Davide Isospin-breaking corrections in Giusti Kaon decays on the lattice OUTLINE ▪ Motivations Current and future ▪ QCD + QED on the lattice: RM123 method status of the first-row CKM unitarity ▪ Leptonic decays of hadrons Amherst, MA, USA K + → µ + ν µ γ ( ) π + → µ + ν µ γ V us ( ) V ud 17 th May 2019 ▪ Future perspectives In collaboration with: V. Lubicz, G. Martinelli, C.T. Sachrajda, F. Sanfilippo, S. Simula and N. Tantalo
ISOSPIN-BREAKING EFFECTS Isospin symmetry is an almost exact property of the strong interactions Isospin-breaking effects are induced by: m u ≠ m d : O[ (m d -m u )/ Λ QCD ] ≈ 1/100 “Strong” “Electromagnetic” O( α em ) ≈ 1/100 Q u ≠ Q d : Since electromagnetic interactions renormalize quark masses the two corrections are intrinsically related Though small, IB effects play often a very important role (quark masses, Mn - Mp, leptonic decay constants, vector form factor) 2
Phenomenological motivations
Phenomenological motivations The determination of some hadronic observables in flavor physics has reached such an accurate degree of experimental and theoretical precision that electromagnetic and strong isospin-breaking effects cannot be neglected anymore 4
The determination of Vus and Vud The relevant processes are V us /V ud leptonic and semileptonic V us K/ π K and π decays π K ( ) ( ) Γ K + → ℓ + ν ℓ γ 2 M K + 1 − m ℓ ( ) 2 M K + 2 ⎛ ⎞ 2 ( ) V us f K = 2 1 + δ EM + δ SU 2 ( ) Γ π + → ℓ + ν ℓ γ ⎜ ⎟ ( ) ( ) ( ) 2 M π + ⎝ ⎠ V ud f π M π + 1 − m ℓ 2 ( ) 2 M K + ,0 5 ( ) = G F Γ K + ,0 → π 0, − ℓ + ν ℓ γ K 0 π − 0 K + ,0 ℓ + δ SU 2 ( ) ( ) 2 ( ) S EW 1 + δ EM K + ,0 π 0 2 192 π 3 C K + ,0 V us f + I K ℓ ( ) 5
Vus and Vud: experimental results ( ) ( ) Γ K + → ℓ + ν ℓ γ 2 M K + 1 − m ℓ ( ) 2 M K + 2 ⎛ ⎞ 2 ( ) V us f K = 2 1 + δ EM + δ SU 2 ( ) Γ π + → ℓ + ν ℓ γ ⎜ ⎟ ( ) ( ) ( ) ⎝ ⎠ 2 M π + V ud f π M π + 1 − m ℓ 2 K/ π ( ) 2 M K + ,0 5 ) = G F ( Γ K + ,0 → π 0, − ℓ + ν ℓ γ K 0 π − 0 K + ,0 ℓ + δ SU 2 ( ) ( ) 2 ( ) S EW 1 + δ EM K + ,0 π 0 2 192 π 3 C K + ,0 V us f + I K ℓ ( ) π K K 0 π − (0) = 0.21654(41) V V us f K f K ( ) ( ) = 0.21654 41 ( ) = 0.27599 38 u s = 0.2760(4) V s f + V us f + 0 u V ud f π V f π ud < 0.2% M. Moulson, arXiv:1704.04104 PDG 6
Vus and Vud: results from lattice QCD f K ± = f K 9 1 + δ SU 2 ( ) f π ± f π f K± / f π ± = 1.1932(19) Nf=2+1+1 f + (0) = 0.9706(27) Nf=2+1+1 f K± / f π ± = 1.1917(37) Nf=2+1 f + (0) = 0.9677(27) Nf=2+1 0.2% 0.3% 7
Electromagnetic and isospin-breaking effects Given the present exper. and theor. (LQCD) accuracy, an important source of uncertainty are long distance electromagnetic and SU(2)-breaking corrections ( ) ( ) Γ K + → ℓ + ν ℓ γ 2 M K + 1 − m ℓ ( ) 2 M K + 2 ⎛ ⎞ 2 ( ) V us f K = 2 1 + δ EM + δ SU 2 ( ) Γ π + → ℓ + ν ℓ γ ⎜ ⎟ K/ π ( ) ( ) ( ) ⎝ ⎠ 2 M π + V ud f π M π + 1 − m ℓ 2 ( ) 2 M K + ,0 5 ( ) = G F Γ K + ,0 → π 0, − ℓ + ν ℓ γ K 0 π − 0 K + ,0 ℓ + δ SU 2 ( ) ( ) 2 ( ) S EW 1 + δ EM K + ,0 π 0 π 2 192 π 3 C K + ,0 V us f + I K ℓ K ( ) At leading order in ChPT both δ EM and δ SU(2) can be expressed in For Γ Kl2 / Γ π l2 terms of physical quantities (e.m. pion mass splitting, f K /f π , …) 25% of error due to higher orders 0.2% on Γ Kl2 / Γ π l2 δ EM = − 0.0069 (17) M.Knecht et al ., 2000; V.Cirigliano and H.Neufeld, 2011 2 ⎛ ⎞ f K + f π + 25% of error due to higher orders ( ) δ SU 2 ( ) = − 1 = − 0.0044 12 ⎜ ⎟ ⎝ ⎠ 0.1% on Γ Kl2 / Γ π l2 f K f π J.Gasser and H.Leutwyler, 1985; V.Cirigliano and H.Neufeld, 2011 8 ChPT is not applicable to D and B decays
Electromagnetic and isospin-breaking effects Given the present exper. and theor. (LQCD) accuracy, an important source of uncertainty are long distance electromagnetic and SU(2)-breaking corrections ( ) ( ) Γ K + → ℓ + ν ℓ γ 2 M K + 1 − m ℓ ( ) 2 M K + 2 ⎛ ⎞ 2 ( ) V us f K = 2 1 + δ EM + δ SU 2 ( ) Γ π + → ℓ + ν ℓ γ ⎜ ⎟ K/ π ( ) ( ) ( ) ⎝ ⎠ 2 M π + V ud f π M π + 1 − m ℓ 2 ( ) 2 M K + ,0 5 ( ) = G F Γ K + ,0 → π 0, − ℓ + ν ℓ γ K 0 π − 0 K + ,0 ℓ + δ SU 2 ( ) ( ) 2 ( ) S EW 1 + δ EM K + ,0 π 0 π 2 192 π 3 C K + ,0 V us f + I K ℓ K ( ) For Γ Kl3 M. Moulson, arXiv:1704.04104 9 15
Unitarity of the CKM first-row V us f K ( ) = 0.27599 38 V ud f π ( ) = 0.21654 41 ( ) V us f + 0 2 ≡ V ud 2 + V us 2 + V ub 2 V u ( ) f K ± f π ± f + 0 V ud from 2 = 0.99884 53 2 = 0.99986 46 9 J. Hardy and ( ) ( ) ≈ 2.2 σ ≈ 0.4 σ V u V u I. S. Towner, 2016 2 = 0.99778 44 2 = 0.99875 37 C.-Y. Seng et al., ( ) ( ) ≈ 3.4 σ ≈ 5 σ V u V u 2018 10
Isospin-breaking effects on the lattice RM123 method
A strategy for Lattice QCD: The isospin-breaking part of the Lagrangian is treated as a perturbation + Expand in: α em m d – m u arXiv:1110.6294 arXiv:1303.4896 RM123 Collaboration 12
1 The (md-mu) expansion - Identify the isospin-breaking term in the QCD action 1 ) − 1 ⎡ ⎤ ( ( ) ( ) uu + dd ( ) uu − dd ∑ ∑ S m u uu + m d dd 2 m u + m 2 m d − m ⎡ ⎤ m = = = ⎢ ⎥ ⎣ ⎦ d u ⎣ ⎦ x x ˆ ( ) − Δ m uu − dd ( ) Ŝ = Σ x ( ū u- ƌ d) = ∑ ⎡ ⎤ m ud uu + dd = S 0 − Δ m S ⎣ ⎦ x Advantage: - Expand the functional integral in powers of Δ m factorized out 0 + Δ m O ˆ 0 1 + Δ m ˆ 0 + Δ m ˆ ( ) S − S S − S S O D φ O e D φ O e ∫ ∫ 1 st 0 + Δ m O ˆ ! ! 0 S O = = O 0 1 + Δ m ˆ 0 + Δ m ˆ 1 + Δ m ˆ ( ) S − S − S S S 0 D φ e D φ e ∫ ∫ 0 for isospin symmetry - At leading order in Δ m the corrections only appear in the valence-quark propagators: (disconnected contractions of ū u and ƌ d vanish due to isospin symmetry) 13
2 The QED expansion ! 2 - Non-compact QED: the dynamical variable is the gauge potential A � (x) − A µ ( x ) = 0 in a fixed covariant gauge ( ) ∇ µ ( p . b . c .) 1 ( ) A ( ) 2 S QED = 1 ∑ ∑ − ∇ µ + ν ( x ) −∇ µ = * ( k ) 2sin( k µ / 2) A ν ( x ) A A ν ( k ) ν 2 2 x ; µ ν k ; µ ν - The photon propagator is IR divergent � subtract the zero momentum mode - Full covariant derivatives are defined introducing QED and QCD links: e f U µ ( x ) q f ( x + ˆ − iaeA µ ( x ) ⎡ ⎤ + q f ( x ) = E µ ( x ) µ ) − q f ( x ) A µ ( x ) → E µ ( x ) = e D µ ⎣ ⎦ QCD QED + − i e A µ ( x ) = 1 − i e A µ ( x ) − 1/ 2 e - Since the expansion leads to: E µ ( x ) = e 2 ( x ) + … 2 A µ + counterterms 24 14
The QED expansion for the quark propagator In the electro-quenched approximation: � 15
The down- and up-quark mass difference QED Δ m ud QED QCD β =1.90, L/a=20 β =1.90, L/a=24 β =1.90, L/a=32 β =1.95, L/a=24 QCD /(m d -m u ) (GeV) QED β =1.95, L/a=32 β =2.10, L/a=48 M K + − M K 0 = 2.07(15) MeV ⎡ ⎤ 3.2 physical point ⎣ ⎦ QCD ⎡ ⎤ M 2 K 0 - M 2 3 ⎣ ⎦ K + and from the experimental value = 2.51(18) GeV m d − m u 2.8 + ] QCD K 2 M K + − M K 0 = − 6.00(15) MeV 0 - M ⎡ ⎤ 2.6 ⎣ ⎦ K 2 [M 2.4 All masses in MSbar at 2 GeV 0 0.01 0.02 0.03 0.04 0.05 m l (GeV) electro-quenched m u = 2.50(17) MeV approximation m d − m u = 2.38(18) MeV m d = 4.88(20) MeV RM123 Collaboration, arXiv:1704.06561 16
QED corrections to hadronic decays
QED corrections to hadronic decays In general the amplitudes are infrared divergent. On the lattice, a natural infrared cutoff is provided by the finite volume. But a delicate procedure to remove it is needed. A method to solve this problem is presented We consider the leptonic decay of a charged pseudoscalar meson, but the method is general (it can be used for semileptonic decays) N. Carrasco, V. Lubicz, G. Martinelli, C.T. Sachrajda, N. Tantalo, C. Tarantino, M. Testa PRD 91 (2015) 074506, arXiv:1502.00257 18
Leptonic decays at tree level Since the masses of the pion and kaon are much ℓ + q 1 W smaller than M W we use the effective Hamiltonian ν ℓ q 2 ( ) ν ℓ γ µ 1 − γ 5 ( ) H eff = G F ( ) q 1 ( ) ℓ q 2 γ µ 1 − γ 5 * V q 1 q 2 ℓ + q 1 2 ν ℓ q 2 1/ a ≪ M W This replacement is necessary in a lattice calculation, since 2 ⎛ ⎞ 2 2 M P ± m ℓ 2 ( ) = G F ) P ± → ℓ ± ν ℓ 2 1 − m ℓ ( 2 ( ) The rate is: ⎡ ⎤ Γ P ± tree 0 8 π V q 1 q 2 f P ⎜ ⎟ ⎣ ⎦ 2 ⎝ ⎠ M P ± In the absence of electromagnetism, the non-perturbative QCD effects are contained in a single number, the pseudoscalar decay constant ℓ + s ( ) ≡ 0 q 2 γ 4 γ 5 q 1 P ( ) = f P ( ) M P ( ) 0 0 0 0 A P K + ℓ + s ν ℓ K + u u ν ℓ In the presence of electromagnetism it is not even possible to give a physical definition of f P J. Gasser and G.R.S. Zarnauskas, PLB 693 (2010) 122 19
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