2006 5/19QCD radiative corrections1 QCD radiative corrections to the leptonic decay of J/ψ Yu, Chaehyun (Korea University)

2006 5/19QCD radiative corrections2 Outline  Dimensional regularization  QCD radiative corrections to the leptonic decay of J/  QCD radiative corrections to the leptonic decay of J/ψ  Summary

2006 5/19QCD radiative corrections3 Dimensional regularization

2006 5/19QCD radiative corrections4 Introduction – Why we need loop correction  LHC era : QCD corrections are inevitable.  New physics through loop diagrams.  Basics of Quantum Field Theory

2006 5/19QCD radiative corrections5Infinity  Simple power conting Logarithmic divergent at as well as  Vacuum polarization Quadratic divergence (power counting) In fact, logarithmic divergence Infrared (IR) divergence Ultraviolet (UV) divergence

2006 5/19QCD radiative corrections6 Vertex correction Assume on-shell condition for external particles Looks impossible !

2006 5/19QCD radiative corrections7 Feynman parametrization by direct calculation by induction by repeated differentiation

2006 5/19QCD radiative corrections8 Shift of integration variable with Use by symmetry by symmetry μ=ν terms survive! Proportional to g μν Determine extra factor by multiplying both sides by g μν Effective mass

2006 5/19QCD radiative corrections9 Wick rotation - trick To avoid poles Divergent !

2006 5/19QCD radiative corrections10 UV convergent when d goes to 0 IR convergent when d >> 4 ⇒ go to convergent region calculate integrals calculate integrals come back to physical region come back to physical regionRegularization  Physical quantities should be finite.  How to treat the infinity → Regularization  Introducing UV cutoff : Cutoff regularization, Pauli-Villars regularization  Analytic continuation of integrals to complex d spacetime dimensions Dimensional regularization Dimensional regularization

2006 5/19QCD radiative corrections11Renormalization  After regularization, infinity hides behind Λ or 1/ε.  The initial Lagrangian is obtained from the classical theory.  Exactly the same in quantum theory?  Normalization of parameters in Lagrangian is changed when we calculate loop-corrections. Physical quantities can be finite by redefinition of parameters in Lagrangian, e.g. mass, charge, normalization of field. Or, we can think that the correct Lagrangian is obtained by adding corrections to the classical Lagrangian, order by order in perturbation theory, so that we keep definitions of mass, charge, normalization of fields → counterterms.

2006 5/19QCD radiative corrections12  Analytical continuation of S-matrix elements to d=4-2ε dimensions.  Physical situation is obtained for d=4.  Infinities of perturbation theory are represented as poles for d=4.  External momenta are still taken to be in 4 dimensions.  Internal loop momenta are taken to be in d dimensions. Dimensional regularization revisited ‘t Hooft-Veltman scheme (HV)

2006 5/19QCD radiative corrections13 Fermion : We have equal number of traces for fermion lines and we will come back to d=4. Scalar : momentum in the loop is taken to be in d dimensions Vector : polarization vector is also taken to be in d dimensions Internal propagator is proportional to g μν where μ,ν have d components. How to treat momenta and polarizatoin vectors? Note that one can have Tr(I)=f(d), but deviations of f(d) from f(4)=4 are never important for Ward identities.

2006 5/19QCD radiative corrections14 Dirac algebra in d dimensions

2006 5/19QCD radiative corrections15 Dirac algebra is scheme-dependent ! ‘t Hooft-Veltman DR scheme (HV) Four-dimensional helicity scheme (FD) Convention DR scheme (CDR)  All momenta and all polarization vectors are in d dimensions.  Momenta and polarization vectors of unobserved particles are d dimensionalal.  Those of observed particles are 4 dimensional.  Only momenta of unobserved particles are continued to d dimensions.  Momenta of observed particles and all polarization vectors are kept in 4 dimensions.

2006 5/19QCD radiative corrections16 What does it mean to regulate a theory?  Divergent Feynman integrals should be finite.  Ward identities of a theory should hold. Gauge invariance

2006 5/19QCD radiative corrections17 γ 5 in dimensional regularization  ε μνρσ is explicitly a four-dimensional tensor  hard to generalize in d dimensions.  hard to generalize in d dimensions.  γ 5 is also a four-dimensional object.  Two most common prescriptions Naïve dimensional scheme (NDR) ‘t Hooft-Veltman scheme (HVDR) Unique and well-defined ambiguous HVDR scheme allows to recover ABJ anomaly, but NDR does not without additional ad hoc prescriptions.

2006 5/19QCD radiative corrections18 Dimension of coupling  QCD Lagrangian in d dimensions dimensionless dimensionless in 4 dim. = d dimensions  Coupling could be dimensionless by introducing renormalization scale

2006 5/19QCD radiative corrections19 Gamma function Analytical continuation Euclidian space

2006 5/19QCD radiative corrections20 Beta function

2006 5/19QCD radiative corrections21 Beta function

2006 5/19QCD radiative corrections22 Spherical coordinates (Euclidean space) In 3 dimensions In d dimensions

2006 5/19QCD radiative corrections23 Fermion self energy One particle irreducible diagrams Two-point function

2006 5/19QCD radiative corrections24 Fermion self energy On-shell renormalization scheme  Pole of propagator : physical mass  Residue of pole : field renormalization constant  1PI is expanded near the physical mass  Denominator of propagator  Full propagator Mass shift Field renormailzation constant

2006 5/19QCD radiative corrections25  Possible tensor structure : g μν,q μ q ν  Ward identity :   No pole at q 2 =0 because unique source of such pole is single massless particle in intermediate state. Gluon self energy One particle irreducible diagrams Gluon propagator

2006 5/19QCD radiative corrections26  Exact propagator has a pole at q 2 =0  Gluon is massless at all orders.  Residue at q 2 =0  renormalization constant  Charge renormalization  At q 2 ≠0, Gluon self energy Color structure are omitted. Ward identity

2006 5/19QCD radiative corrections27 α s corrections to amplitude Field renormalization constant for each external quark line Field renormalization constant for each external gluon line Vertex corrections Charge renormalization, vertex correction

2006 5/19QCD radiative corrections28 Amplitude and cross section Tree level Square of amplitudes at next-to-leading order

2006 5/19QCD radiative corrections29 Infrared divergence  There would be still IR divergences after one-loop corrections.  Remaining IR divergence can be canceled by soft Bremsstrahlung. IR divergence at next-to-leading order Leading order + Bremsstahlung  Same order in α s  If k is soft, we cannot differentiate 2-body final states from 3-body final states. soft momentum k

2006 5/19QCD radiative corrections30 QCD radiative corrections to the leptonic decay of J/ QCD radiative corrections to the leptonic decay of J/ψ

2006 5/19QCD radiative corrections31  cc bound state  Heavy quarkonium  Most simple structure J/ψ meson Factorization : short-distance part (QCD) + long-distance part (wave function at the origin, nonperturbative)  Narrow width : easy to detect  Charge-conjugation symmetry forbids two-gluon final state.  Color conservation forbids one-gluon final state.  J/ψ can decay three gluons or two leptons via virtual photon  O(α s ) corrections to leptonic decay is instructive

2006 5/19QCD radiative corrections32 Diagrams up to the α s order

2006 5/19QCD radiative corrections33 Fermion self energy = C s Feynman gauge

2006 5/19QCD radiative corrections34 Scalar integrals  Define scalar integrals  Tensor integrals are reduced into scalar integrals using  Scaleless integrals are ignored in DR

2006 5/19QCD radiative corrections35 Quark field renormalization constant  Differentiate numerator  Differentiate denominator p 2 =m 2 is not imposed !  Quark field renomralization constant

2006 5/19QCD radiative corrections36 Vertex correction Momentum of J/ψ Relative momentum  We ignore relative momentum to preserve leading terms in v.  In tree level, (-iee c γ μ )

2006 5/19QCD radiative corrections37 Partial fraction expansion  Partial fraction expansion  Inversion of k  All integrals in three-point function are reduced to scalar integrals  Tensor integrals

2006 5/19QCD radiative corrections38 Vertex correction

2006 5/19QCD radiative corrections39 Direct calculation of scalar integrals

2006 5/19QCD radiative corrections40 Scalar integral reduction method  If f(k) is convergent, Translational invariance All scalar integrals are reduced to I 0

2006 5/19QCD radiative corrections41 UV and IR divergence  Scalar integral reduction mixes UV and IR divergences.  Tagging before reduction UV UV UV IR IR

2006 5/19QCD radiative corrections42  One-loop corrections One-loop corrections

2006 5/19QCD radiative corrections43Summary  Dimensional regularization is most convenient tool.  Scalar-integral reduction method is useful for leptonic decay of J/ψ.  UV and IR divergences cancel separately.