PlayingPlaying withwith QCDQCD I:I: EffectiveEffective FieldField TheoriesTheories

EduardoEduardo S.S. FragaFraga

Instituto de Física Universidade Federal do Rio de Janeiro IntroductionIntroduction andand motivationmotivation

JAS 2009 2 • Basic question: the structure of matter (“building blocks”) and the nature of fundamental interactions.

(Univ. Toronto) • New states of (“bulk”, medium) matter - need of extreme conditions to create new kinds of “plasmas”, “gases”, “liquids”, “crystals”. • Our focus: strong interactions and quark matter, QCD phase structure. • Need of sophisticated field theory formulation & diverse complementary techniques. Here: effective models. Mendes’s lectures: lattice simulations.

• EW sector: Eboli’s & Funchal’s lectures

JAS 2009 3 (M. Lahanas) Combining effective models & lattice: QCD you can play with!

eff. models + = Phase transitions & real-time dynamics!

Examples: • QCD phase diagram and the nature of strong interactions Sigma models, NJL, Polyakov Loop Model, quasiparticles and the QGP + Lattice QCD thermodynamics

• EW transition, cold baryogenesis, CP violation, tachyonic preheating Higgs+inflaton eff. model, 2PI effective action, 1/N-NLO approx. + Classical Lattice simulations

• Real-time evolution, dissipation, finite-size effects, transient effects Nucleation, spinodal decomposition, Langevin evolution + Real-time Lattice simulations

JAS 2009 4 Problem from theory side: really tough calculations, resummation methods, valid for high energy scales, misses vacuum pushing perturbative methods to the limit !

Problem from Lattice side: inclusion of dynamical quarks, realistic finite quark masses, finite µ problem tough, time consuming, very expensive computationally, etc.

But we can also play with effective models! And combine all that! Actually, we need to… Effective models are:

• valid in different (complementary) regions of phase space • very flexible and less constrained than the full theory • simpler in terms of tensor and group structures • much simpler (and faster) to simulate on the lattice • often directly connected to statistical mechanics systems • … JAS 2009 5 OutlineOutline

 1st Lecture • Introduction and motivation

• Evidence for quarks, gluons & color

• The QCD Lagrangian

• Asymptotic freedom & the vacuum of QCD

• Effective field theory basic recipe

• Summary

JAS 2009 6  2nd Lecture • Phase transitions in QCD Why? Where? How?

• A very simple framework: the bag model

• Symmetries in the QCD action: general picture

• SU(Nc), Z(Nc) and Polyakov loops

• Adding massless quarks (chiral symmetry)

• Summary

JAS 2009 7  3rd Lecture

• Effective models: general idea • Effective theory for the deconfinement transition using the Polyakov loop

• Effective theory for the chiral transition: the linear σ model

• Nonzero quark mass effects • Combining chiral and deconfinement transitions • Summary

JAS 2009 8  4th Lecture

• Finite T x finite µ: pQCD, lattice, sign problem, etc • Nuclear EoS: relativistic and non-relativistic (brief)

• pQCD at nonzero T and µ (brief)

• Cold pQCD at high density for massless quarks

• Nonzero mass effects

• Compact stars and QCD at high density • Summary

JAS 2009 9  5th Lecture [if time allows…] • Sample calculation in finite-temperature field theory within the λφ4 theory, building blocks of several effective models

• On the blackboard!

• Or… example of effective model: very intense magnetic field effects on the chiral transition.

JAS 2009 10  Things that will not be covered in these lectures • Saturation, parton evolution, small-x physics, pomerons, etc. • Sum rules of QCD • Light-cone QCD, AdS/QCD, etc. • Effective field theories in its “nuclear” aspect • Chiral perturbation theory (just mentioned) • Color superconductivity (just a little) • Fragmentation functions, hadronization, etc. • Quantum Hadrodynamics • Gribov copies, IR-QCD, etc. (see Tereza’s lectures) • Resummation and other technicalities of in-medium QCD (just mentioned) • Topological objects: instantons, θ-vacua, sphalerons, etc. (just mentioned) • …

JAS 2009 11 EvidenceEvidence forfor quarks,quarks, gluonsgluons && colorcolor

JAS 2009 12 Deep inelastic scattering

• Just like in Rutherford’s experiments with (~7 MeV) α particles on gold sheet targets, one can use high-energy electrons (~20 GeV at SLAC) to probe the proton! 1968: 1st evidence for quarks.

(DESY)

Differential cross-section for γ* absorption

F2: proton structure function; measure of quark distribution

Bjorken scaling: For free independent pointlike partons, F2=F2(x) JAS 2009 13 Exercise: Prove it! But things are more complicated…

(NIKHEF) JAS 2009 14

(Kass, 2003) Pieces of evidence for color

The hadron-muon branching ratio [overall Nc factor]

Flavors that contribute:

Crossing the different quark thresholds, one has approximate steps:

JAS 2009 (PDG) 15 0 2 π -> γ γ [overall Nc factor]

(PDG)

Nc=1, e.g., is totally off:

JAS 2009 16 Pieces of evidence for gluons scaling violation and parton evolution (deviations from the 3-jet events simplified quark model) (gluon radiation)

(HERA)

JAS 2009 17 (LEP) A “picture” of the proton (a real mess…) Parton distributions:

(Univ. Oxford)

(HERA)

JAS 2009 18 TheThe QCDQCD LagrangianLagrangian

JAS 2009 19 • In the usual approach to QED, the theory is obtained by quantization of classical electrodynamics. Gauge invariance is, then, shown and used as a side feature.

• On the other hand, one could start with free electrons and assume local U(1) invariance to “give birth” to the electromagnetic field -> the gauge principle!

• To build the QCD Lagrangian, we can start with Nc spin-1/2 free fermions that live in the fundamental

(vector) representation of SU(Nc).

• New quantum number: color - in QCD, Nc=3.

JAS 2009 20 Free quark Lagrangian (1 flavor):

Global unitary SU(Nc) transformations:

Remarks:

α • ψ behaves like a vector under SU(Nc)

• Ω are represented by unitary Nc x Nc matrices • Usually ψ has many indices (color, flavor, spin, …)

JAS 2009 21 For infinitesimal transformations:

ωa: infinitesimal (constant) parameters

a T : generators of SU(Nc)

abc f : structure constants of SU(Nc)

2 a = 1, 2, 3, …, (Nc - 1)

JAS 2009 22 Generalizing to local transformations is simple:

However, the free quark Lagrangian is no longer invariant:

So, we have to build a “covariant derivative” (transforms like ψ), which implies the introduction of a Nc x Nc matrix field: (gauge field)

which transforms under SU(Nc) in a way that suits the definition of the covariant derivative.

JAS 2009 23 That means:

Exercise: imposing gauge invariance, show that the gauge field has to transform as

or, for infinitesimal transformations:

JAS 2009 24 • If the field Aµ is to be identified with dynamical degrees of freedom (gluons), we must add a gauge-invariant kinetic term to the Lagrangian.

• Motivated by QED, we introduce

• By construction:

so that the trace of F2 is an invariant scalar:

JAS 2009 25 The final form of the QCD Lagrangian is given by

Remarks:

2 • Dynamical fields: Nf flavors of quarks in Nc colors & (Nc -1) gluons. • In the functional (Fadeev-Popov) quantization procedure, there will be ghost fields, etc. which are omitted here.

• mf is the quark mass matrix.

JAS 2009 26 Feynman rules:

JAS 2009 27 Remarks:

• Compared to QED, QCD is a lot more complicated due to its non-Abelian nature.

• Gluons do interact with each other, and this brings new phenomena (like confinement).

• Multi-loop calculations become rapidly cumbersome.

• As always, perturbation theory is valid only in a particular domain in momentum.

• All this calls for efficient approximation schemes: one of them is effective field theory and effective model building.

JAS 2009 28 AsymptoticAsymptotic freedomfreedom && thethe vacuumvacuum ofof QCDQCD • The behavior of the coupling with the momentum scale follows from the renormalization group (RG) flow equation:

(Λ: mass scale)

• The qualitative behavior is given by the sign of the beta function. To one loop:

• β is negative for Nf < (11/2)Nc = 16.5 -> asymptotic freedom !

• This is the effect of anti-screening due to gluon self-interactions (opposite to the screening by fermionic fluctuations in QED). As long as Nf < 16.5, gluons win !

JAS 2009 30 • In line with the phenomenon of quark confinement into color singlets (hadrons)

• Matter becomes simpler at very high energies, but very complicated in the opposite limit…

• The vacuum (bound states: hadrons) and other low-energy phenomena cannot be described by perturbative QCD! • Here, there is a clear need for effective models…

JAS 2009 31 EffectiveEffective fieldfield theorytheory basicbasic reciperecipe

JAS 2009 32 Basic recipe for building EFT models:

• Keep relevant symmetries

• Try to include in the effective action all terms allowed by the chosen symmetries

• Mimic of QCD (or other first-principle theory) at low energy using a simpler field theory

• Analytic results: estimates, qualitative behavior, etc. Often not too precise (except for χPT to higher orders)

• No need of renormalizability; natural cutoff (domain of validity)

• Just part of the story - combined with lattice QCD may provide good insight

JAS 2009 33 Summary • The building blocks of hadrons are quarks & gluons, although color is confined into singlet states.

• QCD is believed to be the fundamental theory of strong interactions. Its asymptotically free nature puts the vacuum out of reach for perturbation theory.

• The Lagrangian of QCD and the Feynman rules associated were built by using the Gauge Principle, starting from the quark matter fields and obtaining gluons as connections.

• A simpler, and sometimes necessary or complementary, approach is provided by effective field theories or effective models, especially when one has to deal with the nonperturbative sector of the theory.

JAS 2009 34  2nd Lecture: • Phase transitions in QCD Why? Where? How?

• A very simple framework: the bag model

• Symmetries in the QCD action: general picture

• SU(Nc), Z(Nc) and Polyakov loops

• Adding massless quarks (chiral symmetry)

• Summary

JAS 2009 35