Physics of vacuum polarization ...
Lectures on the physics of vacuum polarization: from GeV to TeV scale
[email protected] www-com.physik.hu-berlin.de/ fjeger/QCD-lectures.pdf ∼
F. JEGERLEHNER
University of Silesia, Katowice
DESY Zeuthen/Humboldt-Universität zu Berlin
Lectures , November 9-13, 2009, FNL, INFN, Frascati
supported by the Alexander von Humboldt Foundaion through the Foundation for Polish Science and by INFN Laboratori Nazionale di Frascati
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
Outline of Lecture: ① Introduction, theory tools, non-perturbative and perturbative aspects
② Vacuum Polarization in Low Energy Physics: g 2 − ③ Vacuum Polarization in High Energy Physics: α(MZ ) and α at ILC scales
④ Other Applications and Problems: new physics, future prospects, the role of DAFNE-2
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
Memories to EURODAFNE, EURIDICE and all that
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
① Introduction, theory tools, non-perturbative and perturbative aspects
1. Intro SM
2. Some Basics in QFT
3. The Origin of Analyticity
4. Properties of the Form Factors
5. Dispersion Relations
6. Dispersion Relations and the Vacuum Polarization
7. Low Energies: e+e− hadrons → 8. High Energies: the electromagnetic form factor in the SM
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
1. Introduction
Theory is the Standard Model:
SU(3) SU(2) U(1) local gauge theory c ⊗ L ⊗ Y broken to SU(3) U(1) QCD ⊗ QED by the Higgs mechanism
Yang Mills 1954, Glashow 1961, Weinberg 1967, Gell-Mann, Fritzsch, Leutwyler 1973, Gross, Wilczek 1973, Politzer 1973, Higgs 1964, Englert, Brout 1964, ... Nobel Prize 1999: G. ’t Hooft, M. Veltman; 2003 Politzer, Gross, Wilczek for elucidating the quantum structure of electroweak interactions in physics and for asymptotic freedom
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
Constituents of matter: SU(3)c Spin 1/2 fermions color
siglets triplets➜ anti-triplets ν u u u ¯u ¯u ¯u ➜ SU(2) 1st family eL L L L L L L L ¯ ¯ ¯ eL− dL dL dL dL dL dL weak iso-spin doublets
sterile νs ➯ νeR uR uR uR ¯uR ¯uR ¯uR weak iso-spin singlets
¯ ¯ ¯ eR− dR dR dR dR dR dR
ν 2nd family µL cL cL cL ¯cL ¯cL ¯cL
µL− sL sL sL ¯sL ¯sL ¯sL
νµR cR cR cR ¯cR ¯cR ¯cR
µR− sR sR sR ¯sR ¯sR ¯sR
ν ¯ ¯ ¯ 3rd family τL tL tL tL tL tL tL ¯ ¯ ¯ τL− bL bL bL bL bL bL
ντR tR tR tR ¯tR ¯tR ¯tR
¯ ¯ ¯ τR− bR bR bR bR bR bR
Leptons Quarks
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
Carrier of Forces: Spin 1 Gauge Bosons Photon γ Octet of ⇔ 0 + W − Z W Gluons Vector Bosons “Cooper Pairs”: Higgs Ghosts ϕ0 + Spin 0 Higgs Boson ϕ− ϕ H Higgs Particle
t X c X u
Also note quark decay pattern:
b X s X d
Note: The u quark is stable, the s and b quarks are metastable. Flavor changing neutral currents (FCNC) at tree level X are forbidden by the GIM–mechanism. Flavor changing neutral current transitions are
allowed, however, as second (or higher) order transitions: e.g. b s is in fact b (u∗, c∗,t∗) s, where the → → → asterix indicates “virtual transition”.
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
Most mysterious SM mass spectrum. Fermions:
Quark mass hierarchy:
t X c X u
b X s X d
Lepton-neutrino mass hierarchy:
ντ νµ νe X X
e
τ X µ X e
e Hierarchy of quark (right top) and lepton masses (right bottom) [ log m in arbitrary units] ∼ f
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
SM tests:
❏ High Energy Frontier, e.g top discovery, triple gauge couplings [LEP,HERA,Tevatron,LHC]
❏ Rare Processes, e.g. µ eγ ( L ), neutrino physics [PSI,BNL,Tevatron,Kamiokande,Homestake,SNO] → 6 µ ❏ High Precision Frontier, e.g. m and m bound from LEP, g 2 [LEP,BNL,Harvard,ILC] t H − Many open questions in the Standard Model
Standard Model very good description
– Less satisfactory explanations
✗ Why 3 families of fundamental fermions ?
✗ Why P, C and CP violations? Why not more CP ? 6 ✗ Why us ? Origin of matter–antimatter asymmetry ?
✗ What Dark Matter ? Why the Cosmological Constant ?
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
❏ However: most extensions of the SM run into serious problems (FVNC’s, CP): if not tuned!!! (R–parity etc.) impose MFV scheme! (Isidori et al.)
❏ another as serious issue: accidental??? custodial symmetry ρ–parameter constraint! (Veltman,Lynn+Nardi,FJ,Czakon et al.)
Cries for new physics
but, please,
don’t touch at all what we have!
In general, SM instable against perturbations!
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
The Parameters of the Standard Model Thomson scattering in four fermion and vector boson processes − − α µ–decay pp,¯ e+e−
Gµ MW SU(2) U(1) L ⊗ Y Higgs mechanism unlike in QED and QCD in SM (SBGT) g2, g1, v parameter interdependence 2 sin Θ MZ ☞ W νe, νN pp,¯ e+e− only 3 independent quantities vf , af (besides fermion masses and mixing parameters) e+e− ff¯ α, Gµ, MZ → e+e− e+e− → ⇓ parameter relationships between very precisely measurable quantities precision tests, possible sign of new physics
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
Low Energy Parametrization and VB masses
QED and four fermion processes at low energy − − Finestructure constant: e.g. Thomson scattering • 1 α = ( 137.0359895(61) )− Fermi constant: µ–decay • 5 2 G = 1.166389(22) 10− GeV− µ × Higgs mechanism:
√ 1 ➤ √ GeV 2Gµ = v2 v = 1/ 2Gµ = 246.22 q Existence of the Higgs condensate verified!
γ Z mixing parameter: νN–scattering • − sin2 Θ = 0.231 0.006 W ± ➤ Masses of vector bosons determined
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
A0 MW πα MW = , MZ = ; A0 = sinΘW cosΘW √2Gµ M = 77.570 1.010 GeV M = 88.390q0.810 GeV W ± Z ± Born level prediction! M = 80.403 0.029 GeV M = 91.1876 0.0021 GeV W ± Z ± (UA2, CDF, LEP) (LEP, SLC)
Experimental result
MW MZ
76 78 80 82 84 86 88 90 92 94
GeV Born approximation “contradicts” data! ➤ Radiative corrections very important!
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
Test of quantum effects
Prototype QED: µ γγ γ γ µ form factors vacuum polarization anomalous magnetic Lamb shift → → moment LEP/SLC version of g 2: −
2 2 2 √2GµMZ sin Θcos Θ = πα (1 + δ) SM: renormalizable theory (’t Hooft 1971)
➤ δ uniquely calculable
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
in principle depends on unverified part of theory
(couplings and masses)
Gauge self couplings: (measured at LEP2)
+ W γ,Z W + γ,Z γ,Z W − W − Top quark: (discovered at Tevatron some time ago) ¯b t¯ W W γ,Z γ,Z
t t Higgs boson: (the hunting is going on) H W, Z W, Z H W, Z W, Z
δ ∆r =∆r( α,G , M , M ,m ,m , ) ≡ µ Z H t b ··· very precisely known | {z } F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
❖ MH unknown, bounds, free parameter (114 GeV < M < 186 GeV and / [160, 170] GeV) H ∈ ❖ mt rather precisely known now (m = 171.3 1.6 GeV) t ± Precision measurement of ∆r ➤ severe restrictions for “unknown physics” ➤ bounds on ∆rnew physics ?
Important fact: subleading radiative corrections (beyond running ∆α and ∆ρtop ) are
10 σ effects
2 f e.g. on precision observables like sin Θeff !!!
Impact for ∆α:
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
Interference between precision test, new physics contribution and hadronic uncertainty:
❖ ∆α = ∆αlep + ∆αhad
❖ ∆αhad has substantial non-perturbative part from low energy hadrons
❖ Uncertainty δ∆αhad does seriously limit precision of predictions! ❖ Mandatory goal for all kinds of precision tests: precise determination of effective fine structure “constant”
➠ Reduction of hadronic uncertainty crucial!
➧ Some remarkable achievements:
F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ...
TOP mass bound
ind +17 mt = 170 10 19 GeV LEP 1995 ± − with 60 GeV ± AFB leptons 0.23068 0.00055 Aτ 0.23240 ± 0.00085 ± Ae 0.23264 0.00096 ± AFB b-quark 0.23235 0.00040 ± AFB c-quark 0.23155 0.00111 < > ± QFB 0.23220 0.00100 ± ALR (SLD) 0.23055 0.00041 LEP + SLD 0.23151 ± 0.00022 250 χ2/dof = 15/6 ] 200 ± mZ = 91 186 2 MeV GeV [ t m 150 mH = 70 - 1000 GeV α-1 = 128.90 ± 0.09 100 0.228 0.23 0.232 0.234 2θlept sin eff TOP discovery at Tevatron 1995 mdir = 171.3 1.6 GeV CDF/D0 2009 ⇒ t ± F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... NC couplings ρℓ = 2g eff − Al 1 sin2 Θf = (1 g /g ) eff 4 − Vl Al -0.031 Preliminary -0.035 mH Vl mt g Al (SLD) -0.039 + − l l + − e e µ+µ− τ+τ− 68% CL -0.043 -0.503 -0.502 -0.501 -0.5 (LEP Electroweak Working Group: J. Alcaraz et al. 99) gAl F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Final 0,l A 0.23099 ± 0.00053 Indirect fb A (SLD) 0.23098 ± 0.00026 Higgs boson mass “measurement” l A (Pτ) 0.23159 ± 0.00041 +35 l mH = 87−26 GeV < > ± Qfb 0.2324 0.0012 CDF/D0 exclude 160-170 GeV 95% C.L. Preliminary 0,b A 0.23217 ± 0.00031 Direct lower bound: fb 0,c ± Afb 0.23206 0.00084 mH > 114 GeV at 95% CL Average 0.23148 ± 0.00017 Indirect upper bound: χ2/d.o.f.: 10.2 / 5 10 3 mH < 186 GeV at 95% CL ] GeV H [ W, Z W, Z H (5) 2 ∆α = 0.02761 ± 0.00036 m had 10 ± mZ= 91.1875 0.0021 GeV ± mt= 174.3 5.1 GeV 0.23 0.232 0.234 2θlept − H sin eff = (1 gVl/gAl)/4 W, Z W, Z (LEP Electroweak Working Group: D. Abbaneo et al. 03) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... 80.6 LEP1, SLD, νN Data − LEP2, pp Data 80.5 68% CL ] GeV [ 80.4 W m 80.3 [ ] mH GeV 95 300 1000 Preliminary 80.2 130 150 170 190 210 [ ] mt GeV (LEP Electroweak Working Group: J. Alcaraz et al. 99) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ Running parameters and the leading RC’s Like in QCD also in QED αQED is a running parameter. Most fundamental low energy parameters α = α(0), GF = Gµ very precise predictions of W and Z gauge boson properties, together with MZ and mt predict everything else. Large radiative corrections in α(E) while Gµ(MZ )= Gµ(0) with high accuracy (no running of VEV v =1/ √2G up to the electroweak scale v 246 GeV. µ ≃ + q e e−: renormalized (running) charge α (s) • QED e2 e2 e2(s)= → 1+(Π (s) Π (0)) γ′ − γ′ need photon vacuum polarization Πγ′ (s); hadrons at low energy contribute non–perturbative physics dramatic loss in precision! ⇒ ⇒ Gµ(E) µ–decay: Fermi constant G • µ G g2 1 µ = 0 + √2 8M 2 ΠW (0) ··· W 0 1 M 2 − W calculable subtraction (renormalization) constant ΠW (0) [mµ MW ] E ≪ MW F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Besides α(s) also the other gauge coupling α2(s) cannot be calculated in pQCD, also here no direct evaluation in terms of experimental data possible. Way out: isospin and flavor separation (FJ 1968) The shift ∆α2 in the g2 SU(2)L coupling α2 = 4π is analogous to ∆α 2 ∆α = Π′ (0) Π′ (M ) 2 3γ − 3γ Z 2 α2 MZ 5 (5) = Σl Ql (ln 2 )+∆α2,hadrons 12π | | ml − 3 where the sum extends over the light leptons and ∆α(5) (s) = 0.0567 0.0006 + 0.006184 ln(s /s)+0.005513 (s/s 1) 2,hadrons ± ·{ 0 · 0 − } is the hadronic contribution of the 5 known light quarks u, d, s, c, b (√s0 = 91.19 GeV). Appears in the relation 1 ∆α sin2 Θ = − 2 +∆ +∆κ sin2 Θ e 1 ∆α νµe,vertex+box e,vertex νµe − 2 lep 2 LEP sin θeff versus sin Θνµe of neutrino scattering Other huge correction in SM: violation of custodial symmetry (weak isospin breaking) showing up in the ρ–parameter: 2 . GNC MW √2Gµ 2 2 ρ = = 2 2 =1+∆ρ ; ∆ρ = 2 3 mt mb . Gµ MZ cos θW 16π | − | LEP’s m prediction Veltman’s Nobel prize → t F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Note on indirect mt measurement by LEP Another version of vacuum polarization effect: gauge boson propagators Leading top mass effect m2 in LEP process e+e− t∗t¯∗ ff¯ (f = t)? ∝ t → → 6 e+ f¯ Z t + + e− f t ··· Direct top corrections to e+e− ff¯ → In fact: ✕ 2 no mt effect ✓ is a renormalization effect, if very precisely known CC coupling Gµ used as an input ➠ 2 . mt effect via ρ = GNC/GCC =1+∆ρ from gauge boson self-energies ΠZ (0) ΠW (0) ∆ρ = 2 2 + sub leading terms MZ − MW ❏ ΠZ (0) and ΠW (0) UV divergent but ∆ρ finite ❏ for mt = mb one finds ∆ρ = 0! (tb–doublet contribution) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Veltman 1977 found that the doublet mass splitting m = m to which the W t 6 b self-energy is sensitive for large splitting becomes proportional to m2 m2 | t − b | √2G ∆ρ = µ 3 m2 m2 + sub leading terms 16π2 | t − b | which measures the weak isospin breaking of the doublet. self–energy correction W νµ − b µ + + W ν¯e t ··· e− − − Top corrections to µ e ν¯eνµ m2 → Note: G m2 t y2 µ t ∝ v2 ∝ t y top Yukawa coupling = mt ψ¯ ψ H t LYukawa − v t t −··· ❖ Leading top mass effect mediated by the longitudinal component of the W -field ❖ Slavonv-Taylor identities (expressing local gauge symmetry) ±µ ± in Feynman gauge reads ∂µW + MW ϕ = 0 ❖ W ± couples with gauge coupling g via the CC fermion current F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❖ charged unphysical Higgs ghosts ϕ± couple to fermions via the Yukawa couplings like mt/v or mb/v ±µ Calculation in unitary gauge no ghosts, W transversal i.e. ∂µW = 0 2 It looks like a mystery how terms proportional to (mt/v) end up in the numerator in place of the g2 (W tb¯ ) coupling and t and b masses in the denominators via the quark propagators. Sometimes calculating a Feynman diagram can end up with a real surprise (Wtb very different from Ztt,Zbb). In unitary gauge: Higgs H only scalar field (physical component) H = m ψ¯ ψ 1 + LYukawa − f f f v Xf LEP has been able to measure the Top-Higgs coupling ! in spite of the fact that the Higgs itself is not involved neither real nor virtual or do we see ghosts? F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Remember large SM quantum corrections: ❏ ∆α large due to large number of degrees of freedom (colored quarks,leptons) and large logs ❏ ∆ρ large single particle power correction, Note: very important novel feature of SM in contrast to QED and QCD 2 the large mt isospin splitting effect is a new feature in QFT ➠ such effect is forbidden in QED and QCD, where the usually anticipated Appelquist-Carrazone theorem holds In “normal” QFT Heavy particles decouple: O(µ/M) as M with µ =energy scale → ∞ or light mass (masses and couplings independent, masses in propagators only → damping effect) ➠ non-decoupling effects direct consequence of fact that in SM masses are generate by SSB (Higgs mechanism) ➠ SSB implies mass-coupling relation : in fact significant non-decoupling effect G m2 y2, y top Yukawa coupling is strong coupling effect µ t ∝ t t → ➠ first observed by Veltman, measured at LEP and triggered top discovery F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... 2. Some Basics in QFT Particle Physics is considered to be fundamental physics . What we mean is that we have some general principles which constrain the framework more than other physical theories. The most crucial step: Special Relativity + Quantum Mechanics = Quantum Field Theory Predicts matter antimatter! Science fiction turned out to be the reality! → A particularly predictive framework provided it is a renormalizable QFT. SM may be understood as kind of unique minimal renormalizable extension of QED + weak CC process (Fermi V-A). F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ Space-time symmetries = symmetries of the dynamics Quantum Mechanics Galilei invariance, unitary symmetry, Hilbert space Schrödinger [NP 1933] equation Heisenberg uncertainty relation[NP 1932] Pauli [NP 1933] equation ( -spin), exclusion principle SU(2) Quantum Field Theory Newton Mechanics rotations, space-translations Galilei invariance Special Relativity Dirac 1928 [NP 1933] Maxwell Electromagnetism SL(2,C) invariance Einstein 1905 [NP 1921] Wigner states + Lorentz [NP 1902] boosts relativistic kinematics etc. (see next) + time-translations ↑ ′ ↑ ′ invariance SL(2,C) + Lorentz/Poincaré invariance P+ ∼P h i F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ Quantum Field Theory: some general properties ❖ Particle-antiparticle crossing symmetry: ANTIMATTER predicted 1928 by Dirac [NP 1933] to each particle an antiparticle must exist charge conjugation C new symmetry ! electron positron [discovered 1932 by C. Anderson [NP 1936]] → ❖ Spin-statistics theorem: (Fierz 1939, Pauli 1940) fermions = 1 odd integer, bosons=integer spin (consequence of Einstein causality) 2 ❖ CPT theorem: (Schwinger 1951, Lüders, Pauli, Bell 1954) product of C, P and T in any order universal symmetry of any local relativistic QFT 1948 : Tomonaga, Schwinger and Feynman [NP 1965] QED renormalizable QFT to all orders in perturbation theory mathematically well-defined charge and mass as only free parameters [the ones in the Lagrangian], very predictive [Lamb shift, g 2, etc.] 20 years after Dirac! − Besides relativistic covariance, main problem Dirac’s Fermi sea [vacuum filled with electrons, positrons as holes in the sea] infinities physical; solution: QFT built on empty vacuum, positron are particles like → an electron related by a symmetry (C). Dogma of an empty vacuum! (quantum fluctuations only) ⇒ F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Scheme: classical local Lagrangian specified by symmetries + quantization rules in conjunction with scattering theory setup (expansion in free fields). Key object: the unitary scattering matrix S which encodes the perturbative interaction processes and is given by (0) i R d4x (x) S = T e Lint . ⊗ = omission of vacuum graphs (proper normalization of S) ⊗ Unitarity requires + + + 1 SS = S S =1 S = S− , ⇔ infers conservation of quantum mechanical transition probabilities Crucial: T –prescription = time ordering of all operators T φ(x)φ(y) = Θ(x0 y0)φ(x)φ(y) Θ(y0 x0)φ(y)φ(x) { } − ± − bosons Einstein locality (v < c) statistics: ❏ → ± fermions Implements scattering picture boundary condition ((LSZ reduction)): fields under the T product up to a possible sign (in case of Fermi statistics) always stand in the order of decreasing time arguments (from left to right). Fields to the left are associated with outgoing states those to the right are associated with incoming states, in relation to scattering matrix elements. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... The full Green functions of the interacting fields like Aµ(x), ψ(x), etc. can be expressed completely in terms of corresponding free fields via the Gell-Mann Low formula (interaction picture) e.g. electromagnetic vertex 0 T Aµ(x)ψα(y)ψ¯β(¯y) 0 = h | | i N 4 (0) n (0) (0) (0) i d x′ L (x′) i 4 4 0 T Aµ (x)ψα (y)ψ¯ (¯y)e int 0 ⊗ = d z1 d zn β R n! h | | i n=0 ··· n (0) (0) (0) (0) o(0) N+1 0 T Aµ (x)ψα (y)ψ¯ (¯y) (z ) (z ) 0P +RO(e ) h | β Lint 1 ··· Lint n | i⊗ n o (0) ➠ (x) the interaction part of the Lagrangian Lint ➠ on right hand side free fields only, everything known ➠ expanding exponential perturbation expansion, works if coupling small ⇒ ➠ formal series not well defined requires regularization and renormalization ➠ Feynman rules allow for straight forward evaluation. Note: path integral quantization ¯ Z J, χ,χ,¯ = V ψ ψe¯ i (Leff +JV +¯χψ+ψχ+··· ) { ···} D µiD D R Z F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... the generating functional of the time-ordered Green functions. Crucial for non-Abelian gauge theories like the SM (’t Hooft) no splitting into free and interaction part (non-gaugeinvariant splitting) Provides: ❏ non-perturbative definition of theory, basic for lattice QCD [discretized] ❏ relationship between Euclidean and Minkowski quantum field theory (Wick rotation) ❏ Osterwalder–Schrader theorem ascertains that the Euclidean correlation functions of fields can be analytically continued to Minkowski space ❏ full Minkowski QFT including its S–matrix, if it exists, can be reconstructed from the knowledge of the Euclidean correlation functions ❏ from a mathematical point of view the Minkowski and the Euclidean version of a QFT are completely equivalent, Special Relativity=O(4) rotation symmetry! Technical aspects: ➠ non-perturbative approach to QFT, ❖ transparent approach to quantization and renormalization of non-Abelian gauge theories F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ States vs Fields Invert field decomposition in terms of creation and annihilation operators Dirac field: a(~p, r) =u ¯(~p, r)γ0 d3x eipx ψ(x) , b+(~p, r) =v ¯(~p, r)γ0 d3x e−ipx ψ(x) . Z Z Photon field: ↔ µ∗ 3 ipx c(~p, λ) = ε (~p, λ)i d x e ∂ 0 Aµ(x) − Z Since these operators create or annihilate scattering states, the above relations provide the bridge between the Green functions, the vacuum expectation values of time–ordered fields, and the scattering matrix elements. S–matrix elements are the residues of the one–particle poles of the time–ordered Green functions. Each external field has 4 possible interpretations as a state: ❖ incoming particle ❖ outgoing particle ❖ incoming antiparticle ❖ outgoing antiparticle One analytic function several physical processes. This is QFT! F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Main consequences: ❖ particle–antiparticle crossing, C–invariance of QED and QCD ❖ spin–statistics theorem: integer spin bosons, half-integer spin fermions ∼ ❖ CPT–invariance of any QFT CP-violation is T [maximal P/ maximal C/ ] ⇒ ↔ What we finally need is T –matrix: S = I +i(2π)4 δ(4)(P P ) T . f − i with unity(no-scattering) split off and the overall four–momentum conservation factored out. Cross-sections 4 (4) (2π) δ (Pf −Pi) dσ = T 2 dµ(p′ )dµ(p′ ) 2 2 fi 1 2 2 λ(s,m1,m2) | | ··· q Life-times τ = 1/Γ: 4 (4) (2π) δ (Pf −Pi) 2 ′ ′ dΓ = Tfi dµ(p )dµ(p ) 2m1 | | 1 2 ··· etc. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... 3. The Origin of Analyticity At the heart of analyticity is the causality. The time ordered Green functions which encode all information of the theory in perturbation theory are given by integrals over products of causal propagators (z = x y) − iS (z) = 0 T ψ(x)ψ¯(y) 0 F h | | i = Θ(x0 y0) 0 ψ(x)ψ¯(y) 0 Θ(y0 x0) 0 ψ¯(y)ψ(x) 0 − h | | i− − h | | i 0 + 0 = Θ(z )iS (z)+Θ( z )iS−(z) − exhibiting a positive frequency part propagating forward in time and a negative frequency part propagating backward in time. The Θ function of time ordering makes the Fourier–transform to be analytic in a half–plane in momentum space. For K(τ = z0)=Θ(z0)iS+(z), for example, we have + + ∞ ∞ iωτ ητ iξτ K˜ (ω)= dτK(τ)e = dτK(τ)e− e Z Z0 −∞ such that K˜ (ω = ξ +iη) is a regular analytic function in the upper half ω–plane η > 0. This of course only works because τ is restricted to be positive, which means causal. In a relativistically covariant world, in fact, we always need two terms, a positive frequency part 0 + Θ(z = t t′) S (z), corresponding to the particle propagating forward in time, and a negative frequency part 0 − Θ( z = t′ t) S−(z), corresponding to the antiparticle propagating backward in time. The two terms − − correspond in momentum space to the two terms of exhibiting the simple poles. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Of course, for a free Dirac field we know what the Stückelberg-Feynman propagator in momentum space looks like q/ + m S˜ (q)= F q2 m2 +iε − and its analytic properties are manifest: as 1 1 1 = q2 m2 +iε q0 ~q 2 + m2 iε q0 + ~q 2 + m2 iε − − − − 1 1 1 = p p 2ω q0 ω +iε − q0 + ω iε p − p p − is an analytic function in q0 with poles at q0 = ( ω iε). Is is the basis for the Wick rotation to the Euclidean ± p − region Im q0 C ⊗ ⊗ ⊗ Re q0 ⊗ R Wick rotation in the complex q0–plane. The poles of the Feynman propagator are indicated by ’s. is an ⊗ C integration contour, R is the radius of the arcs. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... More precisely: analyticity of a function f˜(q0, ~q ) in q0 implies that the contour integral dq0 f˜(q0, ~q ) =0 (R) IC for the closed path C(R) vanishes. If the function f˜(q0, ~q ) falls off sufficiently fast at infinity, then the contribution from the two “arcs” goes to zero when the radius of the contour R . In this case we obtain → ∞ i ∞ − ∞ dq0 f˜(q0, ~q )+ dq0 f˜(q0, ~q )=0 Z +iZ −∞ ∞ or +i + ∞ ∞ ∞ dq0 f˜(q0, ~q )= dq0 f˜(q0, ~q )= i dqd f˜( iqd, ~q ) , − − Z Zi Z −∞ − ∞ −∞ which is the Wick rotation. At least in perturbation theory, one can prove that the conditions required to allow us to perform a Wick rotation are fulfilled. We notice that the Euclidean Feynman propagator obtained by the Wick rotation 1 1 q2 m2 +iε →−q2 + m2 − has no singularities (poles) and an iε–prescription is not needed any longer. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Analyticity is an extremely important basic property of a QFT and a powerful instrument which helps to solve seemingly purely “technical” problems as we will see. For example it allows us to perform a Wick rotation to Euclidean space and in Euclidean space a QFT looks like a classical statistical system and one can apply the methods of statistical physics to QFT. In particular the numerical approach to the intrinsically non–perturbative QCD via lattice QCD is based on analyticity. The objects which manifestly exhibit the analyticity properties and are providing the bridge to the Euclidean world are the time ordered Green functions. Note that by far not all objects of interest in a QFT are analytic. For example, any solution of the homogeneous (no source) Klein-Gordon equation ( 2 + m2 ) ∆(x y; m2)=0 , x − like the so called positive frequency part ∆+ or the causal commutator ∆ of a free scalar field ϕ(x), defined by < 0 ϕ(x),ϕ(y) 0 > = i∆+(x y; m2) | | − [ϕ(x),ϕ(y)] = i∆(x y; m2) , − which, given the properties of the free field, may easily be evaluated to have a representation + 2 3 4 0 2 2 ipz ∆ (z; m ) = i (2π)− d p Θ(p ) δ(p m )e− − − Z 2 3 4 0 2 2 ipz ∆(z; m ) = i (2π)− d p ǫ(p ) δ(p m )e− . − − Z F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Thus, in momentum space, as solutions of (p2 m2) ∆(˜ p)=0 , − only singular ones exist. For the positive frequency part and the causal commutator they read Θ(p0) δ(p2 m2) and ǫ(p0) δ(p2 m2) , − − respectively. The Feynman propagator, in contrast, satisfies an inhomogeneous (with point source) Klein-Gordon equation ( 2 + m2 )∆ (x y; m2)= δ(4)(x y) . x F − − − The δ function comes from differentiating the Θ function factors of the T product. Now we have 0 T ϕ(x),ϕ(y) 0 = i∆ (x y; m2) h | { }| i F − with 2 4 4 1 ipz ∆ (z; m )=(2π)− d p e− F p2 m2 +iε Z − and in momentum space (p2 m2) ∆˜ (p)=1 , − F obviously has analytic solutions, a particular one being the scalar Feynman propagator 1 1 = i πδ(p2 m2) . p2 m2 +iε P p2 m2 − − − − F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... The iε prescription used here precisely correspond to the boundary condition imposed by the time ordering prescription T in configuration space. The symbol denotes the principal value; the right hand side exhibits the P splitting into real and imaginary part. Note: ✓ 1/(p2 m2) makes sense in Minkowski and Euclidean region − ✕ while δ(p2 m2) 0 p2 < 0 does not − ≡ ∀ Note: Causality is fundamental in any physical theory (predictability of the future) [unlike in economics] i.e. also non-relativistic theories are causal ❏ non-relativistic causality ➠ analyticity in half-plain ❏ relativistic causality particle (forward) antiparticle (backward) yields analyticity in entire energy plane ❏ only in relativistic theory Wick rotation is possible (equivalence Euclidean vs Minkowski) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... 4. Properties of the Form Factors We again consider the interaction of a lepton in an external field: the relevant T –matrix element is T = e µ A˜ext(q) fi Jfi µ with µ µ µ µ =u ¯ Γ u = f j (0) i = ℓ−(p ) j (0) ℓ−(p ) . Jfi 2 1 h | em | i h 2 | em | 1 i By the crossing property we have the following channels: 2 2 ❖ Elastic ℓ− scattering: s = q =(p p ) 0 2 − 1 ≤ ❖ Elastic ℓ+ scattering: s = q2 =(p p )2 0 1 − 2 ≤ ❖ Annihilation (or pair–creation) channel: s = q2 =(p + p )2 4m2 1 2 ≥ ℓ 2 The domain 0 4 (4) i T ∗ T =Σ(2π) δ (P P ) T ∗ T , if − fi n − i nf ni Zn which derives from unitarity of S and the definition of the T –matrix, taking f S+S i and inserting a complete h | | i set of intermediate states, tells us that for s< 4m2 there is no physical state n allowed by energy and ℓ | i momentum conservation and thus 2 Tfi = Tif∗ for s< 4mℓ , F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ext which means that the current matrix element is hermitian. As the electromagnetic potential Aµ (x) is real its Fourier transform satisfies ext ext A˜ ( q)= A˜∗ (q) µ − µ and hence µ µ 2 = ∗ for s< 4m . Jfi Jif ℓ If we interchange initial and final state the four–vectors p1 and p2 are interchanged such that q changes sign: µ q q. The unitarity relation for the form factor decomposition of u¯ Π u thus reads (u = u(p ,r )) →− 2 γℓℓ 1 i i i q u¯ γµF (q2)+iσµν ν F (q2) u 2 E 2m M 1 q = u¯ γµF (q2) iσµν ν F (q2) u ∗ 1 E − 2m M 2 n+ µ+ 2 µν+ qν 2 o+ = u γ F ∗(q )+iσ F ∗ (q ) u¯ 2 E 2m M 1 µ 2 µν qν 2 =u ¯ γ F ∗(q )+iσ F ∗ (q ) u . 2 E 2m M 1 The last equality follows using u+ =u ¯ γ0, u¯+ = γ0u , γ+ = γ , γ0γ γ0 = γ , γ0γµ+γ0 = γµ 2 2 1 1 5 5 5 − 5 and γ0σµν+γ0 = σµν. Unitarity thus implies that the form factors are real 2 Im F (s)i =0 for s< 4me F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... 2 2 below the threshold of pair production s =4me. For s 4me the form factors are complex; they are analytic in ≥ 2 the complex s–plane with a cut along the positive axis starting at s =4me. In the annihilation channel (p = p2, p+ = p1) − − µ µ 0 jem(0) p , p+ =Σ 0 jem(0) n n p , p+ , h | | − i h | | ih | − i Zn + where the lowest state n contributing to the sum is an e e− state at threshold : E+ = E = me and − | i 2 ~p+ = ~p =0 such that s =4me. Because of the causal iε–prescription in the time–ordered Green functions − the amplitudes change sign when s s∗ and hence → Fi(s∗)= Fi∗(s) , which is the Schwarz reflection principle. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... 5. Dispersion Relations Causality together with unitarity imply analyticity of the form factors in the complex s–plane except for the cut along the positive real axis starting at s 4m2. Cauchy’s integral theorem tells us that the contour integral, for ≥ ℓ the contour shown in the Figure, satisfies C 1 ds′F (s′) Fi(s)= . 2πi s′ s IC − Im s R C Re s s|0 Analyticity domain and Cauchy contour for the lepton form factor (vacuum polarization). is a circle of radius R C 2 C with a cut along the positive real axis for s > s0 =4m where m is the mass of the lightest particles which can be pair–produced F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Since F ∗(s)= F (s∗) the contribution along the cut may be written as lim (F (s +iε) F (s iε))=2i Im F (s); s real , s> 0 ε 0 → − − and hence for R → ∞ ∞ 1 Im F (s′) F (s)= lim F (s +iε)= lim ds′ + . ε 0 ε 0 ∞ → π → s′ s iε C 4mZ 2 − − In all cases where F (s) falls off sufficiently rapidly as s the boundary term vanishes and the integral | | → ∞ C∞ converges. This may be checked order by order in perturbation theory. In this case the “un–subtracted” dispersion relation (DR) ∞ 1 Im F (s′) F (s)= lim ds′ ε 0 π → s′ s iε 4mZ 2 − − uniquely determines the function by its imaginary part. A technique based on DRs is frequently used for the calculation of Feynman integrals, because the calculation of the imaginary part is simpler in general. The real part which actually is the object to be calculated is given by the principal value ( ) integral P ∞ 1 Im F (s′) Re F (s)= ds′ , π P s′ s 4mZ 2 − which is also known under the name Hilbert transform. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... For our form factors the fall off condition is satisfied for the Pauli form factor FM but not for the Dirac form factor FE. In the latter case the fall off condition is not satisfied because FE(0) = 1 (charge renormalization condition = subtraction condition). However, performing a subtraction of F (0), one finds that (F (s) F (0))/s E E − E satisfies the “subtracted” dispersion relations ∞ F (s) F (0) 1 Im F (s′) − = lim ds′ , ε 0 s π → s′(s′ s iε) 4mZ 2 − − which exhibits one additional power of s′ in the denominator and hence improves the damping of the integrand at large s′ by one additional power. Order by order in perturbation theory the dispersion integral is convergent for the Dirac form factor. A very similar relation is satisfied by the vacuum polarization amplitude which we will discuss in the following section. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... 6. Dispersion Relations and the Vacuum Polarization − − + + − − − + + − + + + − − + + − − − + + + − + + r − − − + + − − − Vacuum polarization causing charge screening by virtual pair creation and re–annihilation Dispersion relations play an important role for taking into account the photon propagator contributions. The related photon self–energy, obtained from the photon propagator by the amputation of the external photon lines, is given by the correlator of two electromagnetic currents and may be interpreted as vacuum polarization for the following reason: F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... µ µ The em Ward-Takahashi identity (current conservation) in QED reds Zf Γ (p, p) on shell = eγ or | − − ′ µ µ eγ δZf + Γ (p, p) =0 − on shell − where the prime denotes the non–trivial part of the vertex fu nction. This relation tells us that some of the diagrams directly cancel. For example, we have (V = γ) V γ V + 1 + 1 = 0 2 2 V implies that in QED (not in the SM) charge renormalization is caused solely by the photon self–energy correction; the fundamental electromagnetic fine structure constant α in fact is a function of the energy scale α α(E) of → a process due to charge screening. The latter is a result of the fact that a naked charge is surrounded by a cloud of virtual particle–antiparticle pairs (dipoles mostly) which line up in the field of the central charge and such lead to a vacuum polarization which screens the central charge. This is illustrated in the figure. From long distances (classical charge) one thus sees less charge than if one comes closer, such that one sees an increasing charge with increasing energy. Diagrammatic representation of a vacuum polarization effect: F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... µ− γ virtual γ µ− µ− pairs µ− + + + ¯ γ∗ e e−,µ µ−, τ τ −,uu,d¯ d, γ∗ → ··· → Feynman diagram describing the vacuum polarization in muon scattering The vacuum polarization affects the photon propagator in that the full or dressed propagator is given by the geometrical progression of self–energy insertions iΠ (q2). The corresponding Dyson summation implies that − γ the free propagator is replaced by the dressed one µν ′ µν µν ig µν ig iDγ (q)= −2 iDγ (q)= 2 − 2 q +iε → q +Πγ (q )+iε modulo unphysical gauge dependent terms. By U(1)em gauge invariance the photon remains massless and 2 2 2 hence we have Π (q )=Π (0) + q Π′ (q ) with Π (0) 0. As a result we obtain γ γ γ γ ≡ µν ′µν ig iDγ (q)= 2 − 2 + gauge terms q (1+Πγ′ (q )) where the “gauge terms” will not contribute to gauge invariant physical quantities, and need not be considered further. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... 2 Including a factor e and considering the renormalized propagator (wave function renormalization factor Zγ) we have µν 2 ′ ig e Z i e2 D µν(q)= γ + gauge terms γ 2− 2 q 1+Πγ′ (q ) which in effect means that the charge has to be replaced by arunning charge 2 2 2 2 e Zγ e e (q )= 2 . → 1+Πγ′ (q ) The wave function renormalization factor Z is fixed by the condition that at q2 0 one obtains the classical γ → charge (charge renormalization in the Thomson limit. Thus the renormalized charge is e2 e2 e2(q2)= → 1+(Π (q2) Π (0)) γ′ − γ′ 2 where the lowest order diagram in perturbation theory which contributes to Πγ′ (q ) is γ γ f¯ f and describes the virtual creation and re–absorption of fermion pairs + + + γ∗ e e−, µ µ−,τ τ −, uu,¯ dd,¯ γ∗. → · · ·→ F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... e2 In terms of the fine structure constant α = 4π reads 2 α 2 α(q )= ; ∆α = Re Π′ (q ) Π′ (0) . 1 ∆α − γ − γ − The various contributions to the shift in the fine structure constant come from the leptons (lep = e, µ and τ ) the 5 light quarks (u, d, s, c, and b and the corresponding hadrons = had) and from the top quark: ∆α =∆α +∆(5)α +∆α + lep had top ··· 2 2 Also W –pairs contribute at q > MW . While the other contributions can be calculated order by order in (5) perturbation theory the hadronic contribution ∆ αhad exhibits low energy strong interaction effects and hence cannot be calculated by perturbative means. Here the dispersion relations play a key role. The leptonic contributions are calculable in perturbation theory. Using our result for the renormalized photon self–energy, at leading order the free lepton loops yield 2 ∆αlep(q )= α 5 yℓ √1 yℓ+1 = yℓ +(1+ ) √1 yℓ ln − 3π 3 2 √1 yℓ 1 ℓ=e,µ,τ − − − | − − | h i = P α 8 + β2 + 1 β (3 β2)ln 1+βℓ 3π 3 ℓ 2 ℓ ℓ 1 βℓ ℓ=e,µ,τ − − | − | P α h 2 2 5 2 2 i 2 2 = 3π ln q /mℓ 3 + O mℓ /q for q mℓ ℓ=e,µ,τ | | − | | ≫ 0.03142P for q2 = M 2 ≃ Z F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... where y =4m2/q2 and β = √1 y are the lepton velocitiesa. This leading contribution is affected by small ℓ ℓ ℓ − ℓ electromagnetic corrections only in the next to leading order. The leptonic contribution is actually known to three loops at which it takes the value 2 4 ∆α (M ) 314.98 10− . leptons Z ≃ × As already mentioned, in contrast, the corresponding free quark loop contribution gets substantially modified by low energy strong interaction effects, which cannot be calculated reliably by perturbative QCD. The evaluation of the hadronic contribution will be discussed later. Vacuum polarization effects are large when large scale changes are involved (large logarithms) and because of the large number of light fermionic degrees of freedom as we infer from the asymptotic form in perturbation theory 2 pert 2 α 2 q 5 2 2 ∆α (q ) Qf Ncf ln | 2| ; q mf . ≃ 3π mf − 3! | | ≫ Xf The figure illustrates the running of the effective charges at lower energies in the space–like region. Typical values are ∆α(5GeV) 3% and ∆α(M ) 6%, where about 50% of the contribution comes from leptons and ∼ Z ∼ ∼ aThe final result for the renormalized photon vacuum polarization for a lepton of mass m then reads 2 α 5 y Π′ (q )= + y − 2(1+ ) (1 − y) G(y) γ ren 3π 3 2 ff 2 2 1 √1 y+1 with and − . y =4m /q G(y)= 2√1 y ln √1 y 1 − − − F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... about 50% from hadrons. Note the sharp increase of the screening correction at relatively low energies. ∼ Shift of the effective fine structure constant ∆α as a function of the energy scale in the space–like region q2 < 0 (E = q2). The vertical bars at selected points indicate the uncertainty − − p Note alternative interpretation of VP: ❖ photon self–energy ❖ vacuum expectation value of the time ordered product of two electromagnetic currents: F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ⊗ ⊗ → One may represent the current correlator as a Källen-Lehmann representation in terms of spectral densities. To this end, let us consider first the Fourier transform of the vacuum expectation value of the product of two currents. b Using translation invariance and inserting a complete set of states n of momentum pn , satisfying the completeness relation d4p n Σ n n =1 (2π)3 | ih | Z Zn where Pn includes, for fixed total momentum pn, integration over the phase space available to particles of all b 0 NoteR that the intermediate states are multi–particle states, in general, and the completeness integral includes an integration over pn, since 0 2 2 pn is not on the mass shell pn 6= mn + ~pn. In general, in addition to a possible discrete part of the spectrum we are dealing with a continuum of states. p F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... possible intermediate physical states n , we have | i i d4x eiqx 0 jµ(x) jν (0) 0 h | | i Z 4 d pn 4 i(q pn)x µ ν = i d x e − Σ 0 j (0) n n j (0) 0 (2π)3 h | | ih | | i Z Z Zn d4p = i n Σ(2π)4 δ(4)(q p ) 0 jµ(0) n n jν(0) 0 (2π)3 − n h | | ih | | i Z Zn = i2π Σ 0 jµ(0) n n jν(0) 0 . h | | ih | | i|pn=q Zn In our case, for the conserved electromagnetic current, only the transversal amplitude is present and, taking into account the physical spectrum condition p2 0, p0 0 we may write the Källen-Lehmann representation ≥ ≥ i d4x eiqx 0 T jµ (x) jν (0) 0 h | em em | i Z ∞ 1 = dm2 ρ(m2) m2 gµν qµqν − q2 m2 +iε Z − 0 2 µν µ ν 2 = q g q q Πˆ ′ (q ) − − γ F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ˆ 2 2 c where Πγ′ (q ) up to a factor e is the photon vacuum polarization function introduced before : 2 2 ˆ 2 Πγ′ (q )= e Πγ′ (q ) . With this bridge to the photon self–energy function Πγ′ we can get its imaginary part by substituting 1 π i δ(q2 m2) q2 m2 +iε →− − − . Thus contracting with 2Θ(q0)g and dividing by g (q2 gµν qµqν )=3q2 we obtain µν µν − 0 ˆ 2 0 2 2Θ(q ) Im Πγ′ (q ) = Θ(q )2π ρ(q ) 1 µ = 2π Σ 0 j (0) n n jµ em(0) 0 . −3q2 h | em | ih | | i|pn=q Zn Again causality implies analyticity and the validity of a dispersion relation. In fact the electromagnetic current c In case of a conserved current, where ρ0 ≡ 0, we may formally derive that ρ1(s) is real and positive ρ1(s) ≥ 0. To this end we consider the element ρ00 ρ00(q) = Σ h0|j0(0)|nihn|j0(0)|0i Z q=pn n ˛ ˛ 0 2 = Σ |h0|j (0)|ni|q=pn ≥ 0 Zn 0 2 2 2 = Θ(q ) Θ(q ) ~q ρ1(q ) from which the statement follows. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... correlator exhibits a logarithmic UV singularity and thus requires one subtraction such that we find 2 ∞ Im 2 q Πγ′ (s) Π′ (q ) Π′ (0) = ds . γ − γ π s (s q2 iε) Z0 − − Here, we need the optical theorem which derives from the unitarity of S translated to T –matrix elements 4 (4) i T ∗ T =Σ(2π) δ (P P ) T ∗ T , if − fi n − i nf ni Zn and the optical theorem, is obtained from this relation in the limit of elastic forward scattering f i where | i→| i 2 Im T =Σ(2π)4 δ(4)(P P ) T 2 . ii n − i | ni| Zn Graphically, this relation may be represented by B, p2 B, p2 2 Im = 2 Im = n n X A, p A, p X A, p1 A, p1 Optical theorem for scattering and propagation. It tells us that the imaginary part of the photon propagator is proportional to the total cross section F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... + σ (e e− γ∗ anything) (“anything” means any possible state). The precise relationship reads tot → → 1 Im Πˆ ′ (s) = R(s) γ 12π 2 s + α(s) Im Π′ (s) = e(s) Im Πˆ ′ (s)= σ (e e− γ∗ anything) = R(s) γ γ e(s)2 tot → → 3 where 4πα(s)2 R(s)= σ / . tot 3s + + The normalization factor is the point cross section (tree level) σ (e e− γ∗ µ µ−) in the limit µµ → → s 4m2 . Taking into account the mass effects the R(s) which corresponds to the production of a lepton pair ≫ µ reads 4m2 2m2 R (s)= 1 ℓ 1+ ℓ , (ℓ = e,µ,τ) ℓ − s s r which may be read of from the imaginary part. This result provides an alternative way to calculate the renormalized vacuum polarization function, namely, via the DR which now takes the form 2 ′ℓ 2 αq ∞ Rℓ(s) Πγ ren(q )= ds 2 3π 4m2 s(s q iε) Z ℓ − − yielding the vacuum polarization due to a lepton–loop. In contrast to the leptonic part, the hadronic contribution cannot be calculated analytically as a perturbative series, F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... + but it can be expressed in terms of the cross section of the reaction e e− hadrons, which is known from → experiments. Via 2 + 4πα(s) R (s)= σ(e e− hadrons)/ . had → 3s we obtain the relevant hadronic vacuum polarization 2 ′had 2 αq ∞ Rhad(s) Πγ ren(q )= ds 2 . 3π 2 s(s q iε) Z4mπ − − At low energies, where the final state necessarily consists of two pions, the cross section is given by the square of the electromagnetic form factor of the pion (undressed from VP effects), 3 1 4m2 2 R (s)= 1 π 1 F (0)(s) 2 , s< 9 m2 , had 4 − s | π | π which directly follows from the corresponding imaginary part of the photon vacuum polarization. There are three differences between the pionic loop integral and those belonging to the lepton loops: ❖ the masses are different ❖ the spins are different ❖ the pion is composite – the Standard Model leptons are elementary The compositeness manifests itself in the occurrence of the form factor Fπ(s), which generates an enhancement: at the ρ peak, F (s) 2 reaches values about 45, while the quark parton model would give about 7. The | π | F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... remaining difference in the expressions for the quantities Rℓ(s) and Rh(s), respectively, originates in the fact that the leptons carry spin 1 , while the spin of the pion vanishes. Near threshold, the angular momentum barrier 2 suppresses the function Rh(s) by three powers of momentum, while Rℓ(s) is proportional to the first power. The suppression largely compensates the enhancement by the form factor – by far the most important property is the mass. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... pQCD not only fails due to strong coupling (non-convergence of expansion) but because of spontaneous chiral symmetry breaking (100% non-perturbative) responsible for the existence of pions and quark condensates, which are missing to all orders in Compilation of αs measurements (Bethke 04). pQCD, i.e. pQCD fails to correctly describe α (M )=0.322 0.030 at M =1.78 GeV the low energy structure of QCD s τ ± τ Must use non-perturbative methods: ❏ Dispersion relations, sum rules ❏ Chiral perturbation theory extended to include spin 1 vector states ❏ QCD inspired models: extended Nambu Jona-Lasinio model (ENJL), hidden local symmetry (HLS) model ❏ large Nc QCD approach (dual to infinite series of narrow vector resonances) ❏ lattice QCD in future F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ Low energy: the ρ peak “Outstanding” non-perturbative low energy hadron spectrum: e+e− π+π− → A precise new KLOE measurement of F 2 with ISR events | π| 2 |F | 40 ¼ KLOE SND CMD-2 30 20 10 2 2 M¼¼ ()GeV 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ππ event with KLOE detector. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... + 7. Hadron-production in e e−–annihilation ❖ at dawn of QCD: 1974 hadron physics in crisis + ❖ since 1969: e e− hadrons cross-sections with storage ring ADONE at Frascati much too high for → theoretical expectations at that time. These experiments measured the ratio + σ (e e− γ∗ hadrons) R(s)= tot → → (mµ=0) σµµ + of the hardonic cross section in units of the µ µ− pair production cross section, and R values in the range from 1 up to 6 were actually measured. ❖ in 1973: R measured at the Cambridge Electron Accelerator (CEA) at Harvard: R 5[6] at 4[5] GeV ∼ ❖ London Conference in 1974: conclusion R raising smoothly form 2 at 2 GeV to about 6 at 5 GeV ❖ on the theory side confusing: about 22 different models were proposed, among them also QCD [Fritzsch+Leutwyler], as a solution of the “unitarity crisis”. ❖ Unitarity predicts a decrease of the total cross–section like 1/s up to logs at high energies. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... The ratio R as of July 1974. The solution was QCD, providing a factor 3 from color degrees of freedom, and a new species of quark charm, which completed the 2nd family of fermions. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Year Accelerator Emax (GeV) Experiments Laboratory 1961-1962 AdA 0.250 LNF Frascati (Italy) 1965-1973 ACO 0.6-1.1 DM1 Orsay (France) 1967-1970 VEPP-2 1.02-1.4 ’spark chamber’ Novosibirsk (Russia) 1967-1993 ADONE 3.0 BCF,γγ, γγ2, MEA, LNF Frascati (Italy) µπ, FENICE 1971-1973 CEA 4,5 Cambridge (USA) 1972-1990 SPEAR 2.4-8 MARKI,CB,MARK2 SLACStanford(USA) 1974-1992 DORIS -11 ARGUS, CB, DASP 2, DESY Hamburg (D) LENA, PLUTO 1975-1984 DCI 3.7 DM1,DM2,M3N,BB Orsay (France) 1975-2000 VEPP-2M 0.4-1.4 OLYA, CMD, CMD-2, Novosibirsk (Russia) + Chronology of e e− facilities ND,SND 1978-1986 PETRA 12-47 PLUTO, CELLO, JADE, DESY Hamburg (D) MARK-J, TASSO 1979-1985 VEPP-4 -11 MD1 Novosibirsk (Russia) 1979- CESR 9-12 CLEO,CUSB Cornell(USA) 1980-1990 PEP -29 MAC,MARK-2 SLACStanford(USA) 1987-1995 TRISTAN 50-64 AMY, TOPAZ, VENUS KEK Tsukuba (Japan) 1989 SLC 90GeV SLD SLACStanford(USA) 1989-2001 BEPC 2.0-4.8 BES,BES-II IHEPBeijing(China) 1989-2000 LEPI/II 110/210 ALEPH,DELPHI,L3, CERNGeneva(CH) OPAL 1999-2007 DAΦNE Φ factory KLOE LNF Frascati (Italy) 1999- PEP-II B factory BABAR SLAC Stanford (USA) 1999- KEKB B factory Belle KEK Tsukuba (Japan) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Thresholds for exclusive multi particle channels below 2 GeV F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ General O(α2) framework for predicting R(s) Quarks are charged particles and couple to the photon via the quark contribution to the electromagnetic current: 2 1 J γ = u¯ γ u d¯ γ d µ 3 i µ i − 3 i µ i i X where i =1, 2, 3 labels the three quark doublets: ui u c t = , , . di d s b The Lagrangian density for strong and electromagnetic inte ractions is given by 1 = + eJ γAµ F F µν . Lstrong&electromagnetic LQCD µ − 4 µν We are interested in the S–matrix element 4 + i R d x ( , + , ) + X S e e− = X T e LQCD int LQED int e e− h | | i h | | i of the process n o + e e− X specifically X = any hadronic state of appropriate QN′s . → e2 1 ➠ α = = (137.036)− small one–photon exchange approximation + perturbative corrections 4π → F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Leading O(α2) approximation + 2 4 (4) ˜ µ ν + X S e e− = i e (2π) δ (pX p+ p ) Dµν(q) X J (0) 0 0 Jlep(0) e e− . h | | i − − − − ×h | had | ih | | i where Dµν denotes the photon propagator e+ γ hadrons e− Hadron production in electron–positron annihilation. ν µ Current conservation of the leptonic current qν Jlep =0 and the hadronic one qµJhad =0 implies that only the gµν –part of the photon propagators contributes to the matrix element: 1 D˜ (q) g . µν →− µν q2 +iε 4 (4) For the relevant T –matrix element, defined by S = 1 +i(2π) δ ( pi) T we find 2 e µ P + − Te+e X = X Jhad(0) 0 0 Jµ lep(0) e e− → q2 h | | ih | | i where the leptonic matrix element is given at leading order by + 0 Jµ lep(0) e e− = v¯(p+, s+) γµ u(p , s ) h | | i − − − F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... and therefore 2 + e µ T (e e− γ∗ X)= v¯(p+, s+) γµ u(p , s ) X J (0) 0 . → → −q2 − − · h | had | i The total cross–section is + 2 σh(s)=Σ dσ(e e− X, s =(p+ + p ) ) → − ZΓ 3 d p 2 2 with the differential one given by ( denoting dµ(p)= (2π)3 2 ω(p) , ω(p)= m + ~p ) 4 (4) p (2π) δ (p+ + p pX ) 2 dσ = − − T dµ(p′ ) 2 2 1 2 λ(s, me,me) | | ··· + For unpolarized e e−-beams, the squarep matrix element is given by the average over the initial spins of e− and e+ and can easily be calculated by standard techniques: 1 e4 T 2 = T 2 = ℓ hµν | | 4 | | q4 µν s± X 1 2 µν µ ν where ℓµν = qµqν q gµν (p+ p )µ(p+ p )ν and h = 0 J (0) X X J (0) 0 . 2 − − − − − − h | | ih | | i One obtains 2 2 α 16π 4 (4) µν σh(s)= ℓµν Σ (2π) δ (p+ + p pX ) h 3 2 2s 1 4m /s X − − − e Z where summation/integration is overp the hadronic final states. The leptonic tensor ℓµν describes the initial state F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... and is fixed. For what concerns the hadronic part we try to be general before we will resort to pQCD which only applies at sufficiently high energy. At this point the optical theorem helps to proceed at the non-perturbative level. As shown above we have s + ImΠ′ (s)= σ (e e− anything) . γ 4πα tot → + In first place, applying the optical theorem to e e−–annihilation we have + + 4 n (4) 2 2 Im T (e e− e e−)forward = (2π) Πi=1 dµ(pi′ ) δ (Pn′ Pi) T (p1′ , , pn′ p+, p ) → ′ − · | ··· | − | QNXs Z which in the given one–photon exchange approximation has the structure shown e+ hadrons e+ γ γ − − e X a)e b) + The optical theorem in e e−–annihilation: a) including the QED part, b) the QCD part isolated. 4 (4) µν The isolated QCD part, representing (2π) δ (p+ + p pX ) h , thus corresponds to the hadronic − − excitations (hadronic vacuum polarization) in the photon propagator. The hadronic part corresponds to the photon µ propagator after amputation of the external photons Aµ(x) J µ(x), where only the hadronic part J (x) is → had F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... of interest here. The hadronic “blob” b) thus is given by the hadronic current correlator Πˆ µν(q) = i d4x ei qx 0 J µ (x)J ν (0) 0 h | had had | i Z µν 2 µ ν 2 = g q q q Πˆ ′ (q ) . − − γ ˆ 2 2 Note that Πγ′ (q ) times e is the usual photon self-energy. We have used transversality which is a consequence of the current conservation. The hadronic photon vacuum polarization thus is given by a single scalar amplitude 2 2 ˆ 2 Πγ′ (q )= e Πγ′ (q ). + The physical thresholds of hadron production in e e− X are → + 0 + + 0 0 2 a X = π π−,ρ,π π π−,ω,K K−,K K , φ, with lowest threshold at s =4 mπ± . The vector + 0 + ··· mesons ρ, ω and φ correspond to π π−, π π π− and KK¯ resonance peaks, respectively. We now may continue our cross–section calculation using 4 (4) µν ˆ µν µ ν 2 µν ˆ 2 (2π) δ (p+ + p pX ) h = 2 Im Π (q)=2 q q q g Im Πγ′ (q ) , − − − aIf we would include higher order electromagnetic interaction, like final state radiation (FSR) from hadrons, the lowest possible state exhibiting a hadron would be 0 with threshold at 2 . In our counting this is a 3 effect not included in our leading order consideration. π γ s = mπ0 O(α ) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... such that, working out kinematics 2 2 2 α 16π 2me ˆ σh(s) = 1+ Im Πγ′ (s) s 4m2 s 1 e − s q2 2 16π α 2 2 Im Πˆ ′ (s) since s 4m m . ≃ s γ ≥ π ≫ e It is common use to represent the hadronic production cross–section σh(s) in units of the leptonic point cross section. This leads to the definition of the so called R–ratio function + . σtot(e e− hadrons) R(s) = + → + σ0(e e− µ µ−) → 2 2 σh(s) 16π α 1 = = 3 s Im Πˆ ′ (s) 4πα2 4πα2 · s γ 3s ˆ 2 = 12 πImΠγ′ (s)=12π ρ1(s) where 1 Im ˆ is the spectral density used in the Källen-Lehmann representation earlier. ρ1(s)= π Πγ′ (s) As we will see below the function R(s) has the asymptotic property R(s) constant ; s → → ∞ ˆ which means that the imaginary part of the vacuum polarization function Πγ′ (s) tends to a non–vanishing constant at time–like infinity. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Exploiting analyticity: The DR, derived earlier, determines the real part of the vacuum polarization amplitude in terms of its imaginary part. In fact the electromagnetic current correlator exhibits a logarithmic UV singularity and thus requires one subtraction such in place of the unsubtracted DR ∞ Im ˆ 2 1 Πγ′ (s) Πˆ ′ (q )= ds , γ π s q2 iε Z0 − − we find 2 ∞ Im ˆ 2 q Πγ′ (s) Πˆ ′ (q ) Πˆ ′ (0) = ds = ∆α(s) . γ − γ π s (s q2 iε) − Z0 − − In a renormalizable QFT UV divergencies in physical amplitudes are at most quadratic. Therefore at most two subtractions (mass– and wavefunction–renormalization, or vertex renormalization) are needed to get any DR finite. In low energy effective theories the number of subtractions needed corresponds to the number of low energy constants at the given order of the expansion (powers in p/Λ; p process specific momentum, Λ, typically ΛQCD or Mp the proton mass). F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... While in perturbative QCD colored, fractionally charged qq¯-pairs are produced, in reality only hadrons are realized as states via non-perturbative hadronization: u¯ π+ π+ d γ π0 d¯ u π− π− + Hadron production in low energy e e−–annihilation: the primarily created quarks must hadronize. The shaded zone indicates strong interactions via gluons which confine the quarks inside hadrons. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Summary: In the one photon exchange approximation, i.e. at O(α2), the total hadronic production cross–section in electron–positron annihilation e+ + − ∗ γ σh(e e γ hadrons) : X hadrons → → e− determines the imaginary part of the photon vacuum polarization amplitude had γ γ : Πˆ µν (q)= qµqν q2 gµν Πˆ ′ (q2) q − γ had 16π2α2 σ = Im Πˆ ′ . h s γ had + Usually one represents the e e− hadrons data in terms of → σh(s) R(s)= = 12 π Im Πˆ ′ . (mµ=0) γ had σµµ The hadronic shift of the effective fine structure constant at scale M is given by − ∗ 2 ∞ e+e γ had α M dsR(s) → → ∆α (M 2)= Re . had − 3π s (s M 2 iε) Z2 4mπ − − (Cabibbo-Gatto Roma 1960!) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ High energy hadron-production in e+e−–annihilation ❖ Asymptotic freedom: weak coupling at high energies (5) Status of αs [Bethke 2009](left) compared with 1989 pre LEP status (right) αs (MZ )=0.11 0.01 (5) ± (corresponding to Λ = 140 60 MeV).[Altarelli 89]. MS ± ❖ perturbative QCD (pQCD) applies for M √s τ ≤ ❖ photon at short distances directly couples to the quarks q = u,d,s,c,b,t + ❖ e e− hadrons via γ∗ qq¯ → → F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❖ observed hadrons only (confinement!) and sharp spikes of vector bosons resonances: J/ψ series, Υ series ❖ main issue hadronization : no detailed understanding ❖ meaning of quark parton cross sections ruled by “quark–hadron duality”a Quark Hadron Duality (QHD): for sufficiently large s the average non–perturbative hadron cross–section equals the perturbative quark cross–section: + + σ(e e− hadrons)(s) σ(e e− qq¯)(s) , → ≃ q → X where the averaging extends from threshold up to the given s value which must lie far enough above a threshold (global duality). Approximately, such duality relations then would hold for energy intervals which start just below the last threshold passed up to s. aQuark–hadron duality was first observed phenomenologically for the structure function in deep inelastic electron–proton scattering (Bloom&Gilman 1970) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Status of quark hadron duality: ❏ qualitatively QHD is clearly seen in the data for appropriate regions of s ❏ it is not yet possible to quantify the accuracy of the conjecture ❏ a quantitative check would require much more precise cross-section measurements ❏ applicability of QHD as a tool in precision physics? not really under control ❏ in large-N QCD, N c QHD becomes exact (infinite series of narrow vector states) c − → ∞ Note: this is one of the most important testing grounds for pQCD. On theory side: calculate ’ ˆ ˆ R(s)=12 π Im Πγ′ had(s) ; Πγ had(q): , where the “blob” is the hadronic one, in pQCD represented by valence quark loops dressed in all possible ways by gluons and possible sea quark loops. The leading plus next to leading QCD perturbation expansion F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... diagrammatically is given by ’ = + ⊗ + + + ⊗ + ⊗ + ⊗ + + + ⊗ ··· 0 Lowest order is (αs) : 2 2 Im = ➠ proportional to free quark–antiquark production cross-section This approximation = Quark Parton Model (QPM) , Free quarks with the strong interaction turned off. By AF the QPM is expected to provide a good approximation in F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... the high energy limit of QCD. Calculation yields 2 2 Qf 2 2 2 Im Πˆ ′ (q )= N 1+2m /q Im B (m ,m ; q ) γ had c 12π2 f 0 f f f X and the imaginary part of scalar two-point function integral 1 B (m ,m ; q2) = Reg dx ln(m x + m (1 x) x (1 x)q2) 0 1 2 − 1 2 − − − Z0 with Im B = π 1 4m2 /q2. The mass dependent threshold factor is a function of the center of mass quark 0 − f velocity q 2 1/2 4mf vf = 1 − s ! and, we may write Q2 R(s) = N f 12π2 1+2m2 /s 1 4m2 /s [s 4m2 ] c 12π2 f − f ≥ f Xf q v = N Q2 f 3 v2 Θ(s 4m2 ) . c f 2 − f − f f X The other possibility to get this result is to calculate the Born cross section for the production of a quark-antiquark F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... + pair σ(e e− qq¯ ), directly, by taking free quarks X =qq ¯ in the matrix element. We then have → X J µ (0) 0 = Q u¯ (p , s ) γµ v (p , s ) h | had | i q q q q q q¯ q¯ µν and h in is given by an expression like ℓµν. Like µ-pair production with adjusted charge and color factor N =3. In quark–parton model which, with N =3, Q = 2 and Q = 1 , predicts the following c c u,c,t 3 d,s,b − 3 constant R–values: Nf 34 5 6 quarks uds udsc udscb udscbt 1 2 R 2 3 2 3 3 5 range 1.8 - 3.73 GeV 4.8 - 10.52 GeV 11.20 - (2 m -10.0) GeV (2 m +10.0) GeV - t t ∞ F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Test of pQCD prediction of R with some more recent data in the non-resonant regions. The spikes show the sharp J/ψ (cc¯ ) and Υ (¯bb) resonances. Including the presently known higher order terms the perturbative result for R(s) is given by v R(s)pert = N Q2 f 3 v2 Θ(s 4m2 ) c f 2 − f − f f X 1+ ac (v )+ a2c + a3c + a4c + × 1 f 2 3 4 ··· F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... where a = α (s)/π and, assuming 4m2 s, i.e. in the massless approximation s f ≪ c1 = 1 3 3 33 123 c = C (R) C (R) β ζ(3) N + N 2 2 −32 2 − 4 0 − 48 f 32 c 365 11 = N β ζ(3) 1.9857 0.1153N 24 − 12 f − 0 ≃ − f c = 6.6368 1.2002N 0.0052N 2 1.2395 ( Q )2/(3 Q2 ) 3 − − f − f − f f Xf Xf 5β c = 135.8 34.4 N +1.88 N 2 0.010 N 3 π2β2 1.9857 0.1153 N + 1 , 4 − f f − f − 0 − f 6β 0 with β = (11 2/3 N )/4, β = (102 38/3 N )/16. All results are in the MS scheme. 0 − f 1 − f Nf = f:4m2 s 1 is the number of active flavors. There is a mass dependent threshold factor in front of the f ≤ curly brackets and the exact mass dependence of the first correction term P 2π2 π2 1 c (v)= (3 + v) 1 3v − 6 − 4 is singular (Coulomb singularity due to soft gluon final state interaction) at threshold. The singular terms of the n–gluon ladder diagrams ··· F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... exponentiate and thereby remove the singularity : 2x 2παs 1+ x 1 e−2x ; x = 3v → − αs αs 2παs 4παs 1 1+ c1(v) + 1+ c1(v) . π ··· → π − 3v 3v 1 exp 4παs − − 3v Since pQCD in any case should not be applied near threshold, the above remark is somewhat academic. In contrast to QED, where Coulomb interaction problems are there and have to be cured appropriately. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Calculations: the coupling αs and the masses mq have to be understood as running parameters 2 2 2 m0f mf (µ ) 2 R ,αs(s0) = R ,αs(µ ) ; µ = √s . s0 ! s ! where √s0 is a reference energy. RG resummation dramatically improves the convergence of perturbative approximations. Perturbative QCD is supposed to work best in the deep Euclidean regiona away from the physical region characterized by the cut in the analyticity plane: Im s asymptotic freedom (pQCD) | Re s ←− s0 Analyticity domain for the photon vacuum polarization function. In the complex s–plane there is a cut along the 2 positive real axis for s > s0 =4m where m is the mass of the lightest particles which can be pair–produced. aThis is directly accessible in Deep Inelastic electron–proton Scattering (DIS). F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... + More “parton physics” in e e−–annihilation: elementary hard process tells us that in a high energy collision of positrons and electrons (in the center of mass frame) q and q¯ are produced with high momentum in opposite directions (back–to–back), q¯ e+ q¯ + γ e ✶ e− θ e− q q + Fermion pair production in e e−–annihilation. The lowest order Feynman diagram (left) and the same process in the c.m. frame (right). The arrows represent the spatial momentum vectors and θ is the production angle of the quark relative to the electron in the c.m. frame. + + The differential cross–section, up to a color factor the same as for e e− µ µ−, reads → 2 dσ + 3 αs 2 2 (e e− qq¯)= Q 1+cos θ dΩ → 4 s f typical for an angular distribution of a spin 1/2 particle. Indeed,X the quark and the antiquark seemingly hadronize individually in that they form jets = bunches of hadrons which concentrate in a relatively narrow angular cone. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Two and three jet event first seen by TASSO at DESY in 1979. It is one of the spectacular and at the beginning completely unexpected predictions of QCD, that the energy get collimated more and more into narrower and narrower jets as the energy increases. However, as the energy increases also the number of primary partons increases such that the energy distributes among a larger number of jest, which then brings back a more isotropic distribution. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ Non–Perturbative Effects, Operator Product Expansion Chiral Symmetry U(N ) U(N ) f V ⊗ f A key role of Left–handed and Right–handed massless fields Nambu 1960 chiral symmetry of strong interaction must be broken spontaneously! only SU(Nf )V hadrons exist [parity partners missing] = pions as Nambu-Goldstone bosons! ⇒ (non-perturbative) if unbroken: nucleons must be massless, parity doubling! contradicting observation! ❏ SBB associated quark condensates ⇒ ❏ how do they contribute to R(s) ❏ such non–perturbative (NP) effects may be included via Operator Product Expansion (OPE)a aThe operator product expansion (Wilson short distance expansion) is a formal expansion of the product of two local field operators A(x) B(y) in powers of the distance (x − y) → 0 in terms of singular coefficient functions and regular composite operators: x + y A(x) B(y) ≃ Ci(x − y) Oi( ) 2 Xi F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... OPE of the electromagnetic current correlator yields αs ′ 4πα 1 11 GG Πˆ NP(Q2) = Q2N 1 a h π i γ 3 q cq · 12 − 18 Q4 q=Xu,d,s a 11 3 m qq¯ + 2 1+ + l a2 h q i 3 2 − 4 qµ Q4 ′ ¯ 4 4 257 1 2 mq q′q′ + a + ζ3 lqµ a h 4 i + 27 3 − 486 − 3 ′ Q ··· q =Xu,d,s where a α (µ2)/π and l ln(Q2/µ2). αs GG and m qq¯ are the scale–invariantly defined ≡ s qµ ≡ h π i h q i condensates. Sum rule estimates of the condensates yield typically (large uncertainties) αs GG (0.389 GeV)4, m qq¯ (0.098 GeV)4 for q = u, d , and m qq¯ (0.218 GeV)4 h π i∼ h q i∼− h q i∼− for q = s . Note that the above expansion is just a parametrization of the high energy tail of NP effects associated with the existence of non–vanishing condensates. The dilemma with the OPE in our context is that it works for large enough Q2 only and in this form fails do describe NP physics at lower Q2. Once it starts to be numerically relevant pQCD starts to fail because of the growth of the strong coupling constant. x y where the operators + represent a complete system of local operators of increasing dimensions. The coefficients may be calculated Oi( 2 ) formally by normal perturbation theory by looking at the Green functions N x + y h0|T A(x) B(y) X|0i = Ci(x − y) h0|T Oi( ) X|0i + RN (x, y) . 2 Xi=0 F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... Note: condensates break dilatation (conformal) invariance µ β(gs) ¯ Θ µ = GG +(1+ γ(gs)) mu uu¯ + md dd + 2gs ··· where β(gs) and γ(gs) are the RG coefficients F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ Some considerations on hadron production at low energy Certainly, quark-antiquark pair production is far from describing experimental cross–section data at low energies, where “infrared slavery” (confinement) is the dominating feature. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... The Fig. shows a compilation of the low energy data in terms of the pion form factor. The latter is defined by 2 πα 3 2 2 3 σ (s)= (β ) F (s) or F (s) =4 R (s)(β )− . ππ 3s π | π | | π | ππ π ❖ also low energy effective theory of QCD: chiral perturbation theory (CHPT) fails (has no ρ) ❖ looks natural to apply vector-meson dominance (VMD); proper QCD implementation Resonance Lagrangian Approach (RLA) ❖ in fact photon mixes with hadronic vector–mesons like the ρ0 ❖ also nearby resonances like ρ0 and ω are mixing (distorting Breit-Wigner shape) ❏ Pions and Scalar QED ❖ Pions can be seen in a particle detector ➠ behave like point particles (relatvistic Wigner state) ❖ on pion level coupling to photon fixed by gauge invariance charged pion field = complex scalar field ϕ free Lagrangian (0) µ 2 = (∂ ϕ)(∂ ϕ)∗ m ϕϕ∗ Lπ µ − π via minimal substitution ∂ ϕ D ϕ =(∂ +ieA (x)) ϕ, which replaces the ordinary by the covariant µ → µ µ µ derivative, and which implies the scalar QED (sQED) Lagrangian sQED (0) µ 2 µ ν = ie(ϕ∗∂ ϕ ϕ∂ ϕ∗)A + e g ϕϕ∗A A . Lπ Lπ − µ − µ µν F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... In sQED the contribution of a pion loop to the photon VP is given by iΠµν (π)(q)= + − γ . and the renormalized transversal photon self–energy reads ′ α 1 Π (π)(q2)= + (1 y) (1 y)2 G(y) γren 6π 3 − − − where y =4m2/q2 and G(y) was given before. For q2 > 4m2 there is an imaginary or absorptive part given by substituting Im π such that G(y) G(y)= 2√1 y → − − ′ α Im Π (π)(q2)= (1 y)3/2 γ 12 − and for large q2 is 1/4 of the corresponding value for a lepton. According to the optical theorem the absorptive + + part may be written in terms of the e e− γ∗ π π− production cross–section σ + − (s) as → → π π ′ s Im Π had(s)= σ (s) γ 4πα had which hence we can read off to be 2 πα 3 σ + − (s)= β π π 3s π F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... β = (1 4m2 /s) pion velocity ➠ sQED predicts π − π p Fπ(s) = 1 independent of s, in view of the ρ-resonance sitting there a complete failure up to 50 : 1 results. Thus sQED only works in conjunction with vector meson dominance model (VDM) or improvements of it ➠ e e F (q2), e2 e2 F (q2) 2 → π → | π | taking care of bound-state nature of pion ( form-factor) → Mandatory constraint: electromagnetic current conservation F (0) = 1 ↔ π Still sQED used in controlling real photon radiation issues: final state radiation (FSR) (Bloch-Nordsieck prescription), Kinoshita-Lee-Nauenberg theorem and all that. Urgently need precise experimental investigation of FSR from hadrons (Venanzoni et al.) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ The Vector Meson Dominance Model ❖ phenomenological fact: photon and ρ meson exhibit direct coupling γ ρ0–mixing ↔ − ❖ implementation: VMD model and extensions like RLA ❖ naive VMD model: replacing the photon propagator as µ ν µν µν µν q q µν 2 i g i g i(g 2 ) i g m + + − q = ρ + , q2 ···→ q2 ···− q2 m2 q2 m2 q2 ··· − ρ ρ − where the ellipses stand for the gauge terms. ❖ no change as q2 0 → ❖ changes asymptotics 1/q2 1/q4 → ❖ directly exhibits ρ0-resonance However, the naive VMD model does not respect chiral symmetry properties. More precisely, VMD relates ❏ had hadronic part of the electromagnetic current jµ (x) ❏ source density J (ρ)(x) of the neutral vector meson ρ0 by M 2 1 B jhad(0) A = ρ B J (ρ)(0) A h | µ | i − 2γ q2 M 2 h | µ | i ρ − ρ where q = p p , p and p the four momenta of the hadronic states A and B, respectively, M is the B − A A B ρ F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... mass of the ρ meson. So far our VMD ansatz only accounts for the isovector part, but the isoscalar contributions mediated by the ω and the φ mesons may be included in exactly the same manner: B B = γ V =ρ0, ω, φ, γ X ··· V A A M 2 1 = V ; = − . 2γ q2 M 2 V − V The vector meson dominance model. A and B denote hadronic states. The key idea is to treat the vector meson resonances like the ρ as elementary fields in a first approximation. Free massive spin 1 vector bosons are described by a Proca field Vµ(x) satisfying the Proca equation 2 2 ν ( + MV ) Vµ(x) ∂µ (∂ν V )=0, which is designed such that it satisfies the Klein-Gordon equation and at − ν the same time eliminates the unwanted spin 0 component: ∂νV =0. In the interacting case this equation is replaced by a current–field identity (CFI) (2 + M 2 ) V (x) ∂ (∂ V ν )= g J (V )(x) V µ − µ ν V µ where the r.h.s. is the source mediating the interaction of the vector meson and gV the coupling strength. The F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... µ (V ) current should be conserved ∂ Jµ (x)=0. The CFI then implies g B V (0) A = V B J (V )(0) A h | µ | i −q2 M 2 h | µ | i − V where terms proportional to qµ have dropped due to current conservation. The VMD assumes that the hadronic electromagnetic current is saturated by vector meson resonancesa 2 had MV jµ (x)= Vµ(x) 2γV V =ρ0, ω, φ, X ··· in particular 2 had Mρ 2 2 ρ(p) j (0) 0 = ε(p, λ)µ , p = M . h | µ | i 2γρ ρ ❏ 2 MV required for dimensional reasons ❏ γV defines coupling constant (convention) ❏ VMD relation derives from the CFI and ansatz The VMD model is known to describe the gross features of the electromagnetic properties of hadrons quite well, a In large Nc QCD all hadrons become infinitely narrow, since all widths are suppressed by powers of 1/Nc, and the VMD model becomes exact with an infinite number of narrow vector meson states. The large-Nc expansion attempts to approach QCD (Nc = 3) by an expansion in 1/Nc. In leading approximation in the SU(∞) theory R(s) would have the form 9π ∞ ee 2 R(s)= Γ Mi δ(s − M ) . α2 i i Xi=0 F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... most prominent example are the nucleon form factors. A way to incorporate vector–mesons ρ, ω, φ, ... in accordance with the basic symmetries of QCD is the Resonance Lagrangian Approach (RLA), an extended version of CHPT, which also implements VMD in a consistent manner. In the flavor SU(3) sector, similar to the pseudoscalar field Φ(x), the SU(3) gauge bosons conveniently may be written as a 3 3 matrix field × 0 ρ + ω8 ρ+ K + √2 √6 ∗ 0 ρ ω8 0 Vµ(x) = TiVµi = ρ − + K − √2 √6 ∗ i 0 ω8 X K∗− K∗ 2 − √6 µ in order to keep track of the appropriate SU(3) weight factors. ❏ ρ0 γ –Mixing − The unstable spin 1 vector mesons are described by a propagator exhibiting a pole M˜ 2 q2 = M 2 i M Γ ρ ≡ pole ρ − ρ ρ in the complex q2-plane with correspondence physical mass real part of location of propagator pole ⇐⇒ width imaginary part of the location of the pole . ⇐⇒ F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... The simple relation between the full propagator and the irreducible self-energy only holds if there is no mixing, like 0 0 for the charged ρ±. In the neutral sector, because of γ ρ mixing, we cannot consider the ρ and γ − propagators separately. They form a 2 2 matrix propagator, so that the pole condition is modified into × 2 Π 0 (s ) 2 γρ P s m 0 Π 0 0 (s ) =0 , P − ρ − ρ ρ P − s Π (s ) P − γγ P ˜ 2 b with sP = Mρ0 . Note: such mixing is not present in the charged channel (e.g. in τ –decay) bThe simplest way to treat this problem is to start from the inverse propagator given by the irreducible self-energies (sum of 1pi diagrams). Again we restrict ourselves to a discussion of the transverse part and we take out a trivial factor −i gµν in order to keep notation as simple as possible. With this convention we have for the inverse γ − ρ propagator the symmetric matrix 2 2 2 1 k + Πγγ (k ) Πγρ(k ) Dˆ − = 0 1 2 2 2 2 Πγρ(k ) k − Mρ + Πρρ(k ) @ A Using 2 × 2 matrix inversion a b 1 1 c −b M = 0 1 ⇒ M − = 0 1 b c ac − b2 −b a @ A @ A F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❏ Vector–Meson Production Cross–Sections + The cross–section for spin 1 boson production in e e−–annihilation is described in full detail in Sect. 14 of my Lausanne Lectures. For massive spin 1 boson the polarization vectors ε(p, λ) satisfy the completeness relation pµpν ε(p, λ)µ ε∗(p, λ)ν = gµν 2 − − Mρ Xλ and the hadronic tensor reads 2 2 4 1 Mρ Mρ pµpν Mρ hµν = ε(p, λ)µ ε∗(p, λ)ν = gµν 2 2 , 3 2γρ 2γρ − − Mρ 2γρ Xλ and one easily calculates the spin average T 2. | | + 0 The result for the e e− ρ cross–section and some useful approximations are the following: → we find for the propagators 1 1 D = ≃ γγ 2 2 2 2 Πγρ(k ) k + Πγγ (k ) k2 + Πγγ (k2) − k2 M2+Πρρ(k2) − ρ 2 2 −Πγρ(k ) −Πγρ(k ) D = ≃ γρ 2 2 2 2 2 2 2 2 2 2 (k + Πγγ (k ))(k − Mρ + Πρρ(k )) − Πγρ(k ) k (k − Mρ ) 1 1 D = ≃ . ρρ 2 2 2 2 2 Πγρ(k ) k − M + Πρρ(k ) k2 − M 2 + Πρρ(k2) − ρ ρ k2+Πγγ (k2) These expressions sum correctly all the reducible bubbles. The approximations indicated are the one-loop results. The extra terms are higher order contributions. F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... ❖ Breit-Wigner resonance: field theory version The field theoretic form of a Breit-Wigner resonance obtained by the Dyson summation of a massive spin 1 transversal part of the propagator in the approximation that the imaginary part of the self–energy yields the width 2 by Im ΠV (MV )= MV ΓV near resonance. 2 12π Γ + − sΓ σ (s)= e e . BW M 2 Γ (s M 2 )2 + M 2 Γ2 R − R R ❖ Breit-Wigner resonance The resonance cross–section from a classical non–relativistic Breit-Wigner resonance is given by 3π ΓΓe+e− σBW (s)= . s (√s M )2 + Γ2 − R 4 ❖ Narrow width resonance The narrow width approximation for a zero width resonance reads 2 12π 2 σNW(s)= Γe+e− δ(s MR) . MR − F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009– Physics of vacuum polarization ... More realistic refined approaches including isospin breaking from m = m and multiple resonances d 6 u ❖ Gounaris-Sakurai model ❖ Resonance Lagrangian models ❖ Analytic approach à la Omnès (Colangelo, Leutwyler et al.) F. Jegerlehner INFN Laboratori Nazionali di Frascati, Frascati, Italy –November 9-13, 2009–