Pos(LATTICE2018)266

PoS(LATTICE2018)266 . µ https://pos.sissa.it/ , ) b ( , and quark masses ) µ ( , s α , ) c ( , , Silvano Simula ) b ( in which these corrections have been in- decays. The separate definition of strong ) 2 3 ` µ K π ( Γ / ) 2 µ , Guido Martinelli ) K [email protected] b [email protected] ( , , )( Γ a [email protected] ( , , Francesco Sanfilippo ) d ( ∗ , Vittorio Lubicz ) ) e b ( )( a ( [email protected] are set equal in QCD and in QCD+QED in some renormalisation scheme and at some scale isospin-breaking effects and those due to electromagnetismadvocate requires conventions a based convention. on We define hadronic and schemes,hadronic in masses which for a example, chosen are setschemes set which of equal have hadronic been in largely quantities, QCD used and to date, in in QCD+QED. which the This renormalised is in contrast with Speaker. cluded. We also discuss the additionalnetic theoretical issues corrections which to arise semileptonic when including decays, electromag- such as The precision of lattice QCD computationsisospin-breaking of corrections, many including quantities has electromagnetism, reached mustis such be to a be precision included made that if in further extractingMaskawa progress matrix fundamental elements, information, from such experimental as measurements. thecluding radiative values We corrections of discuss in the Cabibbo-Kobayashi- leptonic framework and forment semileptonic in- of decays infrared of divergences. hadrons, including We the brieflythe treat- review first isospin numerical breaking results in for leptonic the decays and ratio present ∗ Dept. of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK Dip. di Matematica e Fisica, Università Roma Tre, Via della Vasca NavaleINFN, 84, Sezione I-00146 di Roma Tre, Via della Vasca Navale 84, I-00146 Rome, Italy Dip. di Fisica, Universita‘ di Roma “Tor Vergata" and INFN Sezione di Tor Vergata, Dipartimento di Fisica, Università di Roma La Sapienza and INFN Sezione di Roma, Copyright owned by the author(s) under the terms of the Creative Commons ) ) ) ) ) e a b c d c The 36th Annual International Symposium on22-28 Lattice July, 2018 Field Theory - LATTICE2018 Michigan State University, East Lansing, Michigan, USA. Attribution-NonCommercial-NoDerivatives 4.0 International License (CC BY-NC-ND 4.0). ( Via della Ricerca Scientifica 1, I-00133E-mail: Roma, Italy Nazario Tantalo ( Rome, Italy ( ( Piazzale Aldo Moro 5, 00185( Roma, Italy Christopher Sachrajda [email protected] [email protected] [email protected] Davide Giusti Radiative corrections to decay amplitudes inQCD lattice PoS(LATTICE2018)266 are (2.2) (2.3) (2.1) O bare quark can be made, 0 a , physical Christopher Sachrajda ,

and 0 m Ω ` c D , S s m , ` m 0 d 0 , m a 0 u a with this action and this is m + = O 4 kin ` S R ` and ∑ is made in the absence of QED. + + Ω s K A g m m S 0 0 . ] and present the ﬁrst numerical results for a a + 3 Ω

, m = is obtained by imposing that some dimensionful ]. Ongoing work in extending the framework to phys Ω f 2 m 0 , with “kin" and "m" labelling the corresponding 4 and so some choice of convention is necessary if s 3 S ` m [ a 1 g f R H ) , m 2 = m When performing QCD simulations in the 4-flavour 0 µ ) 0 Ω + K π a α ( m m ( we can, for example, choose the four 0 f kin Γ 0 s a O S / a however, what the QCD and QED contributions to g ) ) 2 = µ f ∑ 2 K 1% ( R ( + and the strong coupling Γ , O -baryon, takes its physical value: f 0 Ω π YM Ω m S m m 0 2 s 0 1 a g a are shifted by and the charged leptons = ) to be those for which the 4 dimensionless ratios: = H f 0 c 1 are the gluon and Maxwell actions respectively and the sums are over the m full R , ]. In order to make further progress in determining the CKM matrix elements A S 0 s 1 S m (QCD+QED) theory what one means by QCD and what one means by radiative , 0 d and m full , below indicate that these quantities are defined in QCD, i.e. without QED. The lattice corresponding to the chosen value of 0 u 0 0 m YM a a or better [ S ) In the The hugely impressive recent progress in lattice QCD computations has led to the determi- 1% ( the ratio of leptonic decay rates take their physical values. Thespacing superscript 0 on the barespacing masses and the subscript 0 on the lattice Of course different choices forleading the to quantities different used lattice to artefactsthe in determine masses predictions. the of hadrons Once QED corrections are included however, from these decays, isospin-breaking effects, including radiativerequires corrections, a must procedure be for included. handling This theof presence of the infrared finite-volume divergences, as effects well dueof as an to our understanding the proposal presence for of how this the might photon. be In achieved this [ talk we review the status corrections becomes convention dependent. To illustrate this consider the action for the full theory: semileptonic decays is reviewed with acase. discussion We of start the however, new with theoretical aseparate issues discussion strong which of isospin arise the breaking in convention-dependent effects question this from of those how due one to might radiative2. corrections. What is QCD? kinetic and mass terms. We imagine calculating some observable we wish to define the QCD and QED contributions separately. nation of many quantities,O including leptonic decay constants and semileptonic form factors, of Radiative corrections 1. Introduction where unambiguous. At the levelseparately of requires a definition.choice of Before bare quark explaining masses this, it mayCalculation be of instructive O to in the recalltheory without absence how QED, of the for each QED: valuemasses of ( flavours of the quarks quantity, e.g. the mass of the PoS(LATTICE2018)266 ) we (2.5) (3.1) (2.4) . α , which O decays in scheme at 2 ` π MS . and a 2 δ , ` Christopher Sachrajda , ) K ) no longer take their ) + 2 . s 0 ) α g a ( , , 2.2 α e ( 2 O ( = ), and the rate is given by ) take their physical values. f O ], and has been followed in µ a 2 ` + 2 K 5 p m 2.2 m ) m K are µ i i f − , ε QCD s i 1 g 1-4) in Eq.( , O i ≡ in Eq.( e 0 s 2 ` ) ( i a = g f p h m R i ) ( m ( a 2 0 K Flavour Physics Lattice Averaging Group µ Z K i a δ | , m s R where the u 2 g K − , 5 f , ) = ) γ e 0 s i 2 0 O µ ( | ε g a g δ γ 2 , π us ¯ ) is a particularly simple paradigm illustrating the s Z + 0 8 | V ( + | 1 f 0 0 = ( 3.1 h 2 F ( m 0 s G ) g K QCD phys i f ) so that the ratios 0 i µ R , a µ f O 0 s ) = , 0 ` = g 0 s m ¯ a , ν i g δ h 0 , − R ( 0 ` + f = ( f 0 m g → Z Z m full − i a = K f ( ) = ) = aO h Γ µ µ m ( ( With the inclusion of QED, in the hadronic scheme (which we advocate s f = Other ways of defining what the QED corrections are clearly possible. An g m O 2GeV. The four dimensionless ratios = is the contribution from the electromagnetic corrections and mass counterterms. The is the Fermi constant. Equation ( µ ]. In this scheme the renormalised coupling and masses are equal in QCD and QCD+QED is the bare coupling in QCD. FLAG has adopted such a definition in the 1 F O 0 s mass itself is changed, leading to a shift in the lattice spacing, δ g G (a). Without QED corrections all the QCD effects are contained in a single number, e.g. for Since hadronic masses are now calculated precisely in lattice simulations and their values are We now consider leptonic decays of pseudoscalar mesons illustrated for Having calibrated the lattice, imagine we wish to make a prediction for an observable Ω 1 decays it is the kaon decay constant 2 ` well known from experimental measurements,hadronic we schemes strongly to suggest define that whatmasses is it are derived meant is quantities by more which QCD. natural are By not to measured contrast, use directly the in3. renormalised experiments. couplings Radiative corrections and in leptonic decays where approach taken in much ofmeasured precision experimentally. flavour On physics. the On right-hand the side left-hand is the side fundamental is quantity a to quantity be which determined is (in a scale of physical values and we can write: Fig. K in some scheme and at some renormalisation scale define QCD by imposing exactlycorrections the now same shift conditions as the above hadronicadd for masses mass QCD used counterterms without for QED. the TheThe calibration QED and to compensate for this we a number of publications, including(FLAG) [ the reviews of the where indirect one has been proposed by Gasser, Rusetsky and Scimemi [ Radiative corrections Hadronic scheme(s): for illustration we take to be of mass dimension 1: where first term on the right-handtinuum side limit can as be does calculated the withindifference between QCD sum. QCD alone (defined Such and as a has above)are separation a and calculated the allows well directly, full us defined i.e. theory"? con- without to The taking answer isospintheory the breaking the and difference corrections between in question: calculations QCD. performed “What If in needed is the in the full the future,The the GRS scheme scheme: can be extended to higher orders in PoS(LATTICE2018)266 . E ∆ , the (3.2) k (b) for 1 ]). 1 [ ]. The first (b). 2 ud 1 ) and known V K / ℓ f − ¯ ℓ ν us , where the sub- , see Fig. ) V 0 γ Γ ` , leading to an inte- Christopher Sachrajda (although in practice ¯ ν k ` / , us 1 V )) → ∼ E K mesons in QCD. (b) one of the ∆ ( ( 1 γ 1 K Γ Γ + and 20 MeV which is possible with the is calculated in perturbation theory ) + ) to obtain ` pt 0 π (b) ¯ pt 0 ν . Γ ]; indeed any measurement of the lep- is the volume. The second term on the ` 3.1 ( Γ 6 − E ∞ V ∆ → → lim ,K V is sufficiently small for the structure depen- ; the result is presented in Ref. [ − K ( decays and obtain the ratio π E decays; the solid circle represents the insertion of 0 E ) + 2 ∆ ` 2 Γ ∆ pt 0 ` π (b). From this diagram we can see that the strong . Moreover, diagrams such as these are infrared Γ K 3 kaon and 1 K f − in lattice simulations, removing the need for a low value of and 0 1 2 Γ Γ ` ( ∞ K − → ¯ l ν lim ] and NA62 experiments. It is now very useful to rewrite the point-like V 11 , ) = is determined we can use ( where the cut-off E 10 K ∆ E f W ( . In order to construct a physical observable which is free of infrared ∆ 1 . In practice one might take 4 1 Γ and the meson and lepton propagators each k < / 2 + γ k 0 E / Γ 1 ¯ u (a) d,s ∼ , satisfies ) and also term(s) containing the non-perturbative QCD effects (here γ itself must be computed non-perturbatively in a lattice simulation since hard modes, − to be neglected E us 0 is the width calculated for a (a) Schematic diagram illustrating the leptonic decays of 1 denote the number of photons in the final state and the energy of the photon in the kaon V K Γ ,K , pt 0 − Γ π We are also developing techniques to compute At this stage, the observable we calculate is If we require a precision of better than about 1%, then radiative corrections must be included 1 scripts 0 rest-frame where right-hand side can be calculatedconvergent, in but contains perturbation a theory term directly proportional in to log infinite-volume. It is infrared dence of tonic decay rate will necessarily includeresolution of the the contributions detector. of We have real proposed photonsrates a with in method for lattice energies computing simulations, below radiative including the corrections the to handling decay of infrared divergences, as we now explain. energy resolution of the KLOE [ observable in the form term is also infrared convergent, since whenexample, the is virtual soft photon it in couples diagrams onlywhereas for to the charge of thewhich meson. do resolve the structure of the kaon, contribute to diagrams such as that in Fig. and one contributing diagram is sketched in Fig. diagrams contributing to the radiative corrections to kinematic factors. Thus once this case Figure 1: the effective weak Hamiltonian. and electroweak interactions no longerare factorise no and longer that contained the in non-perturbative the strong single interactions number Radiative corrections it is more useful to compare the rates for divergent, as can be seen byphoton simple propagator power counting; in thegrand region which of behaves small as photon 1 momenta divergences we must include real photons in the final state [ NA NA NA NA NA NA Error Error Error 0.60454 0.35336 0.52146 PoS(LATTICE2018)266 0.00073713 0.00043621 0.00061469 Value Value 0.945 Value 3.4005 5.5006 -2.3945 0.48734 -0.00904 -0.62217 -1.8029 y = m1 + m2 * M0 =m1+m2* y 0.23574 0.99028 y = m1 + m2 * M0 =m1+m2* y y = m1 + m2 * M0 =m1+m2* y 0.010591 0.0015021 R R R m1 m2 Chisq m1 m2 m1 m2 Chisq Chisq , π 3 ]. ) ` R 4 are r 0.0030 δ ( (3.4) (3.3) s and pt 0 using 0 ]. The β Γ C ) 7 and 2 − µ A40.20 K 0 0.0025 π R ( Γ π δ Γ R K are the decay δ π R K / δ R ) ) π δ , 2 0 0.0020 ( ]. using the relation K µ f 3 K π 2 A40.24 Christopher Sachrajda ( ) K 2 Γ / L , and for R 2 , a π 0.0015 , δ ) K . The coefficients With this definition of the π ` R . in Ref. [ K ..., δ ) R ) ` masses and coupling equal to A40.32 δ + 0.0010 r 16 ( ( ) + 0 ` L MS ˜ r 1 C π ( A40.40 ( 1 m 2 0.0005 0122 C . ! and 0 = 1.90 ~ 580 MeV ~ 320 MeV β π K µ ) 2 µ 2 M M − ` ) + m r m L ( = 0.0000 1 π − − , obtained using a range of quark masses, π

0.015 0.010 0.005 0.000

-0.005 -0.010 -0.015 m P 2 π

2 K δ R K π ( C m m R K R δ

log which we will use for illustration. Although the 4 δ ) ` 3 π will be presented in a paper currently in preparation. 3 K ` ¯ r ν ) m m 0.05 ( 2 − 0 ] showing the results from the individual ensembles as ` 2 µ ˜ = 1.90 = 1.95 = 2.10 β β β

C 4 phys π + s ) ) ( 0 0 π continuum limit continuum fit at fit at fit at ( ( K π Γ = m s ) + f f m ` 0.04 → r ( us ud 0 and 0 V V ¯ ) K C

2 and meson masses (as indicated) after the universal terms proportional to , i.e. independent of the structure of the meson beyond the value of µ = β K ) = 0.03 ) PDG ) ( L 2 2 is the mass of the final-state charged lepton Γ ( µ µ ` pt 0 = 1.90, L/a =40corr.) (FVE =1.90, L/a =24corr.) (FVE =1.95, L/a =32corr.) (FVE =1.95, L/a =48corr.) (FVE =2.10, L/a (GeV) π K ], where the unknown low-energy constants cancel in the ratio. β β β β m ud ( ( Γ 8 m [ Γ Γ universal ) 0.02 to regulate the zero-mode in the photon propagator in a finite volume [ and 21 have been subtracted. are L ( K L ) m ` / r / we present two plots from [ ( ` 0.01 are the physical masses and (in spite of the discussion in Sec. We present the explicit expressions for 1 0112 . 2 m . C 0 π and 1 ) , Left-hand plot: results for the correction = 1.90, L/a =20corr.) (FVE =1.90, L/a =24corr.) (FVE =1.90, L/a =32corr.) (FVE =1.90, L/a 2 ] = physical point physical β β β K − L L ` m and r / K , = . The calculation has been performed in the electro-quenched approximation. Full details, 1 0.00 ) π We are now generalising the framework developed for leptonic decays and described in Sec. In Fig. As a first numerical study of the method, we have computed the ratio We use QED ( ` β π m r computed in QCD, so that the leading structure-dependent finite-volume effects in 0.000 K

[

-0.005 -0.010 -0.015 O

(

R δ 0 R K ˜ f correction we can compare directlyδ with the same quantity obtained in chiral perturbation theory to semileptonic decays, such as where constants obtained in iso-symmetric QCD withthose the in renormalized the full QCD+QED theoryUsing numerous extrapolated twisted-mass to ensembles infinite we volume find and to the continuum limit [ well as the extrapolation tosubtraction the of the physical universal point terms) (left-hand ofand results plot) obtained and at the four different volume volumesand dependence at the (after the separate results same for masses 4. Radiative corrections in semileptonic decays volumes with the twisted-massto formulation the for physical the point. fermions. The Right-hand cross plot: corresponds Volume to dependence the of extrapolation results for Figure 2: Radiative corrections finite-volume effects then take the form: where of the twisted mass formulation of lattice fermions. We define the correction separately, obtained at the same log C PoS(LATTICE2018)266 ]. : 9 ν p (4.2) (4.1) , and 2 + , where ` ` dq p . π as well as / ` + ds , are physical = π µ 0 2 s 2 π q ! p d f 2 dq π ) dq ` / E / − M π and Γ decays. ∆ 2 corrections has not + K 2 , 2 ( − ds q Christopher Sachrajda 3 0 p d q 1 ` 2 L 2 K Γ / K d f = 2 M dq d q − + µ ` ) , are not calculable in lattice π K pt 0 p K Γ ds , where 2 π 2 ) + d 2 π + dq ℓ − q → p ¯ ν ℓ π ( (

+ B , ∞ 0 γ ) f → 2 lim V q and ( + + f ! ππ ` + 5 π pt 0 µ → q Γ ds 2 B 2 2 π d M dq 2 − q − 2 K ` π 0 0 M ¯ Γ ) ds K 2 2 2 d q ( decay amplitudes can be calculated. dq 0 f

ππ ∞ -decays, such as = B → i lim → ) V corrections depend on the derivatives of the form factors, K K which have a lower energy than that of the external pion-lepton pair. The = p L One of the diagrams contributing to radiative corrections to ( ` 3 / π K | and we follow the same procedure as for leptonic decays and write: Γ ds 2 u 2 2 d µ ) ` γ corrections do not depend on derivatives of the form factors with respect to the meson ¯ dq s p | L Figure 3: ) is the cut-off on the energy of the real photon. The infrared divergences cancel separately / + π π p E p ( ∆ π A second significant difference with leptonic decays is the presence of intermediate states in We have determined the integrands/summands which need to be input into the Poisson sum- In QCD the amplitude depends on two form factors h = ( ` π masses; such derivatives are not physical. The explicit calculation of the 1 quantities and in principle can behand determined the from 1 experiment or lattice calculations. On the other calculations, whereas the diagram of Fig. not just on theon form external factors kinematic themselves. variables other than This hadron is masses. a The general derivatives feature for amplitudes which depend yet been performed. mation formula to determine thedecays finite-volume corrections. is An that important the difference 1 with leptonic where in each of the two terms on the right-hand side. The number of such stateson depends the on volume. the choice The of problemstate, the of including kinematic intermediate those variables states containing with additional energies hadrons,of smaller is heavy than particularly mesons. severe that for of For semileptonic thestates much decays external to of handle the effectively. phase-space This therefor is appear analogous charmless to to two-body be the fact too that many the lighter amplitudes intermediate and CP-asymmetries unphysical contributions to the correlationtemporal function separation from of the such weak states Hamiltonian and growand the exponentially operators lepton. at with the The the sink presence which annihilate ofin the such the pion contributions calculation and of the long-distance need effects to and subtract is them a is manifestation a of general the feature Maiani-Testa theorem [ Radiative corrections main ideas presented above arenow also explain. applicable in this case, several new features arise which we When including radiative corrections, the natural observable to consider is s PoS(LATTICE2018)266 and 65 if we ) 95 ` ν L µ calculation γ ¯ ` )( s µ L γ ¯ u Christopher Sachrajda ab initio = ( -regularisation in which W 1 W O (2009) 627 Erratum: [Eur. Phys. J. (2006) 76 64 (2003) 97 doi:10.1140/epjc/s2003-01383-1 632 32 (2008) 413 doi:10.1143/PTP.120.413 (2011) 7 doi:10.1016/j.physletb.2011.04.038 6 120 , where for kaon decays . Up to now we have matched the operators in the 700 W 1 (1990) 585. doi:10.1016/0370-2693(90)90695-3 )) (1937) 54. doi:10.1103/PhysRev.52.54 2 O k 52 245 Z − W and are currently in the process of extending this to include M M 2 W ) (2018) no.7, 072001 doi:10.1103/PhysRevLett.120.072001 (2017) no.2, 112 doi:10.1140/epjc/s10052-016-4509-7 M α ( log ( 2 120 77 π O α k ( to facilitate the subtraction of the unphysical intermediate states? / + ` 2 1 W π s M [KLOE Collaboration], Eur. Phys. J. C [KLOE Collaboration], Phys. Lett. B and CKM i j V.L., G.M., S.S. and C.T. thank MIUR (Italy) for partial support under Con- 2 V q 2 indicates that the operator is to be evaluated in the F et al. et al. G √ , Eur. Phys. J. C W (2015) no.7, 074506 doi:10.1103/PhysRevD.91.074506 [arXiv:1502.00257 [hep-lat]]. = et al. 91 eff H doi:10.1016/j.physletb.2005.11.008 [hep-ex/0509045]. (2010) 703] doi:10.1140/epjc/s10052-009-1217-6, 10.1140/epjc/s10052-009-1177-x [arXiv:0907.3594 [hep-ex]]. [arXiv:1102.0563 [hep-ph]]. [arXiv:0804.2044 [hep-ph]]. C. Tarantino, Phys. Rev. Lett. (2017) no.3, 034504 doi:10.1103/PhysRevD.95.034504 [arXiv:1611.08497 [hep-lat]]. [arXiv:1711.06537 [hep-lat]]. [hep-ph/0305260]. Rev. D [arXiv:1607.00299 [hep-lat]]. For semileptonic decays, in addition to the renormalization, some important practical issues We are successfully developing and implementing a framework for the [9] L. Maiani and M. Testa, Phys. Lett. B [8] V. Cirigliano and H. Neufeld, Phys. Lett. B [4] D. Giusti, V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Simula, N. Tantalo and [5] J. Gasser, A. Rusetsky and I. Scimemi, Eur. Phys. J.[6] C F. Bloch and A. Nordsieck, Phys. Rev. [7] M. Hayakawa and S. Uno, Prog. Theor. Phys. [3] V. Lubicz, G. Martinelli, C. T. Sachrajda, F. Sanfilippo, S. Simula and N. Tantalo, Phys. Rev. D [2] N. Carrasco, V. Lubicz, G. Martinelli, C. T. Sachrajda, N. Tantalo, C. Tarantino and M. Testa, Phys. [1] S. Aoki [11] F. Ambrosino [10] F. Ambrosino the superscript lattice and W-regularisations at need to be investigated, including the subtractioncontributions of in the unphysical an (exponentially growing actual inphase-space computation. time) needed to This obtain precise requires determinations of acan the be phenomenological CKM imposed matrix understanding on elements; what of useful the cuts the photon propagator is the strong interactions. Acknowledgements tract No. PRIN 2015P5SBHT. G.M. alsoGrant acknowledges partial No. support from 267985 ERC Ideas “DaMeSyFla".L000296/1 Advanced and No. C.S. ST/P000711/1 was and partially by an supported Emeritus by FellowshipReferences from STFC the (UK) Leverhulme Trust. Grants No. ST/ are to determine the CKMthe matrix improvement elements of to the a calculation(s) precisiondecays of is is better the that renormalization. 1% The or effective so. Hamiltonian The for priority these for Radiative corrections 5. Future prospects of radiative corrections to leptonic and semileptonic decays. Such a framework is necessary