B K decays in a finite volume Akaki Rusetsky, University of Bonn In collaboration with A. Agadjanov, V. Bernard and U.-G. Meißner arxiv:1605.03386, Nucl. Phys. B (in print) 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.1
Plan Introduction: effective Lagrangian and form factors Non-relativistic EFT: essentials Lüscher-Lellouch formula Continuation to the pole and fixing the photon virtuality Limit of an infinitely narrow resonance Conclusions, outlook A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.2
The problem Rare B decays provide one of the best opportunities in search of BSM physics Theoretical uncertainties arise in the calculated form factors lattice QCD In general, the decay products contain resonances, e.g. B K l + l. A procedure for the extraction of the resonance form factors on the lattice should be defined Analytic continuation to the resonance pole should be studied. What is the photon virtuality q 2 at the pole? The case of coupled channels should be investigated A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.3
The effective weak Lagrangian forb K l + l decay H eff = 4G F 2 V tsv tb 10 i=1 C i W i Seven form factors: W 7 = m be 16π 2 sσµν1 2 (1+γ 5)bF µν W 9 = W 10 = e 2 16π 2 sγµ1 2 (1 γ 5)b lγ µ l e 2 16π 2 sγµ1 2 (1 γ 5)b lγ µ γ 5 l V(q 2 ), A 0 (q 2 ), A 1 (q 2 ), A 2 (q 2 ), T 1 (q 2 ), T 2 (q 2 ), T 3 (q 2 ) Photon virtuality: q 2 = (p B p K ) 2 A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.4
Matrix elements on the lattice The K is unstable: need to scan the energy of the πk pair, photon 3-momentum fixed πk pair in the CM frame: no mixing Asymmetric box: L L L, varying L, fixed L Projection onto the irreps: O (±) E = 1 2 (O 1 O 2 ), O A1 = O 3 Extraction of the form factors: V(+) 1 2 s(γ 1 +iγ 2 )b B = 2iEV(q2 ) m B +E, etc A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.5
Non-relativistic EFT The Lagrangian contains non-relativistic field operators, particle number is conserved 4-particle interaction Lagrangian: L I = C 0 Φ 1 Φ 2 Φ 1Φ 2 +... The propagator with the relativistic dispersion law: D(p) = 1 2w(p) 1 w(p) p 0 i0, w(p) = M 2 +p 2 Threshold expansion applied in loops power counting EFT in a finite volume: same Lagrangian, loops modified d 3 p (2π) 3 1 L 3 p (finite box with periodic b.c.) Provides the relation between the infinite / finite volume A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.6
Relativistic vs non-relativistic Matching is performed on mass shell. Consequently, only on-shell vertices appear in the perturbative expansion Threshold expansion automatically puts the amplitudes in the loops on energy shell: d 3 k (2π) 3 f(k 2 ) k 2 q 2 0 = d 3 k (2π) 3 ( f(q 2 0 ) k 2 q 2 0 ) + regular }{{} =0, threshold expansion = +... 2 2 f(k ) f(q ) 0 All results are the same as in the relativistic framework + skeleton expansion, come at a less effort A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.7
Multi-channel amplitude in a finite volume T L = 8π s f(e) 1 [t p 1 τ 1 (t 2 +τ 2 )+s 2 1 ε τ 1τ 2 t] 1 p1 c p ε s ε τ 1 τ 2 t 2 1 p1 p 2 c ε s ε τ 1 τ 2 t 1 p 2 [t 2 τ 2 (t 1 +τ 1 ) s 2 ε τ 1τ 2 t] t i = tanδ i (p), t = t 2 t 1 : scattering phases c ε = cosε, s ε = sinε: mixing angle τ i = tanφ i : expressed through Lüscher zeta-functions Lüscher equation in the two-channel case: f(e) = (t 1 +τ 1 )(t 2 +τ 2 )+s 2 εt(τ 2 τ 1 ) = 0 Factorization of the amplitude near the eigenvalue: T αβ L (E) f αf β E n E + A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.8
Lüscher-Lellouch formula, two-channel case γ O K K B O F 1 M + X 1 X 2 π η F 1 M + + O X 1 F M 2 O + FM X 2 M 2 F(E n, q ) = V 1 p1 τ1 1 8πE f F 1 1 +p 2 τ2 1 f F 2 2 E=En A 1 = F 1 (c 2 ε cosδ 1e iδ 1 +s 2 ε cosδ 2e iδ 2 )+ A 2 = F 2 (c 2 ε cosδ 2 e iδ 2 +s 2 ε cosδ 1 e iδ 1 )+ p1 p2 p 2 F2 c ε s ε (cosδ 1 e iδ 1 cosδ 2 e iδ 2 ) p 1 F1 c ε s ε (cosδ 1 e iδ 1 cosδ 2 e iδ 2 ) F 1, F2 : Irreducible amplitudes, exponentially suppressed volume dependence Agrees with: M. Hansen and S. Sharpe, PRD 86 (2012) 016007 A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.9
Extraction of the form factors at the pole T -matrix on the second Riemann sheet: T αβ II (s) h αh β s R s + Form factors at the pole: residues in the pertinent Green functions F R (E R,q) = i 8πE (p 1h 1 F 1 p 2 h 2 F 2 ) E ER Universal: do not depend on the process! Uniquely defined! A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.10
Photon virtuality (q 0,q) ( m 2 B +q 2,q) (E R,0) q 2 = (E R m 2 B +q2 ) 2 q 2, complex! Keeping q 2 real and performing the limit s ER 2 correspond to the resonance pole! does not A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.11
Infinitely small width 1-channel case: F R = F up to normalization in the limit Γ 0 2-channel case: normalization different in different channels... Define ũ 1 = t 1 1 ( p 1 c ε F 1 + p 2 s ε F 2 ) ũ 2 = t 1 2 ( p 2 c ε F2 p 1 s ε F1 ) ũ 1, ũ 2 are low energy polynomials: do not contain the small energy scale Γ At the resonance, the decay matrix element factorizes ũ 1 = O(Γ 1/2 ) and ũ 2 = O(1) A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.12
The form factor in the limitγ 0 F R (E R,q) = 1 (r 1 ũ 1 +r 2 ũ 2 ) 4π E ER r 2 1 = t 2 1 t 2 +i 2is 2 ε h (E), r 2 2 = t 2 2 t 1 i+2is 2 ε h (E) h(e) = (t 1 i)(t 2 +i)+2is 2 ε(t 2 t 1 ) Resonance emerges in one channel: t 1 diverges, t 2 stays finite, t 1 = O(Γ 1 ) F R (E R,q) = Γ 0 2En V F(E n,q)+o(γ 1/2 ) A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.13
Conclusions, outlook Using non-relativistic EFT in a finite volume, a framework for the extraction of the B K l + l form factors from the lattice data is constructed The multi-channel Lüscher-Lellouch formula is reproduced. The extraction of the form factors at the resonance pole is carried out It is shown that, at the resonance, the photon virtuality q 2 must be complex The limit Γ 0 is studied in detail in the multi-channel case A. Rusetsky, 34th International Symposium on Lattice Field Theory, 24-30 July 2016, Southampton p.14