Symmetry and Confinement B. Caudy & JG, arXiv:0712.0999 B. Lucini & JG, in progress Jeff Greensite Quarks and Hadrons in Strong QCD St. Goar, Germany What is Confinement? Juliet: "What's in a name? That which we call a rose By any other name would smell as sweet." Romeo and Juliet (II, ii, 1-2) What are people trying to prove, in order to “prove” confinement? And what do they mean by that word? 1 linear static quark potential, rising to infinity most order parameters 2. colorless asymptotic particle states common terminology These are not quite the same thing, which raises some semantic issues: What are people trying to prove, in order to “prove” confinement? And what do they mean by that word? 1 linear static quark potential, rising to infinity most order parameters 2. colorless asymptotic particle states common terminology These are not quite the same thing, which raises some semantic issues: against #1 - in real QCD, with quarks, the static potential rises and then levels off, due to string breaking. so is real QCD not confining? What are people trying to prove, in order to “prove” confinement? And what do they mean by that word? 1 linear static quark potential, rising to infinity most order parameters 2. colorless asymptotic particle states common terminology These are not quite the same thing, which raises some semantic issues: against #1 - in real QCD, with quarks, the static potential rises and then levels off, due to string breaking. so is real QCD not confining? against #2 - asymptotic particle states are also colorless in a Higgs theory, where there is no linear potential at all. so are Higgs theories confining? The Fradkin-Shenker-Osterwalder-Seiler Theorem Consider an SU(2) gauge-Higgs theory with lattice action 1 1 S = β Tr[UUU †U †] + γ Tr[φ†(x)U (x)φ(x + µ)] 2 2 µ x,µ pl!aq ! " It has a phase diagram something like this: (Campos, 1997) The theorem says that there is no complete separation between the Higgs-like and the confinement-like regions. More precisely: between a point ”a” deep in the confinement-like regime ( β , γ 1 ) , and a point ! “b” deep in the Higgs regime ( β , γ 1 ) , ! there is a path from a to b such that all Green’s functions of all local, gauge-invariant operators A(x )B(x )C(x )... ! 1 2 3 " vary analytically along the path. b This rules out an abrupt transition from a colorless to a color-charged spectrum. a Creation operators for three colorless vector mesons: φ!†(x)Uµ(x)φ!(x + µ) , Re φ!†(x)Uµ(x)ϕ!(x + µ) , Im φ!†(x)Uµ(x)ϕ!(x + µ) ! " ! " where ϕ!(x) = σ2φ!(x) Higgs-like region: “W-bosons” Confinement-like region: Mesons (with scalar constituents) So “color confinement” in the asymptotic spectrum doesn’t distinguish between Higgs and “real QCD”-like dynamics. Question Can the confinement phase of a gauge theory be regarded as the symmetric (or broken) realization of a gauge symmetry? This idea appears in a number of popular approaches, in particular: i) Dual superconductivity ii) the Kugo-Ojima criterion iii) Coulomb-gauge confinement Naively, symmetry breaking of gauge invariance violates Elitzur’s Theorem: Local gauge symmetries do not break spontaneously. In the absence of gauge fixing, ϕ = 0 regardless of the shape of the Higgs potential. ! " However, although local symmetries can’t break spontaneously, it is still possible to break a global subgroup of the local symmetry. Subgroups of this kind are typically what remains of the local symmetry after a gauge choice. Landau gauge Remnant symmetries are homogenous gauge transformations g(x) = g There are also inhomogenous transformations 1 g(x) = exp[iΛa(!; x) σ ] (Hata, 1983) 2 a 1 Λa(!; x) = !a xµ g (A ! )a + O(g2) µ − ∂2 µ × µ analogous to abelian transformations Aµ Aµ + ∂µφ with → µ φ(x) = c + "µx 〈φ〉≠0 ߭ remnant gauge symmetry breaking. Coulomb gauge In addition to g ( x ) = g there is a much larger, time-dependent remnant symmetry g(x, t) = g(t) Define T L(x, T ) = P exp i dtA0(x, t) ! "0 # Then T r[ L ( x , T )] = T r [ g L ( x , T ) g † ] for constant transformations, but Tr[L(x, T )] = Tr[g(0)L(x, T )g†(T )] ! 〈Tr[L]〉 in Coulomb gauge probes the breaking of a remnant gauge symmetry which is different from that probed by 〈φ〉 in Landau gauge. It is insensitive to the breaking of the homogenous g(x)=g symmetry. In principle, 〈Tr[L]〉 and 〈φ〉 could show transitions (0 → non-zero) at different places in the space of couplings. Do different global subgroups of the gauge group break in different places in the phase diagram? If they do, then there is an ambiguity is the phrase “spontaneously broken gauge symmetry”. Precision requires specifying which symmetry is actually broken. With this in mind, we revisit three confinement criteria which are based on the symmetric or broken realization of a gauge symmetry. I. The Kugo-Ojima Criterion Kugo and Ojima introduce a function u ab ( p 2 ) defined by p p uab(p2) g µ ν = µν − p2 ! " 4 ip(x y) a b d x e − 0 T [D c (x)g(A c) (y) 0 " | µ ν × | $ # where ca(x) is the ghost field in a covariant gauge. They then show that the expectation value of charge vanishes in any physical state ph ys Q a ph ys = 0 (confinement?) ! | | " providing the following conditions are satisfied: 1. Remnant symmetry under g(x) = g is unbroken 2. The criterion u ab (0) = δ ab is satisfied. − It turns out that (2) implies that the spatially inhomogenous remnant symmetry in Landau gauge is also unbroken (Hata, Kugo). Therefore, the Kugo-Ojima scenario requires that the entire remnant gauge symmetry in Landau gauge is unbroken, i.e. 〈φ〉 = 0 . II. The Coulomb Criterion Marinari, Parisi, Paciello, Taglienti (1993) Olejnik, Zwanziger, JG (2004) The idea is to show that ޼ The Coulomb energy of an isolated color charge is infinite; ޽ The color Coulomb potential is confining. It turns out that both of these are implied by unbroken remnant gauge symmetry g(x, t) = g(t) which means that Tr L(x, T ) = 0 ! " #$ Isolated Charge a a Ψq = q (x)Ψ0 propagation in time a (H E )T a G(T ) = Ψ e− − 0 Ψ ! q | | q " Tr L(x, T ) ∝ ! " #$ infinite energy if G(T)=0 , which implies〈Tr[L]〉 = 0 Color-Coulomb Potential d Vcoul(R) = lim log Tr[L(x, T )L†(y, T )] − T 0 dT → ! " Vcoul(R) goes flat at G(R) → ∞ (no confinement) if 〈Tr[L]〉 ≠ 0 So both conditions require unbroken remnant gauge symmetry. Remnant Symmetry Breaking in the Gauge-Higgs Model Landau Gauge Coulomb Gauge order parameters 1 1 φ = φ(x) U(t) = U (x, t) V V 0 x 3 x " " ! ! L 1 1 t 1 QL = Tr[φφ†] Q = Tr[U(t)U †(t)] 2 C L 2 ! # t t=1 ! " $ "" symmetric phase # # 1 1 Q QL C ∝ V ∝ V3 broken phase lim QC > 0 lim QL > 0 V V →∞ →∞ We see a thermodynamic transition (or sharp crossover) for β > 2.0 " " ! ↑ ↑ /(001!,(-) $%&'(&)*)&+!,(-) . 1.2 " 2.2 # ! 4 4 Plaquette Energy, "=2.2, 16 Lattice Plaquette Energy, "=1.2, 16 Lattice 0.64 0.55 0.63 0.5 0.62 0.45 0.61 P 0.6 P 0.4 0.59 0.35 0.58 0.3 0.57 0.25 0.56 0.4 0.6 0.8 1 1.2 1.4 0.5 1 1.5 2 2.5 3 ! ! For β > 2.0 , the Landau and Coulomb gauge transitions happen at about the same γ . For β < 2.0 , these transitions happen at different γ . 4 "=1.2, 14 lattice 0.6 QL Q 0.5 C 0.4 Q 0.3 0.2 0.1 0 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 γ! We need to pinpoint the location of the transitions... Plotting Q vs lattice extension L shows that indeed and below the transition, and Q ≈ constant above.... Landau gauge, !=1.2 10 1 0.1 L Q 0.01 "=1.0 "=1.1 "=1.2 0.001 "=1.3 "=1.35 "=1.4 "=1.45 0.0001 1 1 10 Q QL L C ∝ V ∝ V3 but a better method is to look for peaks in the susceptibilities. Let Q = Q and define susceptibilities ! " ! χ = V 2 Q2 Q2 L ! L" − L ! # χ = V 2 Q2 Q2 C 3 ! "C " − C ! # " Landau gauge, #=1.2 Coulomb gauge, #=1.2 6 1.0*10 7.0*103 L= 6 L= 6 9.0*105 L= 8 L= 8 L=10 3 5 6.0*10 L=10 8.0*10 L=12 L=12 L=14 5 3 L=14 7.0*10 5.0*10 L=16 6.0*105 4.0*103 5 ! 5.0*10 ! 3 4.0*105 3.0*10 3.0*105 γ 2.0*103 2.0*105 3 1.0*105 1.0*10 0 0.0*10 0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 0.0*10 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 3 " " The final result: Gauge Symmetry-Breaking Transition Lines 3 Coulomb transition Landau transition 2.5 2 higgs-like region ! 1.5 1 0.5 confinement-like region 0 0 0.5 1 1.5 2 "β Coulomb and Landau remnant gauge symmetries break in different places, and they break in the absence of any discontinuity in the spectrum, or in the Green’s functions.