/(2n ))] is of 1/2 use of the equations of motion. The covariant deriva- order δ . The function ξn develops an ultraviolet sin- tives are given by Dt h± = ∂t h± ∓ieA0h± and Dh± = ∇h± ± ieAh±,wheree denotes the electromagnetic 1 The basis of operators containing two space derivatives can be coupling. What concerns the one-pion–one-kaon sec- chosen such that none of them contributes to the energy shift and tor, we only list the terms needed to evaluate the decay decay width at next-to-leading order in isospin symmetry breaking − + width and the energy shift of the π K atom at order [16]. J. Schweizer / Physics Letters B 587 (2004) 33–40 35 − + → − + ¯ ±;± Fig. 1. Non-relativistic π K π K scattering amplitude. The blob describes the vector form factor of the pion and kaon. TNR denotes the truncated amplitude. → = 2 + 2 1/2 gularity as d 3, with ωi (p) (Mi p ) . The 3-momentum p de- notes the center of mass momentum of the incom- α 2µ+ ξn = Λ(µ) − 1 + 2 ln + ln ing particles, q the one of the outgoing particles. The n µ effective Lagrangian in Eqs. (1) and (2), allows us 1 − + → 0 0 + ψ(n)− ψ(1) − , to evaluate the non-relativistic π K π K and − + → − + n π K π K scattering amplitudes at threshold − 1 at order δ. In the isospin symmetry limit, the effective Λ(µ) = µ2(d 3) − ln 4π − Γ (1) , (6) − couplings C1, C2 and C3 are d 3 √ with ψ(n) = Γ (n)/Γ (n) and the running scale µ. 2π + − 2 2π − C1 = a + a ,C2 =− a , At order δ4, the total energy shift may be split into µ+ 0 0 µ+ 0 a strong part and an electromagnetic part, according to = 2π + C3 a0 , (10) = h + em µ+ En En En . (7) 2 + = 1/2 + For the discussion of the electromagnetic energy shift, where the S-wave scattering lengths a0 1/3(a0 − 2a3/2) and a = 1/3(a1/2 −a3/2) are defined in QCD, we refer to Section 4. The strong S-wave energy shift 0 0 . 0 0 . reads at mu = md and Mπ = Mπ+ , MK = MK+ . By substi- 3 3 2 2 2 tuting these relations into the expression for the de- h α µ+ αµ+ 2 µ0k0 2 E =− C − C ξn − C C cay width (5) and the strong energy shift (8), one ob- n πn3 1 2π 1 4π2 2 3 tains the Deser-type formulae [1,2]. We demonstrate + O δ5 . (8) the matching at next-to-leading order in δ by means of the π−K+ → π−K+ amplitude. In the presence The results for the decay width (5) and energy shift (8) of virtual photons, we first have to subtract the one- are valid at next-to-leading order in isospin symmetry photon exchange diagram from the full amplitude, as breaking. displayed in Fig. 1. The coupling constant C1 is deter- ¯ ±;± mined by the truncated part TNR , which contains an infrared singular Coulomb phase θ as d → 3, 3. Matching the low-energy constants c ±;± ±;± T¯ (p; p) = e2iαθc Tˆ (p; p), NR NR The coupling constants Ci can be determined µ+ d−3 1 1 through matching the non-relativistic and the relativis- θc = µ − ln 4π + Γ (1) |p| d − 3 2 tic amplitudes at threshold.
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