arXiv subjects
Publications and source records attributed to Matteo Fael.
We compute the matching coefficients between QCD and non-relativistic QCD for external vector, axial-vector, scalar and pseudo-scalar currents up to three-loop order. We concentrate on the non-singlet contributions and present precise numerical results with an accuracy of about ten digits. For the vector current the results from arXiv:1401.3004 are confirmed, increasing the accuracy by several orders of magnitude.
We compute the three-loop non-singlet corrections to the photon-quark form factors taking into account the full dependence on the virtuality of the photon and the quark mass. We combine the method of differential equations in an effective way with expansions around regular and singular points. This allows us to obtain results for the form factors with an accuracy of about eight to twelve digits in the whole kinematic range.
We compute the relation between the pole mass and the kinetic mass of a heavy quark to three loops. Using the known relation between the pole and the $\overline{\rm MS}$ mass we obtain precise conversion relations between the $\overline{\rm MS}$ and kinetic masses. The kinetic mass is defined via the moments of the spectral function for the scattering involving a heavy quark close to threshold. This requires the computation of the imaginary part of a forward scattering amplitude up to three-loop order. We discuss in detail the expansion procedure and the reduction to master integrals. For the latter analytic results are provided. We apply our result both to charm and bottom quark masses. In the latter case we compute and include finite charm quark mass effects. Furthermore, we determine the large-$β_0$ result for the conversion formula at four-loop order. For the bottom quark we estimate the uncertainty in the conversion between the $\overline{\rm MS}$ and kinetic masses to about 15 MeV which is an improvement by a factor two to three as compared to the two-loop formula. The improved precision is crucial for the extraction of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ at Belle II.
In these proceedings we discuss the relation between the kinetic and the on-shell schemes for the bottom and the charm quarks and present the methods for the calculation of the mass relation to higher orders in perturbative QCD. The bottom mass in the kinetic scheme is a pivotal input parameter in the inclusive determination of $|V_{cb}|$ from $B\to X_c \ell ν_\ell$ decays. By combining the relation between the kinetic and the on-shell mass with well-know results for the $\overline{\mathrm{MS}}$-on-shell conversion, we obtain a prediction for $m_b^\mathrm{kin}$ based on precise determinations of $\overline{m}_b(\overline{m}_b)$.
We discuss a practical approach to compute master integrals entering physical quantities which depend on one parameter. As an example we consider four-loop QCD corrections to the relation between a heavy quark mass defined in the $\overline{\rm MS}$ and on-shell scheme in the presence of a second heavy quark.
We compute corrections of order $α_s^3$ to the decay $b \to c \ell \barν$ taking into account massive charm quarks. In the on-shell scheme large three-loop corrections are found. However, in the kinetic scheme the three-loop corrections are below 1\% and thus perturbation theory is well under control. We furthermore provide results for the order $α_s^3$ corrections to $b \to u \ell \barν$ and the third-order QED corrections to the muon decay which will be important input for reducing the uncertainty of the Fermi coupling constant $G_F$.
We describe a method to numerically compute multi-loop integrals, depending on one dimensionless parameter $x$ and the dimension $d$, in the whole kinematic range of $x$. The method is based on differential equations, which, however, do not require any special form, and series expansions around singular and regular points. This method provides results well suited for fast numerical evaluation and sufficiently precise for phenomenological applications. We apply the approach to four-loop on-shell integrals and compute the coefficient function of eight colour structures in the relation between the mass of a heavy quark defined in the $\overline{\rm MS}$ and the on-shell scheme allowing for a second non-zero quark mass. We also obtain analytic results for these eight coefficient functions in terms of harmonic polylogarithms and iterated integrals. This allows for a validation of the numerical accuracy.
We compute two-loop corrections to the vector current matching coefficient involving two heavy quark masses. The result is applied to the computation of the $Υ(1S)$ decay width into an electron or muon pair. We complement the next-to-next-to-next-to-leading order corrections of Ref. arXiv:1401.3005 by charm quark mass effects up to second order in perturbation theory. Furthermore, we apply the formalism to $Γ(J/Ψ\to \ell^+\ell^-)$ and compare to the experimental data.
We compute three-loop corrections to the relation between the heavy quark masses defined in the pole and kinetic schemes. Using known relations between the pole and $\overline{\rm MS}$ quark masses we can establish precise relations between the kinetic and $\overline{\rm MS}$ charm and bottom masses. As compared to two loops, the precision is improved by a factor two to three. Our results constitute important ingredients for the precise determination of the Cabibbo-Kobayashi-Maskawa matrix element $|V_{cb}|$ at Belle~II.
We consider the on-shell mass and wave function renormalization constants $Z_m^{\rm OS}$ and $Z_2^{\rm OS}$ up to three-loop order allowing for a second non-zero quark mass. We obtain analytic results in terms of harmonic polylogarithms and iterated integrals with the additional letters $\sqrt{1-τ^2}$ and $\sqrt{1-τ^2}/τ$ which extends the findings from Ref. [1] where only numerical expressions are presented. Furthermore, we provide terms of order ${\cal O}(ε^2)$ and ${\cal O}(ε)$ at two- and three-loop order which are crucial ingrediants for a future four-loop calculation. Compact results for the expansions around the zero-mass, equal-mass and large-mass cases allow for a fast high-precision numerical evaluation.
We present in a full analytic form the partial widths for the lepton flavour violating decays $μ^\pm \to e^\pm e^+ e^-$ and $τ^\pm \to \ell^\pm \ell'^{+} \ell'^{-}$, with $\ell,\ell'=μ,e$, mediated by neutrino oscillations in the one-loop diagrams. Compared to the first result by Petcov in [1], obtained in the zero momentum limit $\mathcal{P}\ll m_ν \ll M_W$, we retain full dependence on $\mathcal{P}$, the momenta and masses of external particles, and we determine the branching ratios in the physical limit $m_ν\ll \mathcal{P} \ll M_W$. We show that the claim presented in [2] that the $τ\to \ell \ell' \ell'$ branching ratios could be as large as $10^{-14}$, as a consequence of keeping the $\mathcal{P}$ dependence, is flawed. We find rates of order $10^{-55}$, even smaller than those obtained in the zero momentum limit, as the latter prediction contains an unphysical logarithmic enhancement.
The increasing interest in the phenomenology of the Standard Model Effective Field Theory (SMEFT), has led to the development of a wide spectrum of public codes which implement automatically different aspects of the SMEFT for phenomenological applications. In order to discuss the present and future of such efforts, the "SMEFT-Tools 2019" Workshop was held at the IPPP Durham on the 12th-14th June 2019. Here we collect and summarize the contents of this workshop.
The Heavy Quark Expansion (HQE) has become an extremely powerful tool in flavor physics. For charm decays, where the expansion parameters $α_s(m_c)$ and $Λ_{\rm QCD}/m_c$ are bigger than for bottom decays, it remains to be seen if the HQE can be applied with similar success. Nevertheless, to make optimal use of the plethora of data already available and coming in the near future, a better understanding of HQE for charm decays is crucial. This paper discusses in detail how the HQE for charm decays is set up, what is the role of four-quark (weak annihilation) operators and how this compares to the well understood bottom decays. Subtleties concerning radiative corrections and the charm mass scheme are briefly discussed. An experimental study of the relevant HQE hadronic matrix elements will then show if the HQE expansion for charm converges well enough. Besides serving as an important cross check for inclusive $B$ decays, in the end, this study might open the road for inclusive $|V_{cs}|$ and $|V_{cd}|$ extractions.
The determination of $V_{cb}$ relies on the Heavy-Quark Expansion and the extraction of the non-perturbative matrix elements from inclusive $b\to c$ decays. The proliferation of these matrix elements complicates their extraction at $1/m_b^4$ and higher, thereby limiting the $V_{cb}$ extraction. Reparametrization invariance links different operators in the Heavy-Quark expansion thus reducing the number of independent operators at $1/m_b^4$ to eight for the total rate. We show that this reduction also holds for spectral moments as long as they are defined by reparametrization invariant weight-functions. This is valid in particular for the leptonic invariant mass spectrum ($q^2$), i.e. the differential rate and its moments. Currently, $V_{cb}$ is determined by fitting the energy and hadronic mass moments, which do not manifest this parameter reduction and depend on the full set of 13 matrix elements up to $1/m_b^4$. In light of this, we propose an experimental analysis of the $q^2$ moments to open the possibility of a model-independent $V_{cb}$ extraction from semileptonic decays including the $1/m_b^4$ terms in a fully data-driven way.
The Standard Model prediction for muon-electron scattering beyond leading order requires the inclusion of QCD contributions which cannot be computed perturbatively. At next-to- and next-to-next-to-leading order, they arise from one- and two-loop diagrams with hadronic vacuum polarization insertions in the photon propagator. We present their evaluation using the dispersive approach with hadronic $e^+e^-$ annihilation data and estimate their uncertainty. We find that these corrections are crucial for the analysis of future high-precision muon-electron scattering data, like those of the recently proposed MUonE experiment at CERN.
The Standard Model prediction for $μ$-$e$ scattering at Next-to-Next-to-Leading Order (NNLO) contains non-perturbative QCD contributions given by diagrams with a hadronic vacuum polarization insertion in the photon propagator. By taking advantage of the hyperspherical integration method, we show that the subset of hadronic NNLO corrections where the vacuum polarization appears inside a loop, the irreducible diagrams, can be calculated employing the hadronic vacuum polarization in the space-like region, without making use of the $R$ ratio and time-like data. We present the analytic expressions of the kernels necessary to evaluate numerically the two types of irreducible diagrams: the two-loop vertex and box corrections. As a cross check, we evaluate these corrections numerically and we compare them with the results given by the traditional dispersive approach and with analytic two-loop vertex results in QED.
These proceedings review the differential decay rates and the branching ratios of the tau and muon decays $τ\to \ell \ell' \ell' ν\barν$ (with $\ell,\ell'=μ,e$) and $μ\to e e e ν\bar ν$ in the Standard Model at NLO. These five-body leptonic decays are a tool to study the Lorentz structure of weak interactions and to test lepton flavour universality. They are also a source of SM background to searches for the lepton-flavour-violating decays $μ\to e e e$ and $τ\to \ell \ell' \ell'$. Even if the shift in the branching ratios induced by radiative corrections turns out to be small and of order 1% --- mainly due to a running effect of the fine structure constant --- locally in the phase space these corrections can reach the 5 - 10% level, depending on the applied cuts. We found for instance that in the phase space region where the neutrino energies are small, and the momenta of the three charged leptons have a similar signature as in $μ\to eee$ and $τ\to \ell \ell'\ell'$, the NLO corrections decrease the leading-order prediction by about 10 - 20%.
New physics realized above the electroweak scale can be encoded in a model independent way in the Wilson coefficients of higher dimensional operators which are invariant under the Standard Model gauge group. In this article, we study the matching of the $SU(3)_C \times SU(2)_L \times U(1)_Y$ gauge invariant dim-6 operators on the standard $B$ physics Hamiltonian relevant for $b \to s$ and $b\to c$ transitions. The matching is performed at the electroweak scale (after spontaneous symmetry breaking) by integrating out the top quark, $W$, $Z$ and the Higgs particle. We first carry out the matching of the dim-6 operators that give a contribution at tree level to the low energy Hamiltonian. In a second step, we identify those gauge invariant operators that do not enter $b \to s$ transitions already at tree level, but can give relevant one-loop matching effects.