K to ππ Decays on the Lattice Elaine Goode University of Southampton September 17, 2010 Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 1 / 23
Introduction Quantitative understanding of K → ππ decays has been an outstanding problem for over 50 years. Study is motivated by the CP violating parameter ǫ ′ /ǫ Also of interest is the ∆ I= 1 / 2 rule Need full non perturbative calculation of decay amplitudes - LQCD Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 2 / 23
Introduction Quantitative understanding of K → ππ decays has been an outstanding problem for over 50 years. Study is motivated by the CP violating parameter ǫ ′ /ǫ Also of interest is the ∆ I= 1 / 2 rule Need full non perturbative calculation of decay amplitudes - LQCD Must overcome problems such as two-particle final state and uncertain chiral extrapolation Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 2 / 23
Introduction Quantitative understanding of K → ππ decays has been an outstanding problem for over 50 years. Study is motivated by the CP violating parameter ǫ ′ /ǫ Also of interest is the ∆ I= 1 / 2 rule Need full non perturbative calculation of decay amplitudes - LQCD Must overcome problems such as two-particle final state and uncertain chiral extrapolation RBC/UKQCD Collaboration - Calculate K → ππ amplitude directly on the lattice To achieve this we use a large lattice, near physical pion mass and Lellouch Luscher factor for the two-particle final state Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 2 / 23
Effective Hamiltonian for weak decay Effective Hamiltonian describes weak interactions and effects of heavy quarks 10 H eff = G F � V i √ CKM C i ( µ ) Q i ( µ ) 2 i =1 C i are Wilson coefficients, Q i are 4-quark operators governing decay. Weak matrix element is 10 � ππ | H eff | K � = G F � V i √ CKM C i ( µ ) � ππ | Q i ( µ ) | K � 2 i =1 Scale dependence of C i and � ππ | Q i ( µ ) | K � must cancel. Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 3 / 23
Isospin Channels Two possible Isospin channels for K → ππ decays: I=0 & I=2 I=0 very difficult All 10 operators contribute ( → 48 different Wick contractions) Operator mixing Signal typically very noisy due to contribution from vacuum Present calculation on 16 3 × 32 lattices with 2+1 flavours of sea quark at unphysical masses ( m π = 420MeV). Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 4 / 23
Isospin Channels Two possible Isospin channels for K → ππ decays: I=0 & I=2 I=0 very difficult All 10 operators contribute ( → 48 different Wick contractions) Operator mixing Signal typically very noisy due to contribution from vacuum Present calculation on 16 3 × 32 lattices with 2+1 flavours of sea quark at unphysical masses ( m π = 420MeV). I=2 comparatively easy! Only three operators to consider (These are linear combinations of the Q i ). No mixing with lower dimensional operators Avoid problem of vacuum subtraction Current calculation on 32 3 × 64 lattices, L s = 32 with 2 + 1 sea quarks at near physical pion mass (valence m π = 145 . 6 MeV) Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 4 / 23
I=2 Operators Classify operators based on how they transform under SU (3) L × SU (3) R . Three operators to consider: Q ∆ I =3 / 2 uu ) L − (¯ � � = (¯ sd ) L (¯ dd ) L + (¯ su ) L (¯ ud ) L (27 , 1) Q ∆ I =3 / 2 uu ) R − (¯ � � = (¯ sd ) L (¯ dd ) R + (¯ su ) L (¯ ud ) R (8 , 8) Q ∆ I =3 / 2 u j u i ) R − (¯ s i d j ) L d j d i ) R s i u j ) L (¯ u j d i ) R � � (8 , 8) mx = (¯ (¯ + (¯ Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 5 / 23
I=2 We use the Wigner-Eckart theorem to relate K + → π + π 0 matrix element to the K + → π + π + matrix element This simplifies operators Q 3 / 2 (27 , 1) → Q ′ 3 / 2 (27 , 1) = (¯ sd ) L (¯ ud ) L The relative normalization is = 3 π + π 0 � � Q 3 / 2 � � K + � π + π + � � Q ′ 3 / 2 � � K + � � � 2 Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 6 / 23
I=2 Momentum For K to ππ decays in the CM frame the pions have non-zero momentum. Fitting to excited states is problematic and results are typically very noisy. One solution is to use twisted boundary conditions. Instead of using periodic boundary conditions in the spatial directions for quark field, choose boundary conditions which cause the quark field to change by a phase e i φ when going through the boundary. The spatial direction is twisted by an amount φ . Instead of momentum 2 π n L , the quark has momentum φ + 2 π n . L Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 7 / 23
I=2 Momentum Set φ = π so that the two-pion ground state has zero total momentum. Twist d-quark only resulting in kaon at rest and pions with equal and opposite momentum For on-shell kinematics we twist in two directions corresponding to √ p = 2 π/ L for each pion. Cosine source reduces number of inversions needed. s p cosine ( x ) = cos( p x x ) cos( p y y ) cos( p z z ) Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 8 / 23
I=2 Simulation Parameters RBC/UKQCD 32 3 × 64, L s = 32 lattices with Domain Wall Fermions and 2+1 dynamical quark flavours, generated on BG/P at ANL. Large lattice volume to accomodate two-particle final state: L = 4 . 51 fm. Simulate with m sea = 0 . 001, m val = 0 . 0001 m sea = 0 . 045 and l l s m val = 0 . 049. s Unitary pion mass ≈ 180 MeV. Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 9 / 23
I=2 Simulation Parameters RBC/UKQCD 32 3 × 64, L s = 32 lattices with Domain Wall Fermions and 2+1 dynamical quark flavours, generated on BG/P at ANL. Large lattice volume to accomodate two-particle final state: L = 4 . 51 fm. Simulate with m sea = 0 . 001, m val = 0 . 0001 m sea = 0 . 045 and l l s m val = 0 . 049. s Unitary pion mass ≈ 180 MeV. Inverse lattice spacing a − 1 = 1 . 4 GeV. Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 9 / 23
I=2 Details of calculation Analyse 47 configurations Combine propagators with P+A b.c. and P-A b.c. to double effective time extent of the lattice Light quark propagators have a source at t π = 0. S-quark has source at t k = 20, 24, 28 and 32 Operator inserted at all times inbetween. Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 10 / 23
I=2 Masses and Energies √ m π m K E ππ ( p = 0) E ππ ( p = 2 π/ L ) Lattice 0.10400(37) 0.3706(13) 2100(10) 0.3687(61) MeV 145.6(5) 519(2) 294(1) 516(9) Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 11 / 23
Extracting the Matrix Element Start with 3-point correlation function � � O ππ ( p , t π ) Q i ( t ) O † C K → ππ = K ( t K ) ∼ M Z ππ Z K exp( − E ππ | t − t π | ) exp( − m K | t − t K | ) with Z ππ = | � 0 | O ππ (0) | ππ � | and Z K = | � 0 | O K (0) | K � | The quotient C K ππ ( t ) M C K ( t K − t ) × C ππ ( t ) ∼ Z K Z ππ should be constant in time. Fit this to extract matrix element. Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 12 / 23
I=2 Matrix Element √ 2 π Quotient of Correlators for (8 , 8) operator. t K = 28, p π = L Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 13 / 23
Two Pion Phase Shift Simple relation exists which relates the two-pion final state energy to the S-wave phase shift (Luscher, M., nucl. Phys. B. 354, p. 531-578) q π = p π L n π − δ ( q π ) = φ ( q π ) , 2 π where π 3 / 2 q tan φ ( q ) = − Z 00 (1 , q 2 ) 1 ( n 2 − q 2 ) − s Z 00 ( s , q 2 ) = � √ 4 π n � m 2 π + p 2 p π determined from two-pion energy: E ππ = 2 π We found δ p π 20(3) MeV -0.44(17) degrees 213(5) MeV -19.6(5.6) degrees Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 14 / 23
Isospin Amplitude A 2 A 2 is physical quantitiy which can be compared with experiment A ( K 0 → ππ ( I = 2)) = A 2 e i δ 2 A 2 related to matrix element via √ � 3 1 ∂φ + ∂δ √ m K E ππ L 3 / 2 a − 3 G F V ud V us A 2 = √ π q π ∂ q π ∂ q π 2 2 � × C i ( µ ) Z ij ( µ ) � ππ | Q j | K � i , j C i ( µ ) are Wilson Coefficients Z ij operator renormalization constants calculated using NPR. In general operators will mix under renormalization. All quantities are real except for Wilson Coefficients which are complex. Elaine Goode (University of Southampton) K to ππ Decays on the Lattice September 17, 2010 15 / 23
Recommend
More recommend
