TOR VERGATA nazario tantalo nazario.tantalo@roma2.infn.it U N I V - PowerPoint PPT Presentation

TOR VERGATA nazario tantalo nazario.tantalo@roma2.infn.it U N I V E R S I T Y O F R O M E School of Mathematics, Physical and Natural Sciences CERN, 07-08-2019 QED corrections to hadronic decays on the lattice

outline • phenomenological relevance of QED radiative corrections 0.000 physical point β = 1.90, L/a = 40 (FVE corr.) continuum limit β = 1.90, L/a = 20 (FVE corr.) β = 1.95, L/a = 24 (FVE corr.) fit at β = 1.90 β = 1.90, L/a = 24 (FVE corr.) β = 1.95, L/a = 32 (FVE corr.) fit at β = 1.95 • QED radiative corrections on the lattice: β = 1.90, L/a = 32 (FVE corr.) β = 2.10, L/a = 48 (FVE corr.) fit at β = 2.10 -0.005 K π • a consistent definition of QED on a finite volume δ R phys m s = m s -0.010 PDG • a prescription to define QCD -0.015 0.00 0.01 0.02 0.03 0.04 0.05 m ud (GeV) • extraction of the physical observable from euclidean correlators (backup) 7 • infrared-safe observables and finite-volume effects 6 5 4 • non-perturbative calculation of the leptonic decay rates of pseudoscalar mesons at O ( α ) 3 2 • non-perturbative calculation of the radiative leptonic 1 decay rates of pseudoscalar mesons (if I have time) 0 0 0.1 0.2 0.3 0.4 • summary & outlooks

are QED radiative corrections phenomenologically relevant? FLAG, arXiv:1902.08191 PDG review, j.rosner, s.stone, r.van de water, 2016 ± ± v.cirigliano et al., Rev.Mod.Phys. 84 (2012) + = + + = + + FLAG average for FLAG average for + ETM 14E ETM 14E FNAL/MILC 14A FNAL/MILC 14A + = HPQCD 13A HPQCD 13A MILC 13A MILC 13A + ETM 10E ETM 10E ℓ FLAG average for = + FLAG average for = + = JLQCD 15C JLQCD 15C RBC/UKQCD 14B RBC/UKQCD 14B RBC/UKQCD 12A RBC/UKQCD 12A Laiho 11 Laiho 11 MILC 10 P − MILC 10 + MILC 10 MILC 10 + JLQCD/TWQCD 10 JLQCD/TWQCD 10 = RBC/UKQCD 10A RBC/UKQCD 10A = MILC 09A MILC 09A MILC 09A MILC 09A MILC 09 MILC 09 MILC 09 MILC 09 Aubin 08 ν ℓ ¯ Aubin 08 RBC/UKQCD 08 RBC/UKQCD 08 HPQCD/UKQCD 07 HPQCD/UKQCD 07 MILC 04 MILC 04 FLAG average for = FLAG average for = = = ETM 14D ETM 14D ETM 09 ETM 09 120 125 130 150 155 160 MeV • in the case of pions and kaons, QED corrections can be • from the last FLAG review we have calculated in χ -pt by estimating the relevant low-energy constants f π ± = 130 . 2(0 . 8) MeV , δ = 0 . 6% , δ QED Γ[ π − → ℓ ¯ ν ( γ )] = 1 . 8% , f K ± = 155 . 7(0 . 3) MeV , δ = 0 . 2% , δ QED Γ[ K − → ℓ ¯ ν ( γ )] = 1 . 1% , f + (0) = 0 . 9706(27) , δ = 0 . 3% δ QED Γ[ K → πℓ ¯ ν ( γ )] = [0 . 5 , 3]% • at this level of precision QED radiative corrections must be included!

are QED radiative corrections phenomenologically relevant? R(D*) ∆ χ 2 = 1.0 contours HFLAV average 0.4 LHCb15 BaBar12 0.35 σ 3 LHCb18 � � B �→ D ( ⋆ ) τ ¯ B ν τ 0.3 R ( D ( ⋆ ) ) = � � B �→ D ( ⋆ ) ℓ ¯ B ν ℓ Belle15 Belle19 0.25 Belle17 HFLAV 0.2 Average of SM predictions ± Spring 2019 R(D) = 0.299 0.003 ± R(D*) = 0.258 0.005 χ P( 2 ) = 27% 0.2 0.3 0.4 0.5 R(D) • presently a ∼ 3 σ discrepancy between SM-theory and experiment • the bulk of the hadronic uncertainties cancel in the ratios but QED radiative corrections are sensitive to the lepton mass and new hadronic quantities are needed at O ( α ) • QED effects are taken into account by using PHOTOS but it is not excluded that an improved treatment can have an impact, s.de Boer et al PRL 120 (2018), s.cal` ı et al 1905.02702 • the analysis of s.de Boer et al PRL 120 (2018) used what in the following is called the point-like effective theory

QED radiative corrections on the lattice including QED radiative corrections into a non-perturbative lattice calculation is a challenging problem! • QED is a long-range unconfined interaction that needs to be consistently defined on a finite volume (or maybe not. . . ) • from the numerical point of view, it is difficult to disentangle QED radiative corrections from the leading QCD contributions but, first of all, what is QCD? • as for any other observable on the lattice, QED radiative corrections have to be extracted from euclidean correlators (backup) • finite-volume effects are potentially very large , e.g. of O ( L − 1 ) in the case of the masses of stable hadrons • in the case of decay rates the problem is much more involved because of the appearance of infrared divergences, O (log( L )) , at intermediate stages of the calculation: the infrared problem !

QED on a finite volume: how? ¯ • it is impossible to have a net electric charge in a periodic box ψψ • this is a consequence of gauss’s law ¯ ψψ � 1 � � � � L 3 d 4 x F µν F µν + ¯ γ µ D f S = ψ f µ + m f ψ f ¯ ψψ 4 ¯ ψψ − ieq f ¯ ∂ k F 0 k ( x ) ψ f γ 0 ψ f ( x ) = 0 � �� � � �� � Ek ( x ) eρ ( x ) ¯ ψψ � � 1 ¯ L 3 d 3 x ρ ( x ) = L 3 d 3 x ∂ k E k ( x ) = 0 ψψ Q = e • one may think to overcome this problem by gauge fixing but large gauge transformations survive a local gauge fixing procedure ( n ∈ Z 4 ) xµnµ 2 πi � 2 πn µ µ Lµ ψ ( x ) �→ e ψ ( x ) , A µ ( x ) �→ A µ ( x ) + L µ xµnµ 2 πi � µ ψ ( x ) ¯ Lµ ψ ( x ) ¯ � ψ ( x ) ¯ ψ (0) �→ e ψ (0) , ψ (0) � = 0 , x � = 0

quenching the zero modes • in order to study charged particles in a periodic box it has been suggested long ago ( duncan et al. 96 ) to quench (a set of) the zero momentum modes of the gauge field , for example � �� � � e − S O D ψ D ¯ T L 3 d 4 x A µ ( x ) �O� = ψ D A µ δ pbc in space µ • by using this procedure one is also quenching large gauge transformations that are no longer a symmetry and charged particles can propagate • the assumption is that the induced modifications on the infrared dynamics of the theory should disappear once the infinite volume limit is taken • the point to note is that the resulting finite volume theory, although it may admit an hamiltonian description, is non-local m.hayakawa, s.uno Prog.Theor.Phys. 120 (2008) BMW, Science 347 (2015), Phys.Lett. B755 (2016) z.davoudi, m.j.savage PRD90 (2014) �� � � � L 3 d 4 x αµ ( t ) Aµ ( t, x ) � L 3 d 3 x A µ ( t, x ) D α µ ( t ) e − QED L : δ �→ pbc in space µ,t

gauge-invariant local theory on the finite volume b.lucini, a.patella, a.ramos, n.t, JHEP 1602(2016) • consider C ⋆ boundary conditions (first suggested by wise and polley 91 ) ψ f ( x + L k ) = C − 1 ¯ ψ T f ( x ) ψ f ( x + L k ) = − ψ T ¯ f ( x ) C U µ ( x + L k ) = U ∗ A µ ( x + L k ) = − A µ ( x ) , µ ( x ) , • the gauge field is anti-periodic ( | p | ≥ π/L ): no zero modes by construction! • this means no large gauge transformations and � � 1 L 3 d 3 x ρ ( x ) = L 3 d 3 x ∂ k E k ( x ) � = 0 Q = e • a fully gauge invariant formulation is possible : technically this is a consequence of the fact that the electrostatic potential is unique with anti-periodic boundary conditions (see backup) ∂ k ∂ k Φ( x ) = δ 3 ( x ) , Φ( x + L k ) = − Φ( x )

QED on a finite volume: many different approaches • QED ∞ : at any fixed order in α radiative corrections can be represented as the convolution of hadronic • QED L : very attractive for its formal simplicity ; correlators with QED kernels , e.g. generally, at O ( α ) the systematics associated with x.feng et al arXiv:1812.09817, LATTICE19 non–localities can be understood � L 3 d 4 x H L QCD ( x ) D L O ( L ) = γ ( x ) � • QED m : formally, the simplest way to solve the d 4 x H QCD ( x ) D γ ( x ) �→ problem in a local framework is to give a mass to the photon ; the L �→ ∞ limit must be taken before restoring gauge invariance ( m γ �→ 0 ) the subtle issue here is the parametrization of the long-distance tails of the hadronic part ; m.endres et al. PRL 117 (2016) m.della morte ALGT19 in fact the proposal is an extension of the spectacular applications of the convolution approach to the g µ − 2 , • QED C a local and fully gauge invariant solution , formally a bit cumbersome , flavour symmetries reduced to discrete subgroups (no spurious operator mixings though) and fully recovered in the infinite volume limit . . . , j.green et al, PRL 115 (2015), . . . a.gerardin ALGT19

QED on a finite volume: many different approaches • QED ∞ : at any fixed order in α radiative corrections can be represented as the convolution of hadronic • QED L : very attractive for its formal simplicity ; correlators with QED kernels , e.g. generally, at O ( α ) the systematics associated with x.feng et al arXiv:1812.09817, LATTICE19 non–localities can be understood � L 3 d 4 x H L QCD ( x ) D L O ( L ) = γ ( x ) � • QED m : formally, the simplest way to solve the d 4 x H QCD ( x ) D γ ( x ) �→ problem in a local framework is to give a mass to the photon ; the L �→ ∞ limit must be taken before restoring gauge invariance ( m γ �→ 0 ) the subtle issue here is the parametrization of the long-distance tails of the hadronic part ; m.endres et al. PRL 117 (2016) m.della morte ALGT19 in fact the proposal is an extension of the spectacular applications of the convolution approach to the g µ − 2 , • QED C a local and fully gauge invariant solution , formally a bit cumbersome , flavour symmetries reduced to discrete subgroups (no spurious operator mixings though) and fully recovered in the infinite volume limit . . . , j.green et al, PRL 115 (2015), . . . a.gerardin ALGT19 • which is the best approach?