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

TOR VERGATA

QED corrections to hadronic decays on the lattice

• utline
• phenomenological relevance of QED radiative corrections
• QED radiative corrections on the lattice:
• a consistent definition of QED on a finite volume
• a prescription to define QCD
• extraction of the physical observable from

• infrared-safe observables and finite-volume effects
• non-perturbative calculation of the leptonic decay rates
• f pseudoscalar mesons at O(α)
• non-perturbative calculation of the radiative leptonic

• summary & outlooks
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• from the last FLAG review we have

• in the case of pions and kaons, QED corrections can be

• at this level of precision QED radiative corrections must be included!

• B → D(⋆)τ ¯

• B
• B → D(⋆)ℓ¯

• 0.2

• presently a ∼ 3σ 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

• QED effects are taken into account by using PHOTOS but it is not excluded that an improved treatment can have an

• the analysis of s.de Boer et al PRL 120 (2018) used what in the following is called the point-like effective theory

• QED is a long-range unconfined interaction that needs

• not. . . )
• from the numerical point of view, it is difficult to

• as for any other observable on the lattice, QED radiative

• finite-volume effects are potentially very large, e.g. of

• in the case of decay rates the problem is much more

• it is impossible to have a net electric charge in a periodic box
• this is a consequence of gauss’s law

• L3 d4x

• γµDf

• ψf
• ∂k F0k(x)
• Ek(x)

• eρ(x)

• L3 d3x ρ(x) =

• L3 d3x ∂kEk(x) = 0

• one may think to overcome this problem by gauge fixing but large gauge transformations survive a local gauge fixing

• in order to study charged particles in a periodic box it has been suggested long

• pbc in space

• µ

• T L3 d4x Aµ(x)
• e−S O
• by using this procedure one is also quenching large gauge transformations that are

• the assumption is that the induced modifications on the infrared dynamics of the

• the point to note is that the resulting finite volume theory, although it may admit an hamiltonian description, is non-local

• µ,t

• L3 d3x Aµ(t, x)
• →
• pbc in space

• L3 d4x αµ(t) Aµ(t,x)

• consider C⋆ boundary conditions (first suggested by wise and polley 91)

• the gauge field is anti-periodic (|p| ≥ π/L): no zero modes by construction!
• this means no large gauge transformations and

• L3 d3x ρ(x) =

• L3 d3x ∂kEk(x) = 0
• a fully gauge invariant formulation is possible: technically this is a consequence of the fact that the electrostatic

• QEDL: very attractive for its formal simplicity;

• QEDm: formally, the simplest way to solve the

• QEDC a local and fully gauge invariant solution,

• QED∞: at any fixed order in α radiative corrections

• L3 d4x HL

• d4x HQCD(x) Dγ(x)

• QEDL: very attractive for its formal simplicity;

• QEDm: formally, the simplest way to solve the

• QEDC a local and fully gauge invariant solution,

• QED∞: at any fixed order in α radiative corrections

• L3 d4x HL

• d4x HQCD(x) Dγ(x)

• which is the best approach?

• QEDL: very attractive for its formal simplicity;

• QEDm: formally, the simplest way to solve the

• QEDC a local and fully gauge invariant solution,

• QED∞: at any fixed order in α radiative corrections

• L3 d4x HL

• d4x HQCD(x) Dγ(x)

• which is the best approach?
• in my opinion this is not the relevant point: what really matters is that one must be able to estimate reliably the

• QED+QCD isospin breaking effects can be calculated by

• the numerical issue here are quark-disconnected diagrams

• one can also perform simulations of QED+QCD at all
• rders in e2 and eventually fit leading isospin breaking

• the numerical issue here is that the very small isospin

• in order to compare results for QED radiative corrections

• indeed, when electromagnetic interactions are taken into

• the QCD action is no longer expected to reproduce

• a natural matching prescription is to use again physical

• another prescription (j.gasser, a.rusetsky and i.scimemi, EPJ

• in RM123+SOTON, PRL 120 (2018), arXiv:1904.08731 we

• this will rapidly became an important issue on which we

• +

• 2

• +
• 2
• I’m now going to describe in some details the non-perturbative lattice calculation of the O(α) QED radiative corrections

• both the theoretical and numerical results discussed below are the outcome of a big effort of the RM123+SOTON

• the problem is much more involved w.r.t. the case of the spectrum because of the appearance of infrared divergences

• let’s consider the infrared-safe observable: at O(α) this is obtained by considering the real contributions with a single

• the finite-volume calculation of the real contribution is an issue: momenta are quantized!

• let’s consider the infrared-safe observable: at O(α) this is obtained by considering the real contributions with a single

• the finite-volume calculation of the real contribution is an issue: momenta are quantized!
• for this reason, by relying on the universality of infrared divergences, it is convenient to rewrite the previous formula as

• −Γpt

• let’s consider the infrared-safe observable: at O(α) this is obtained by considering the real contributions with a single

• the finite-volume calculation of the real contribution is an issue: momenta are quantized!
• for this reason, by relying on the universality of infrared divergences, it is convenient to rewrite the previous formula as

• Γpt

• + e2

• in the limit of very small photon energies ΓSD

• ur result for Γ[K− → µ¯

• with this method, our result for

• this can (remember the caveat concerning the definition
• f QCD) be compared with the result currently quoted

• 0.015
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• infrared divergences can be computed in the so called point-like effective theory

• −D2

• φP + fP
• 2iGF VCKM Dµφ†

• ,

• properly matched effective field theories have, by definition, the same infrared behaviour of the fundamental theory:

• 1 − r2

• infrared divergences can be computed in the so called point-like effective theory

• −D2

• φP + fP
• 2iGF VCKM Dµφ†

• ,

• properly matched effective field theories have, by definition, the same infrared behaviour of the fundamental theory:

• 1 − r2

• structure-dependent terms can also be understood in the effective field theory language, e.g.

• by exploiting the full set of constraints coming from the WIs and from the e.o.m one can rigorously show that in the

• let’s look a bit more in details to the master formula

• Γpt

• + e2

• let’s look a bit more in details to the master formula

• Γpt

• + e2

• concerning the point-like calculation in infinite volume, we have generalized the results obtained in the early days of

• let’s look a bit more in details to the master formula

• Γpt

• + e2

• concerning the point-like calculation in infinite volume, we have generalized the results obtained in the early days of

• concerning the real SD contribution, we have used χpt results, v.cirigliano and i.rosell, PRL 99 (2007), to show (see below for

• let’s look a bit more in details to the master formula

• Γpt

• + e2

• concerning the point-like calculation in infinite volume, we have generalized the results obtained in the early days of

• concerning the real SD contribution, we have used χpt results, v.cirigliano and i.rosell, PRL 99 (2007), to show (see below for

• concerning the point-like finite volume contribution we have calculated the universal infrared logs but also the O(L−1)

• k

• d3k

• dk0

• 1

• we have performed the lattice calculation by using the previously mentioned RM123 method, i.e. by expanding the

• by using this method we managed to obtain excellent numerical signals for the correlators corresponding to the

• we have computed non–perturbatively the required renormalization constants in the RI′-MOM scheme and matched

• we have performed the lattice calculation by using the previously mentioned RM123 method, i.e. by expanding the

• by using this method we managed to obtain excellent numerical signals for the correlators corresponding to the

• we have computed non–perturbatively the required renormalization constants in the RI′-MOM scheme and matched

• we have not computed the contributions corresponding to charged sea-quarks; this is the so called electroquenched

• ur result for Γ[K− → µ¯

• by defining

• our result are

• this can (remember the caveat concerning the definition
• f QCD) be compared with the result currently quoted

• at O(α) the systematics associated with the quenching of the zero modes can be understood, for example

• k

• d4x eikx T 0|jµ

• at O(α) the systematics associated with the quenching of the zero modes can be understood, for example

• k

• d4x eikx T 0|jµ

• the ultraviolet behaviour of this object can be understood by taking

• k

• at O(α) the systematics associated with the quenching of the zero modes can be understood, for example

• k

• d4x eikx T 0|jµ

• the ultraviolet behaviour of this object can be understood by taking

• k

• in the local theory the diagram has a logarithmic divergence (absent with a propagating W) that renormalizes GF ; the

• 1

• 0.015
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• I now move to the discussion of the non-perturbative

• as we have seen, in the region of small (soft) photon

• in the region of experimentally detectable (hard) photon

• in the case of light pseudoscalar mesons one can rely on

• in the case of heavy–light mesons nothing is known from

• the non-perturbative information needed to compute the

• d4y eik·y T0|jα

• − iFV

• FA +

• these can be expressed as functions of xγ (and of mP )

• the infrared divergent contribution (in red) is universal: it is

• for both light and heavy mesons (the plot on the left corresponds to the K while the one on the right to the Ds) we

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• in the case of light mesons (the plots correspond to the K) the structure-dependent form factors are very small and in

• remarkably we are able to cover the full kinematical range 0 ≤ xγ ≤ 1

• in the case of heavy mesons there is a strong enhancement
• f the structure-dependent form factors
• this can be understood by using the argument of d.becirevic et al

• in between the electromagnetic and the weak currents

P r e l i m i n a r y

• we are not alone in the world
• the RBC/UKQCD collaboration started a project to compute

• we are finalizing the phenomenological analysis of our data on

• QED radiative corrections are phenomenologically relevant

• in fact, if the precision is already at the percent level is useless

• including QED radiative corrections in a lattice simulation is a

• soft divergences and, more generally, large finite volume

• non-perturbative corrections have to be extracted from

• it is highly non trivial to probe electrically charged states

• f the theory

• the RM123+SOTON collaboration developed a method to

• with this method a log(L) divergence is turned into a 1/L2

• the RM123+SOTON collaboration provided the first

• and is going to provide soon phenomenologically relevant

• other collaborations started their work on the subject and this

• utlooks
• the calculation of the QED corrections to (radiative) leptonic

• cutoff effects are the problem there but strategies to cope with

• the problem is more challenging in the case of semileptonic

• nevertheless, the RM123+SOTON method can be extended

• the infrared divergence is again proportional to the leading
• rder decay rate (obvious) and the O(L−1) corrections are

• utlooks

∗

RC C R ∗

• besides being an attractive theoretical possibility, we have

• hadron masses can be computed in a fully gauge invariant and

• the RC⋆ collaboration has developed an open-source code,
• penQ*D, that allows to perform full-simulations of

backup material

• power-law finite volume effects arise when internal states can go
• n-shell, e.g.

• k

• d3k

• dk0

• power-law finite volume effects arise when internal states can go
• n-shell, e.g.

• k

• d3k

• dk0

• k

• d3k

• gO(p) + O(k)

• A

• power-law finite volume effects arise when internal states can go
• n-shell, e.g.

• k

• d3k

• dk0

• k

• d3k

• gO(p) + O(k)

• =

• 1

• ,

• power-law finite volume effects arise when internal states can go
• n-shell, e.g.

• k

• d3k

• dk0

• k

• d3k

• gO(p) + O(k)

• =

• 1

• ,

• n

• d3n

• 1

• k

• d3k

• dk0

• 1

• the key point of our method is the universality of infrared divergences
• to see how this works, let’s consider the contribution to the decay rate coming from the diagrams shown in the figure

• d4k

• infrared divergences (and power-law finite volume effects) come from the singularity at k2 = 0 of the integrand
• the tensor Lαµ is a regular function, it contains the numerator of the lepton propagator and the appropriate

• the hadronic tensor is a QCD quantity

• d4x eik·x T 0| Jα

• it satisfies the WIs coming from QED gauge invariance, e.g.

• and, given the kinematics of the process, it is singular only at the

• the hadronic tensor is a QCD quantity

• d4x eik·x T 0| Jα

• it satisfies the WIs coming from QED gauge invariance, e.g.

• and, given the kinematics of the process, it is singular only at the

• the singularity can be isolated by considering the point-like tensor, built in such a way to satisfy the same WIs of the full

• δαµ −

• ,

• the hadronic tensor is a QCD quantity

• d4x eik·x T 0| Jα

• it satisfies the WIs coming from QED gauge invariance, e.g.

• and, given the kinematics of the process, it is singular only at the

• the singularity can be isolated by considering the point-like tensor, built in such a way to satisfy the same WIs of the full

• δαµ −

• ,

• the structure dependent contributions are regular and, since there is no constant two-index tensor orthogonal to k,

• p · k δαµ − kαpµ

• at O(e2) with massive charged particles, singularities arise only at

• the blobs on the right are QCD vertexes, e.g.

• d4xd4ye−ipy−ikxT 0|P (y)jµ(x)P †(0)|0 ,

• d4ye−ipyT 0|P (y)P †(0)|0 ,

• gauge WIs constrain the first two terms in the expansion, e.g.

• at O(e2) with massive charged particles, singularities arise only at

• the blobs on the right are QCD vertexes, e.g.

• d4xd4ye−ipy−ikxT 0|P (y)jµ(x)P †(0)|0 ,

• d4ye−ipyT 0|P (y)P †(0)|0 ,

• gauge WIs constrain the first two terms in the expansion, e.g.

• ΓV − Γpt

• (L) = ΓSD

• once QCD has been defined, QED radiative corrections can be calculated directly or by expanding the lattice

• e−Sfull O
• e−Sfull

• e−SQCD

• e−SQCD

• the building-blocks for the graphical notation, used as a device to do calculations, are the corrections to the quark

• once QCD has been defined, QED radiative corrections can be calculated directly or by expanding the lattice

• e−Sfull O
• e−Sfull

• e−SQCD

• e−SQCD

• vacuum polarization effects are the numerical issue with our method

• it is always a good idea to address the issue of analytical

• the reason is that correlators (Schwinger’s functions)

• euclidean reduction formulae work straightforwardly
• nly for the lightest states, i.e. the leading exponentials

• problems arise when one is interested in processes

• the first step in a lattice calculation of a new observable

• the lightest state appearing in a correlator is readily

• d4q

• d3q

• M2 + |p − q|2

• M2 + |p|2

• M2 + |p − q|2

• M2 + |p|2

• problems of analytical continuation do arise in the case of

• the internal meson-lepton pair, and eventually

• this is a big issue, particularly in the case of B decays because
• f the presence of many kinematically-allowed multi-hadron

• problems of analytical continuation do arise in the case of

• the internal meson-lepton pair, and eventually

• this is a big issue, particularly in the case of B decays because
• f the presence of many kinematically-allowed multi-hadron

• the problem does not arise at the point (on the boundary of the

• in this particular kinematical configuration, by calling

• the starting point is the hadronic tensor (p2 = m2

• d4y eik·y T0|jα

• this can be conveniently decomposed in terms of form-factors as follows

• k2gµα − kµkα

• (p · k − k2)kµ − k2(p − k)µ

• (p · k − k2)gµα − (p − k)µkα

• gµα +

• the choice of the basis is of course not unique and, moreover, the separation of the point-like contribution can also

• in the case of real photons, k2 = 0, the previous expressions simplify as follows

• p · kgµα − (p − k)µkα

• gµα +

• the form factors H1,2 do not enter into the physical decay rate for P → ℓ¯

• the projector P µν(k, n) is such that

• in fact P µν(k, n) is nothing but the numerator of the photon propagator in the Coulomb’s gauge that forbids the

• −i

• p · kgµα − (p − k)µkα
• by introducing the polarization vectors as follows (that depend upon n and k)

• the projector P µν(k, n) can be rewritten in terms of the transverse polarization vectors ǫ1,2 as follows
• r=1,2

• explicit expressions for the transverse polarization vectors are given below

• k2

• k2

• k2

• k2

• k2

• in light of the previous discussion, one can either use the (formally) covariant expressions given above for P µν(k, n) or

• in particular, since the axial and vector part of the weak current can be computed separately, we have

• FA +

• + pα ǫr · p

• the infrared problem has been analyzed by many

• electrically-charged asymptotic states are not

• the perturbative expansion of decay-rates and

• the block & nordsieck approach consists in lifting

• bservables
• at any fixed order in α, infrared-safe observables

• in this framework, infrared divergences appear at

• 2 b.p.s

• 3 b.p.s

• d4k

• mP

• ,

• log
• mP

• + log

• = cIR log

• concerning the perturbative point-like calculation in infinite volume, we have generalized the results obtained in the early

• Γpt

• = Γ0

• 3 log
• m2

• + log(r2

• ,

• notice that ΓV (L) and Γpt

• the divergence can be reabsorbed into a renormalization of GF , both in the full theory and in the point-like effective

• we have analyzed the renormalization of the four-fermion weak operator on the lattice in details and calculated

• we have then matched the non-perturbative results to the so-called W-regularization at O(α) (a.sirlin, NPB 196 (1982);

• 1 +

• OW-reg

• i=1

• indeed, this is the scheme conventionally used to extract GF from the muon decay

• 1 −

• concerning the real structure dependent contributions, the relevant hadronic quantity is

• d4x eikx T 0|jµ

• by using the χpt results (v.cirigliano and i.rosell, PRL 99 (2007)) for these quantities, we have estimated the structure

• ΓR(mγ, E) − Γpt

• < 0.002

• concerning the real structure dependent contributions, the relevant hadronic quantity is

• d4x eikx T 0|jµ

• in the last part of the talk I will show the preliminary results of a fully non-perturbative calculation of the structure

• we performed an analytical calculation of Γpt

• where

• (1 + r2

• ,

• 2 log
• m2

• +

• +

• notice: the lepton wave-function contribution, Γℓℓ

• electrically charged states can be probed by considering (Dirac’s factor)

• d3y Φ(y−x)∂kAk(t,y)
• Θ(t,x)

• these interpolating operators are invariant under U(1) local gauge transformations

• d3y Φ(y−x)∂k∂kα(t,y) Θ(t, x) = e−iqf α(t,x) Θ(t, x)
• the gauge factor is not unique, for example one can consider

• for any consistent gauge-fixing condition one can build the Dirac factor that provides the unique gauge-invariant

• notice though: interpolating operators can be non–local in space but must be localized in time!

• in the compact formulation the path-integral is well

• by choosing an unconventional normalization for the

• x,µν

• 1 − Vµν(x)
• +

• x,µν
• 1 − Uµν(x)
• +
• f,x

• Dirac’s interpolating operators can then be implemented as analytical functions of the link variables, e.g.

• s=−xk

• s=0

• the mass of, say, the charged kaon can be extracted from the fully gauge invariant correlator
• x

• e−∆(L)t

• the numerical results presented in this

• after the inclusion of QED radiative

• 3 different lattice spacings with

• several sea quark masses and volumes