Status and Prospects for Lattice Computations of Nonleptonic and - PowerPoint PPT Presentation

Status and Prospects for Lattice Computations of Nonleptonic and Rare Kaon Decays Chris Sachrajda School of Physics and Astronomy University of Southampton Southampton SO17 1BJ UK NA62 Kaon Physics Handbook MITP 11 - 22 January 2016 Chris Sachrajda MITP, 12th January 2016 1

Munich June 2015, Outline of talk In June 2015, I gave a talk at a MIAPP workshop on New Directions in Lattice Flavour Physics with the following outline: Introduction 1 Status of RBC-UKQCD Collaboration’s calculations of K → ππ decay 2 amplitudes. ∗ Electromagnetic corrections to decay amplitudes. 3 Long-distance contributions to flavour changing processes 4 �� d 4 x d 4 y � f | T [ Q 1 ( x ) Q 2 ( y )] | i � . (i) K L - K S mass difference (and ǫ K ) (ii) (Rare kaon decays) ∗ RBC=Riken Research Center, Brookhaven National Laboratory, Columbia University; UKQCD = Edinburgh + Southampton. Chris Sachrajda MITP, 12th January 2016 2

Mainz, January 2016, Outline of talk Introduction 1 Rare Kaon Decays K → πℓ + ℓ − . 2 Rare Kaon Decays K + → π + ν ¯ ν . 3 Status of RBC-UKQCD Collaboration’s calculations of K → ππ decay 4 amplitudes. ∗ Electromagnetic corrections to decay amplitudes ⇒ Guido Martinelli’s Talk. 5 ∗ RBC=Riken Research Center, Brookhaven National Laboratory, Columbia University; UKQCD = Edinburgh + Southampton. Chris Sachrajda MITP, 12th January 2016 3

2. Rare Kaon Decays: K L → π 0 ℓ + ℓ − N.H.Christ, X.Feng, A.Portelli and C.T.Sachrajda, arXiv:1507.03094 Some comments from F .Mescia, C.Smith, S.Trine hep-ph/0606081 : Rare kaon decays which are dominated by short-distance FCNC processes, K → πν ¯ ν in particular, provide a potentially valuable window on new physics at high-energy scales. The decays K L → π 0 e + e − and K L → π 0 µ + µ − are also considered promising because the long-distance effects are reasonably under control using ChPT. They are sensitive to different combinations of short-distance FCNC effects and hence in principle provide additional discrimination to the neutrino modes. A challenge for the lattice community is therefore to calculate the long-distance effects reliably (and to determine the Low Energy Constants of ChPT). We, the RBC-UKQCD collaboration, are attempting to meet this challenge but will need the help of the wider kaon physics community to do this as effectively as possible. Chris Sachrajda MITP, 12th January 2016 4

K L → π 0 ℓ + ℓ − There are three main contributions to the amplitude: Short distance contributions: 1 F .Mescia, C,Smith, S.Trine hep-ph/0606081 H eff = − G F α V ∗ s γ µ d ) (¯ ℓγ µ ℓ ) + y 7 A (¯ s γ µ d ) (¯ ℓγ µ γ 5 ℓ ) } + h . c . √ ts V td { y 7 V (¯ 2 Direct CP-violating contribution. In BSM theories other effective interactions are possible. Long-distance indirect CP-violating contribution 2 A ICPV ( K L → π 0 ℓ + ℓ − ) = ǫ A ( K 1 → π 0 ℓ + ℓ − ) . The two-photon CP-conserving contribution K L → π 0 ( γ ∗ γ ∗ → ℓ + ℓ − ) . 3 (a) s − d − γ ,Z W u,c,t ν W W + + d u,c,t s − (b) (c) π 0 γ − γ + K L K S K L ε π 0 γ + Chris Sachrajda MITP, 12th January 2016 5

K L → π 0 ℓ + ℓ − cont. The current phenomenological status for the SM predictions is nicely summarised by: V.Cirigliano et al., arXiv1107.6001 � � 2 � � Im λ t � � Im λ t 10 − 12 × 15 . 7 | a S | 2 ± 6 . 2 | a S | Br ( K L → π 0 e + e − ) CPV = + 2 . 4 10 − 4 10 − 4 � � 2 � � Im λ t � � Im λ t 10 − 12 × 3 . 7 | a S | 2 ± 1 . 6 | a S | Br ( K L → π 0 µ + µ − ) CPV = + 1 . 0 10 − 4 10 − 4 λ t = V td V ∗ ts and Im λ t ≃ 1 . 35 × 10 − 4 . | a S | , the amplitude for K S → π 0 ℓ + ℓ − at q 2 = 0 as defined below, is expected to be O ( 1 ) but the sign of a S is unknown. | a S | = 1 . 06 + 0 . 26 − 0 . 21 . For ℓ = e the two-photon contribution is negligible. Taking the positive sign (?) the prediction is Br ( K L → π 0 e + e − ) CPV ( 3 . 1 ± 0 . 9 ) × 10 − 11 = Br ( K L → π 0 µ + µ − ) CPV ( 1 . 4 ± 0 . 5 ) × 10 − 11 = ( 5 . 2 ± 1 . 6 ) × 10 − 12 . Br ( K L → π 0 µ + µ − ) CPC = The current experimental limits (KTeV) are: Br ( K L → π 0 µ + µ − ) < 3 . 8 × 10 − 10 . Br ( K L → π 0 e + e − ) < 2 . 8 × 10 − 10 and Chris Sachrajda MITP, 12th January 2016 6

CPC Decays: K S → π 0 ℓ + ℓ − and K + → π + ℓ + ℓ − G.Isidori, G.Martinelli and P .Turchetti, hep-lat/0506026 We now turn to the CPC decays K S → π 0 ℓ + ℓ − and K + → π + ℓ + ℓ − and consider � d 4 x e − iq · x � π ( p ) | T { J µ T µ i = em ( x ) Q i ( 0 ) } | K ( k ) � , where Q i is an operator from the ∆ S = 1 effective weak Hamiltonian. EM gauge invariance implies that i = ω i ( q 2 ) � π ) q µ � q 2 ( p + k ) µ − ( m 2 T µ K − m 2 . ( 4 π ) 2 Within ChPT the low energy constants a + and a S are defined by � � 1 2 N V ∗ a = √ C 1 ω 1 ( 0 ) + C 2 ω 2 ( 0 ) + f + ( 0 ) C 7 V us V ud sin 2 θ W 2 where Q 1 , 2 are the two current-current GIM subtracted operators and the C i are the Wilson coefficients. ( C 7 V is proportional to y 7 V above). G.D’Ambrosio, G.Ecker, G.Isidori and J.Portoles, hep-ph/9808289 Phenomenological values: a + = − 0 . 578 ± 0 . 016 and | a S | = 1 . 06 + 0 . 26 − 0 . 21 . What can we achieve in lattice simulations? Chris Sachrajda MITP, 12th January 2016 7

Minkowski and Euclidean Correlation Functions The generic non-local matrix elements which we wish to evaluate is � ∞ dt x d 3 x � π ( p ) | T [ J ( 0 ) H ( x ) ] | K ( k ) � X ≡ −∞ � π ( p ) | J ( 0 ) | n � � n | H ( 0 ) | K ( k ) � � π ( p ) | H ( 0 ) | n s � � n s | J ( 0 ) | K ( k ) � � � = − i , i E K − E n + i ǫ E n s − E π + i ǫ n n s {| n �} and {| n s �} represent complete sets of non-strange and strange states. In Euclidean space we calculate correlation functions of the form � T b √ Z K e − E K | t K | √ Z π e − E π t π p , t π ) T [ J ( 0 ) H ( t x ) ] φ † C ≡ dt x � φ π ( � K ( t K ) � ≡ X E , 2 m K 2 E π − T a where X E = X E − + X E + and � π ( p ) | J ( 0 ) | n � � n | H ( 0 ) | K ( k ) � � 1 − e ( E K − E n ) T a � � = − and X E − E K − E n n � π ( p ) | H ( 0 ) | n s � � n s | J ( 0 ) | K ( k ) � � 1 − e − ( E ns − E π ) T b � � X E + = . E n s − E π n s Chris Sachrajda MITP, 12th January 2016 8

4-pt Euclidean Correlation Functions n S n J J K H K H π π t H t J t J t H t K t π t K t π - T a T b - T a T b In Euclidean space we calculate correlation functions of the form � T b √ Z K e − E K | t K | √ Z π e − E π t π p , t π ) T [ J ( 0 ) H ( t x ) ] φ † C ≡ dt x � φ π ( � K ( t K ) � ≡ X E , 2 m K 2 E π − T a where X E = X E − + X E + and � π ( p ) | J ( 0 ) | n � � n | H ( 0 ) | K � � 1 − e ( E K − E n ) T a � � = − X E − and E K − E n n � π ( p ) | H ( 0 ) | n s � � n s | J ( 0 ) | K � � 1 − e − ( E ns − E π ) T b � � = . X E + E n s − E π n s In practice we may need to modify the above formulae to recognise the discrete nature of the lattice. For E K > E n there are unphysical exponentially growing terms which need to be subtracted! This is a common feature in calculations of long-distance effects in Euclidean space. This requires the consideration of π , ππ and πππ intermediate states. Chris Sachrajda MITP, 12th January 2016 9

Removal of single pion intermediate state For illustration, I consider the kaon to be at rest. � 1 − e ( E K − E n ) T a � X E − = − � � π ( p ) | J ( 0 ) | n � � n | H ( 0 ) | K � n E K − E n We use two methods to remove the contribution from the single pion state. We determine the matrix elements � π | H | K � and � π | J | π � and the energies 1 from two and three-point correlations functions and then perform the subtraction directly. We add a term c S ¯ sd to the effective Hamiltonian, with c S chosen for each 2 momentum so that � π | H − c S ¯ sd | K � = 0 . The demonstration that the addition of a term proportional to ¯ sd does not change the physical amplitude can be found in our paper arXiv:1507.03094 . Chris Sachrajda MITP, 12th January 2016 10

Removal of the two-pion divergence ℓ p k q k − ℓ µ In the continuum, space-time symmetries protect us from two-pion intermediate states: � π ( p 1 ) | J µ | π ( p 2 ) π ( p 3 ) � = ǫ µνρσ p ν 1 p ρ 2 p σ 3 F ( s , t , u ) After integrating over the momenta of the two intermediate pions, the only independent vectors are k , p and ǫ γ and so the indices of the Levi-Civita tensor cannot be saturated. This still leaves lattice artefacts two-pion contributions ( ∝ a 2 ) amplified by the growing exponential factors. While we expect these to be very small (as is the case for ∆ m K ), this will have to be confirmed numerically. Chris Sachrajda MITP, 12th January 2016 11