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 ′ D ψ (D ψ) = U(D ψ) 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 ∂ U) U · −→ · · − g
a a [D ,Dν]= igT F − · ν
a a a abc b c NAG-invariance introduces F ν = ∂ Aν ∂ν A + gf A Aν self-interactions among the − gauge fields. They are respon- sible for the anti-screening that generates asymptotic freedom
1 a νa Y M = F F + ψ¯(iγ D m)ψ • L − 4 ν · −
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 3 evjas-2013 The Y-M Field Strength
a a a abc b c F = ∂ A ∂ν A + gf A A ν ν − ν
The price you pay for the non-abelian nature of the theory is a non- • a linear term in F ν
This term causes major departures from the general QED (abelian) • behavior
In QED the field strength F a is by itself gauge invariant • ν 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.
Strongly interacting particles must occur in Nature as families • of 2I + 1 I-multiplet members – and (approximately) degenerate in mass
Nucleons seem OK... what about pions ? •
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 17 evjas-2013
Observations on the Tracks of Slow Mesons in Photographic Emulsions C. M. G. Lattes, G. P. S. Occhialini and C. F. Powell (1947)
ptcl τ(seg) cτ(m) alcance µ 2.2 · 10−6 660 γ = 10; v = 0.995c ; terra! Tempos, alcances: π± 2.6 · 10−8 8 CERN’60,4.5GeV; γcτ = 250m π0 8.4 · 10−17 prompt 0
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 18 evjas-2013 Pions
The Yukawa (1935) postulated “nuclear force carriers”, •
π+ π− π0 p p pn np nn pions acting as ladder operators (π± ⇔ J±) of SU(2) but the symmetry calls for a 3rd (neutral) pion (π0 ⇔ J0)
Where is the third member of the I = 1 iso-triplet ? • Lattes & Gardner (Berkeley, 1950) π0 γ γ (calorimetry) • →
m = 140 MeV m 0 = 135 MeV • π± π Pions seem OK too... next, from spectroscopy to π N interactions • −
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 19 evjas-2013
Isospin sym.-based S.I. phenomenology
an attempt at pion-nucleon dynamics
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 20 evjas-2013 Pion-Nucleon phenomenology
An isospin-invariant model for nuclear physics • Comment: In the absence of an underlying theory, it is common practice in tentative model building to construct effective L’s that hold the assumed symmetry principles.
An isospin symmetric free for the nucleon can be written • L p ¯ ¯i µ ¯i j ψ = ψ = (¯p, n¯) free = i ψ γ ∂µ ψi ψ Mi ψj n L −
where the mass matrix M is very close to the identity 2×2 I
mp 0 mn mp M = − =0.0014 m 0 mn n and therefore (approximately) commutes with the SU(2) isospin generators.
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 21 evjas-2013 Pion-Nucleon phenomenology
Pion fields are introduced as an isospin-one triplet state • isospin eigenstates charge eigenstates
π π+ =( π + iπ )/√2 1 − 1 2 π = π π− =( π iπ )/√2 2 − 1 − 2 π π0 = π 3 3
(cartesian coords.) (l = 1 spher. harmonics)
to effectively mediate nucleon-nucleon interactions (Yukawa).
(fermion-fermion-scalar vertices are now commonly referred to as Yukawa interactions)
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 22 evjas-2013 The Interaction Lagrangean
By introducing an isospin-invariant pion-nucleon interaction • † σ int = gN N π L 2 we get, in a leading order perturbation theory approach,
L = p†pπ0 n†nπ0 √2 p†nπ+ √2 n†pπ− g − − − with symmetry based predictions for the relative strengths of the various pion-nucleon couplings.
Of course, we now know better... This is an overly simplified way of • representing the strong interactions which are not even perturba- tive at this level. So we stop here for now...
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 23 evjas-2013
Inspecting the Weak Interactions
this is where isospin will play a more significant role...
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 24 evjas-2013 Radioactivity and the Neutrino
Becquerel (1896), Rutherford (1901++) • Nuclear beta decay observed, and interpreted as
general AZ A(Z+1)+ β− → 214 214 − 3 3 − examples Pb82 Bi83 + β H1 He2 + β → → reductions n p + β−(+ν) u d + e− + ν → →
Chadwick (1914) there is • Expected measures the β− missing Observed energy spectrum: energy ! Number of events
Beta particle energy Pauli (1930) proposes that a particle escapes undetected. • In his letter; “A desperate remedy ... test and judge.”
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 25 evjas-2013 The Neutrino
Fermi (1932); “neutrinos” and the 4-fermion theory for charged cur- • rent (flavor changing) interaction
p will evolve to the u n e e flavordynamics of d − GF charged currents W ν¯e ν¯e
Cowan and Reines (1956); direct observation of ν’s from a nearby • fission reactor, AZ A(Z+1)+ β− + ν → enabling them to detectν ¯ + p e+ + n →
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 26 evjas-2013 Parity and the Neutrino
C.S. Wu et al. (1956) studied the correlation J pˆe in polarized • · 60Co β decays. (Proposal: C.N.Yang, T.D.Lee – Nobel 1957)
J · pˆe > 0 under P 66 P-invariance implies identical β− J → J emission rates in the parallel and • J anti-parallel hemispheres pˆe → −pˆe
J · pˆe < 0
An observed correlation between the 60Co nuclear spin J and the β− • emission directionp ˆe revealed P -violation in the weak interactions. The first example of a “forward-backward asymmetry”.
Quantitative analysis using the charged current model suggested: • Charged currents are purely left handed, while their anti-currents are purely right handed i.e. maximal P/ !!
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 27 evjas-2013 Weak Isospin – Universality
Using the isospin tools we just learned, consider β− decay and muon decay. •
Note: from Lederman, Schwartz, Steinberger — AGS (Brookhaven) 1962 — Nobel prize, 1989. we know that neutrinos come in different “flavors”
νµ will evolve to the νµ e e flavordynamics of − GF charged currents W ν¯e ν¯e
νe νµ suggesting lepton isospin doublet states ψe = and ψµ = e L L
Question: are the semileptonic GF (n p) and purely leptonic • → GF ( νµ) universal ? “Fermi constant” measurements: → answer YES! universality 5 2 ∼ * Gµ = 1.16637(1) × 10− GeV− in the W.I. is showing up, but with a “hickup”... * Gτ = Gµ = Gβ/0.974(3) and from the Fermi constant we extract 1/2 v = (GF √2)− = 246.2 GeV Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 28 evjas-2013 Cabibbo’s “Rotation”
Motivated by discrepancies between ∆S = 0 and 1 transitions • Semileptonic Decays
∆S = 0 |∆S| = 1
d u s u d d u u n → p e ν¯ Λ → p e ν¯ (π+ → π0 e ν¯) (K+ → π0 e ν¯)
u u K-decay see only 5% of d¯ s¯ the expected rate! π+ → µ+ ν K+ → µ+ ν
2 ∆S = 0 d u G cos θc Cabibbo’s proposal → ∼ 2 • ∆S = 1 s u G sin θc | | → ∼
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 29 evjas-2013 Mixing Angles
Quark “flavor” mixing • u c – Two iso-generations holding four quark flavors and d s – Weak (interaction) eigenstates are not equal to the mass (propagation) eigenstates – Being two different bases in isospin state, a rotation connects them and generates q-flavor mixing
Using the Cabibbo hypothesis over the mass eigenstates • ′ d cos θc sin θc d ′ = s − sin θc cos θc s int mass we obtain the interaction eigenstates u c and d cos θc + s sin θc s cos θc − d sin θc o ,β,π and K decays yield sin θc =0.2243(16) or θc = 13 •
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 30 evjas-2013 GIM (1970)
Using the interaction (rotated) eigenstates cos θc • s¯ W + + u c 0 and K u ν d cos θc + s sin θc s cos θc − d sin θc W − − d sin θc Quark mixing and the cancellation of FCNC • – The reason for proposing
cos θc a 4th quark (charm) s¯ W + + – Example: K0(ds¯) + − 0 −→ K c ν W − − d − sin θc
Note: the cancellation is not exact, d and u have diff. masses, hence propagators not identical. But the loop momentum integration is domi- nated by the W mass pole where d and u masses are negligible.
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 31 evjas-2013 CKM (1973)
Introduce a 3rd generation of flavors to explain CP-violation • “bottom” (FNAL, 1977), and “top” (FNAL, 1995)
SU(2)L charge ↑ SU(2)L charge ↓ u u d d c = Uu c ; s = Ud s t t b b L M L M interaction — propagation interaction — propagation
(uL) uM (dL) dM g2 g2Vud
• The CKM matrix (magnitudes – uncertainties not included):
V V V .974 .226 .004 u c t ud us ub U †Ud = V V V ≈ .226 .973 .042 d s b u cd cs cb Vtd Vts Vtb .009 .041 .999 Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 32 evjas-2013 Extracting CKM magnitudes
from R. Van Kooten, Swieca lectures 2009
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 33 evjas-2013 Comment on Isospin Symmetry
Created to interpret the nucleon S.I. • – turned out to be wrong... – one of the most productive “errors” ever!
This idea played a fundamental role in • – organizing quarks and leptons in generations – structuring the nature of their various transitions – developing the concept of internal symmetries (and consequently their spontaneous breaking) – eventually these became gauged internal symmetries
What made this surprising path possible ? • – or, why were we so lucky ?
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 34 evjas-2013
A fortunate accident(?) from C.Quigg’s TASI’00 lectures
Isospin invariance in the nuclear interactions was made manifest only by • a hitherto unexplained feature of Nature; mu md ≈ In the (u,d) subspace, QCD is not sensitive to (mu md) effects: • O −
(mu md (MeV ) << (ΛQCD 300MeV ) − ≈O ≈ QCD is flavor blind: identical masses imply identical strong dynamics. This is a feature present only in the first generation of quarks: •
mt >> mb >> mc >> ms >> md mu ≈ and rotations in (u,d) subspace (e.g. interchanging p’s and n’s) manifestly appeared as an invariance of the strong forces.
In the absence of such an accident (say; (mu md) ΛQCD), there would • − ≈ have been no direct clues to internal symmetries... yet again, the mystery of mass... An electron and a neutrino in a same multiplet ??
Arthur Maciel, C. Jord˜ao,SP (Jan. 2013) 35