B K decays in a finite volume

Transcription

1 B K decays in a finite volume Akaki Rusetsky, University of Bonn In collaboration with A. Agadjanov, V. Bernard and U.-G. Meißner arxiv: , Nucl. Phys. B (in print) 34th International Symposium on Lattice Field Theory, July 2016, Southampton A. Rusetsky, 34th International Symposium on Lattice Field Theory, July 2016, Southampton p.1

2 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, July 2016, Southampton p.2

3 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, July 2016, Southampton p.3

4 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, July 2016, Southampton p.4

5 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, July 2016, Southampton p.5

6 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 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, July 2016, Southampton p.6

7 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 = 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, July 2016, Southampton p.7

8 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, July 2016, Southampton p.8

9 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) A. Rusetsky, 34th International Symposium on Lattice Field Theory, July 2016, Southampton p.9

10 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, July 2016, Southampton p.10

11 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, July 2016, Southampton p.11

12 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, July 2016, Southampton p.12

13 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, July 2016, Southampton p.13

14 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, July 2016, Southampton p.14