evjas-2013 Moving on to Non-Abelian Gauge Theories from ψ(x) → eiα(x)ψ(x) to iαa(x)·T a (ψ(x))m → e (ψ(x))n mn or, equivalently, “Yang-Mills Theories” Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 0 evjas-2013 C. N. Yang and R. Mills Phys. Rev. 96, v.1, p.191 (1954) A two-fold generalization over QED and the concept of isospin: • – fields are multi-dimensional in some new internal space (ψ(x) ψn(x)) → Generalize isospin-like global rotations to local transformations • – i.e. different at different points in space-time. Generalize QED from commutative (abelian) phase rotations to any non- • commutative continuous group. We restrict ourselves to “Lie Groups” – continuous groups whose trans- • formations can be written as an integration of infinitesimal steps; αa a N a a finite group element: limN (1 + i · T ) = exp(iα (x) · T ) →∞ N infinitesimal element: ( 1+ iǫa · T a + O(ǫ2) ) hermitean “generators”: [T a, T b] = if abcT c fabc are the “structure constants” Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 1 evjas-2013 The for Y-M Theories L Scalar, invariant under the generalized gauge transformations • a a a ψi(x) Uij (α(x), nˆ )ψj (x) Uij = exp( iT α (x))ij −→ − · Same story, the fermion free field derivatives will break locality, • and we need the covariant derivative i, j are isospin indices a a [Dµ]ij = δij ∂µ ig[T ]ij Aµ(x) a are QCD-color indices − · quarks: a = 1...3 gluons: a = 1...8 a where g is the coupling strength between ψ(x) and Aµ(x) So we now have • µ µ = ψ¯i( iγ [Dµ]ij mδij )ψj = ψ¯(iγ Dµ m)ψ L − − where we still need: a – the transformation conditions over Aµ(x) that will guarantee gauge invariance a – the (gauge invariant) kinetic term for the gauge fields Aµ(x) Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 2 evjas-2013 Exercise [important! see e.g. T. Muta sec. 2.1.2] Complete the steps towards the gauge invariant Y M • L ′ Dµψ (Dµψ) = U(Dµψ) starting point −→The covariant derivative of a field transforms like the field itself (this is a necessary condition for preserving the invariance) a a a ′a a a i † † T A T A = U (T A U ∂µU) U · µ −→ · µ · µ − g a a [Dµ,Dν]= igT F − · µν a a a abc b c NAG-invariance introduces Fµν = ∂µAν ∂ν Aµ + gf AµAν self-interactions among the − gauge fields. They are respon- sible for the anti-screening that generates asymptotic freedom 1 a µνa µ Y M = F F + ψ¯(iγ Dµ m)ψ • L − 4 µν · − Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 3 evjas-2013 The Y-M Field Strength a a a abc b c F = ∂µA ∂ν A + gf A A µν ν − µ µ ν The price you pay for the non-abelian nature of the theory is a non- • a linear term in Fµν This term causes major departures from the general QED (abelian) • behavior In QED the field strength F a is by itself gauge invariant • µν In QCD it is NOT (but the F F term in is, of course) • · L And this is because gluons are (color)-charged objects • Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 4 evjas-2013 Comment Next, in a QFT (QCD) program would be: • – gauge fixing (*) (*) covariant gauge fixing, and ghost fields to cancel unphysical – propagators degs. of freedom in gluon propagation – Feynman rules – the β function – the running CC Sorry, we will not – asymptotic freedom cover QCD – confinement etc. etc... Instead, we proceed on our qualitative/intuitive • route towards (EWSB in) the Standard Model Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 5 evjas-2013 QCD References Various good books, including these (getting old but still among the • best) – “Foundations of QCD” – T. Muta (World Scientific) – “QCD and Collider Physics” – Ellis, Stirling, Webber (Cambridge U. P.) Recommended For the basic aspects of the β-function, the running CC and asymp- • totic freedom, Webber’s 2008 CERN lecture-1 (video+slides) – http://videolectures.net/cernacademictraining08 webber qcd/ (is perhaps the very best introduction that I know) Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 6 evjas-2013 Comparing QED vs QCD AGT vs NAGT an intuitive view of asymptotic freedom and confinement Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 7 evjas-2013 QED and charge screening ex: P.& S. sec. 7.5 + − Virtual e e pairs are electric dipoles of length 1/me • ∼ The vacuum polarization around a charged ptcle acts like a dielectric • medium – causes a partial charge screening The “dielectric constant” is dependent on the momentum transfer • As you penetrate the screening cloud, you see more of the “bare” charge. • Example measurements: – q2(Hidrogen atom): the fine structure constant, α 1/137 ≈ – q2(Z-pole at LEP): the effective charge grows to α 1/128 ≈ Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 8 evjas-2013 QCD and charge anti-screening The cloud of virtual qq pairs is QED-like • The cloud of virtual gluons has no QED analog ¯br • b r interaction ¯br gb¯ gr¯ b b no interaction The gauge (gluon) field component of the cloud the closer you get, the higher • the probability that this happens de-localizes the charge, spreads it out in space Overall, the gluon effect is dominant, and color • charge increases with distance Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 9 evjas-2013 The QCD running CC World Summary of αs (2012) Siegfried Bethke, arXiv:1210.0325 • input measurements Q dependence 0.5 τ-decays April 2012 Lattice α τ 3 s(Q) decays (N LO) DIS Lattice QCD (NNLO) + - e e annihilation 0.4 DIS jets (NLO) Heavy Quarkonia (NLO) Z pole fits – e+e jets & shapes (res. NNLO) 0.11 0.12 0.13 Z pole fit (N3LO) α (Μ ) pp –> jets (NLO) s Ζ 0.3 probing small distance scales (x) → ) 2 α 2 α 2 (Q QCD(Q ) QED(Q ) 0.2 eff α ↑ Landau pole 0.1 α (Μ ) = 0.1184 ± 0.0007 QCD s Z ↑ asymptotic confinement freedom → 1 10 100 Q [GeV] large momentum transfer (Q2) → Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 10 evjas-2013 CMS: the inclusive jets Xsecn A multijet event in CMS Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 11 evjas-2013 Introduzindo ΛQCD A fun¸c˜ao-β de QCD ´edada por • 3 3 2∂g(Q) g g Liberdade β(g)= ∂ ln(−Q2/M2) = b0 16π2 b1 (16π2)2 + ... − − assint´otica onde b0 = 11 2nf /3, b1 = 102 38nf /3, ... − − quarks, gluons Novamente, em “leading logs”, hard-scatter • (perturbativo) αs(M) αs(Q)= b α (M) 0 s − 2 2 1+ 4π ln( Q /M ) ΛQCD Definindo-se a escala Λ tal que • hadroniza¸c˜ao 2 (fragmenta¸c˜ao) b0αs(M) ln( M ) 1 tem-se 4π Λ2 ≡ Mostra a evolu¸c˜ao Estruturas 2π de Λ para αs(Q)= QCD b0 ln(Q/Λ) −→ Q > Λ, a regi˜ao hadrˆonicas perturbativa. O valor atribu´ıdo a ΛQCD depende • explicitamente do esquema de re- Confinamento normaliza¸c˜ao, mas ser´aem geral ΛQCD 300MeV ≈ Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 12 evjas-2013 Isospin, and the role of internal symmetries a very fortunate accident Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 13 evjas-2013 Introductory “Language” Internal space; example, the space generated by a set • t (or multiplet) of scalar fields Φ(x) =(φ1,φ2, ...φN ) Internal operations; example, the rotation group • O(N) Φ Φ′ = G Φ G O(N), Gt G =1 −→ · ∈ · or, in the case of complex scalar fields, the unitary group G SU(N) with G† G =1 ∈ · Internal symmetry operations; if all states undergo • the same operation, the physics observables remain invariant Global symmetry operations; the symmetry operation • has to be performed identically everywhere in space-time Local symmetry operations; the symmetry operations • can be different at different points in space-time Local invariance:; G = G(x) [∂µ,G] =0 Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 14 evjas-2013 Isospin Symmetries – S.I. J. Chadwick (1932), discovers the neutron – a companion to the • proton in the atomic nucleus. Nearly degenerate in mass; mn mp − =1.4 10−3 mn × – A different state of the same particle ? the “nucleon”... Heisenberg (1932), charge independence of the nuclear forces • (i.e. “switch off” EM and the p and n become indistinguishable) Evidence? consider mirror nuclei spectroscopy: • Tritium: 3H(pnn) M=8.48MeV while a crude estimate for the Helium: 3He(ppn) M=7.72MeV extra Coulomb repulsion in ppn is ∆E = α/r ; r 2fm ∆E = 0.73MeV diff (binding E) ∆ = 0.76 MeV c ≈ ⇒ c Hyp: the interchange p n is a symmetry operation for the S.I. • ⇔ Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 15 evjas-2013 Isospin Symmetries – S.I. Name “isospin” introduced by E. Wigner in 1937 • The Heisenberg hypothesis (in analogy with spin) • – an invariance under rotations in “isospin” space p I = 1/2 – the Nucleon is an isospin doublet state N = I3 = ±1/2 n – the proton and neutron are two isospin eigenstates of the 1 0 Nucleon; p = , n = 0 1 – and electromagnetism provides the p n discrimination or, equivalently, breaks the isospin symmetry. Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 16 evjas-2013 Consequences a *global* symmetry p p Given N = , then any other state N ′ = eiI·Θ • n n is an equivalent state to N for the S.I.