Towards a Solution of 2+1 (pure) QCD in the Planar Limit

Transcription

1 Towards a Solution of 2+1 (pure) QCD in the Planar Limit Rob Leigh University of Illinois hep-th/ with D. Minic

2 Outline the solution of QCD is certainly one of the grand problems of theoretical physics. it is probable that if a solution is to be found, it will be in the large N limit what ingredients do we need? identify proper variables and the non-perturbative vacuum -- the Master field demonstrate important observable consequences e.g., signals of confinement: area law, string tension, mass gap in 2+1, there is a Hamiltonian formulation, in temporal gauge, which admits a useful parameterization in terms of gauge invariant quantities these may be thought of as local versions of Wilson loops this formulation has an auxiliary gauge invariance whose consequences are crucial the large N, large t Hooft coupling limit may be related to a matrix model, which can be used to reveal the spectrum 2

3 Confinement in 1+1 QCD in the 1970 s, t Hooft showed that confinement can be seen directly by computing Feynman diagrams (large N) the pole of the quark propagator moves off to infinity, because of an IR divergence poles appear in multi-particle channels the partition function of pure QCD on a Riemann surface may be computed exactly, as a sum over SU(N) representations this partition function may be re-interpreted, term by term, as contributions from a QCD string theory thus, in 2d, pure QCD == string theory in turn, this may be related directly to (Das-Jevicki) collective field theory, and to a one-matrix model WOW! Witten! Gross & Taylor Minahan & Polychronakos; etc. 3

4 A Flowchart for 2d QCD QCD2 Gross & Taylor QCD string scaling limit maps:! " M 2d Collective Field Theory T-duality, F-transform oscillators M " #($) 1d matrix model 4

5 Gauge Invariant Formalism [da a µ ] QCD is written as a path integral V ol G eis[a] could change to gauge-invariant variables Φ[A] traditional choice is to take {Φ} to be Wilson loops [dφ] 1 det dφ da e is no path integral formulation known use loop space or Schwinger-Dyson approach certain configurations could dominate path integral such configurations, at large N, would act as classical objects get factorization of amplitudes ΦΦ... Φ Φ... this is essentially the concept of the master field knowledge of master field is equivalent to knowledge of the true vacuum 5

6 2+1 QCD in the Hamiltonian Formalism we consider 2+1 SU(N) Yang-Mills theory with Hamiltonian H Y M = 1 ( T r gy 2 M Π 2 i + 1 ) 2 gy 2 B 2 M we choose the temporal or Hamiltonian gauge, A 0 = 0, leaving the gauge fields A = (A 1 + ia 2 )/2, Ā = (A 1 ia 2 )/2 dynamical z = x 1 ix 2, z = x 1 ix 2 Π i E i is the momentum conjugate to A i A i = it a A a i [ t a, t b] = if abc t c g 2 Y M has dimensions of mass T r t a t b = 1 2 δab time-independent gauge transformations preserve the gauge condition, and the gauge fields transform as a connection A gag 1 gg 1, Ā gāg 1 gg 1, g(z, z) SU(N) Gauss law implies that observables are gauge invariant 6

7 Wilson Loops? a possible basis of gauge invariant operators might be the Wilson loops these are non-local, depending on loops C these are related in a complicated way to what we may regard as natural observables for example, in the large momentum limit (weak coupling), we should expect to see gluons emerging in the spectrum the physics of these loops are in principal determined by loop equations rather difficult to interpret many questions e.g., what is the nature of the vacuum? Φ(C) = T rp e i H C(Adz+Ād z ) there is a better way... 7

8 Karabali-Nair Parameterization it is possible to parameterize the gauge fields as A = MM 1, Ā = M M where M SL(N, C) this looks pure gauge, but M is not unitary remarkably, M transforms linearly under gauge transformations A traceless det M = 1 gauge invariant variables may be written simply note that these are local fields. Roughly, M may be thought of as analogous to an open Wilson line, and H a closed loop the Wilson loop evaluates to M gm H = M M Φ(C) = T rp e i H C(Adz+Ād z ) = T rp e i H C dependence on C is an artifact; one can use the local H variables instead. 8 dz HH 1

9 Holomorphic Invariance one might wonder if the parameterization is well-defined does H capture all of the physics? Is the parameterization one-to-one? in fact, there is a new holomorphic invariance acting on M on the right, which is not seen by the original gauge fields M(z, z) M(z, z)h ( z) the appearance of this can be seen by attempting to invert the defining relations ( ) M(z, z) = 1 d 2 w G(z, w)a(w, w) +... V ( z) so one must ensure that all results are holomorphic invariant one could simply fix the gauge invariance on physical states H(z, z) h(z)h(z, z)h ( z) this will have important consequences... M (z, z) h(z)m (z, z) V = 1, and then enforce holomorphic x G(x, y) = δ (2) (x y) x Ḡ(x, y) = δ (2) (x y) 9

10 The Jacobian now, a change of variables is not too remarkable, classically. However, in this particular case, the path integral Jacobian of the transformation can be worked out -- in fact it is given in terms of the level 2c A hermitian Wess-Zumino-Witten model S W ZW [H] = 1 2π dµ[c] = σ dµ[h]e 2c AS W ZW [H] d 2 z T r H 1 HH 1 i H+ 12π this is both gauge and holomorphic invariant thus the inner product on states can be written as an overlap integral of gauge and holomorphic invariant wave functionals with non-trivial measure 1 2 = dµ[h]e 2c AS W ZW [H] Ψ 1Ψ 2 one can redefine to canonical norm via Φ = e c AS W ZW [H] Ψ d 3 xɛ µνλ T r H 1 µ HH 1 ν HH 1 λ H dµ[h] = dµ(m,m ) V ol G 10

11 The Hamiltonian it is natural to introduce the current J = c A π HH 1 the YM Hamiltonian can then be rewritten in terms of J ( H KN [J] = m x J a (x) δ δj a (x) + x,y Ω ab (x, y) δ δj a (x) δ δj b (y) ) + π mc A x J is a connection for holomorphic invariance: J hjh 1 + π c A hh 1 J a J a where we have introduced the t Hooft coupling Ω ab (x, y) = c A π 2 δ ab (x y) 2 i f abc J c (x) π (x y) m = g2 Y M c A 2π the derivation of the Hamiltonian has involved a careful gauge-invariant regularization wavefunctions can be regarded as (holomorphic invariant) functionals of J 11

12 Vacuum Wave-functional the kinetic part of the Hamiltonian admits the wavefunctional is normalizable, given the non-trivial measure Ψ = 1 (this putting in the potential energy, we may find a wavefunctional (by an expansion in B/m 2 ) Ψ exp 1 2g 2 Y M ( ) 1 B(x) m + m 2 2 this wavefunctional seems to be sensible at both large and small k x,y B(y) R B 2 /k At large k, we find Ψ e 1 2g Y 2 M invariant 2-point function of gluons, appropriate to the conformally AA gy 2 M / k In the low momentum region, the momentum factor is cut off Ψ = e 1 2g 2 Y M m R T r B 2 12

13 Area Law and String Tension the low momentum wave-functional provides a probability measure equivalent to the partition function of the Euclidean 2-dimensional Yang- Mills theory with effective YM coupling g2d 2 = mg2 Y M this can be used to deduce the area law for the Wilson loop (fund. rep.) Φ exp( σa) where the string tension is σ = g4 Y M C 2(G)C 2 (R) 4π = g 4 Y M N 2 1 8π this formula agrees extremely well (~3%) with lattice simulations it is valid at any N, and has a sensible form for large N it is intuitively consistent with the existence of a mass gap similar observables may be computed, interpolating between long and short distances 13

14 Comparison to Lattice Data σ/g 2 Y M Lattice Theory SU(2) SU(3) SU(4) SU(5) Teper 98 14

15 Large N and the Master Field at large N, correlation functions factorize -- vacuum is singled out limit is expected to be controlled by a master field, an matrix configuration correlation functions are to be computed in terms of classical objects determined by the master field knowledge of the master field is equivalent to knowing the correct nonperturbative vacuum at large N (we will phrase things this way) the master field should capture the physics of both asymptotic freedom and confinement (i.e., the high and low momentum limits) small perturbations around this configuration should lead to the spectrum of glueballs expect a tower of non-interacting massive colourless particles our basic claim is that the vacuum wave-functional given above becomes exact at large N can expand around H = 1; large t Hooft coupling, there is a reduction of the system to a matrix model, which allows us to derive the spectrum the holomorphic invariance is a crucial ingredient 15

16 we are led to parameterize H by Matrix Variables H = e ϕ ϕ = ϕ a t a and expand in powers of ϕ the KN Hamiltonian may be regarded as a co"ective field theory and rewritten in terms of ϕ to proceed, one may expand (essentially, J is bosonized) H 1 H = ϕ [ ϕ, ϕ] [[ ϕ, ϕ] ϕ] +... = ta ϕ b e ba [ϕ] ϕ ab ϕ c f abc then for example we find S W ZW = 1 4π ϕ a ϕ b (g + ib) ab [ϕ] g [ϕ] = e [ϕ] e [ ϕ] h abc = f def e ad e be e cf h abc = b ab,c b cb,a b ac,b and (with appropriate regularization) H KN [ϕ] = x P a δ [ϕ](x) δϕ a (x) + x,y Q ab δ [ϕ](x, y) δϕ a (x) δ δϕ b (y) + π mc A x J a J a 16

17 Canonical Basis now one should rotate to the canonical basis this induces changes in the collective Hamiltonian the formal expressions are H 0 = c A mπ x Φ = e c AS W ZW [H] Ψ H = e c AS W ZW He c AS W ZW H 2 + H 1 + H 0 H 1 = (e ab ϕ b ) (e ac ϕ c ) c A x H 2 = Q ab δ [ϕ](x, y) x,y δϕ a (x) [P a [ϕ](x) 2c A x y δ δϕ b (y) Q ab [ϕ](x, y) δs W ZW δϕ b (y) P a [ϕ](x) δs [ W ZW δϕ a (x) c A Q ab [ϕ](x, y) x,y ] δ δϕ a (x) δ 2 S W ZW δϕ a (x)δϕ b (y) c A δs W ZW δϕ a (x) δs W ZW δϕ b (y) ] when expanded in powers of ϕ, one finds H = g2 Y M 2 x π a (x) y C(x, y)π a (y)+ m2 2g 2 Y M ϕ a ϕ a + 2 g 2 Y M ϕ a ( ) ϕ a +... where π is the momentum conjugate to ϕ 1, and C(x, y) F.T. k 2 17

18 Comments H = g2 Y M 2 x π a (x) y C(x, y)π a (y)+ m2 2g 2 Y M ϕ a ϕ a + 2 g 2 Y M ϕ a ( ) ϕ a +... it is clear from this expression that we have not achieved canonical normalization for the field ϕ the dispersion relation though is of a simple form (k) = k 2 ( E 2 k 2 m 2) note too that this seems to describe N 2 1 degrees of freedom this is sensible from the point of view of the perturbative vacuum the holomorphic invariance has not been taken in to account; this will reduce #d.o.f. to o(n) there are also non-quadratic corrections to the Hamiltonian, which are not very well-behaved from the point of view of locality however, we will shortly understand that the second term should be regarded as a mass term and this should cut off the infrared -- would be nice to demonstrate an explicit resummation however, the quadratic part of the Hamiltonian captures the planar vacuum 18

19 Matrix Variables to analyze this theory further, we reduce to an effective matrix model, proceeding in several steps 1. perform the field redefinition (non-local, but non-singular) this results in a Hamiltonian density 1 2 φ a ( k) = k k g 2 Y M ϕ a ( k). T r ( π 2 + m 2 rφ 2 φ[p a, [P a, φ]] +... ) let us focus on the mass term : if we trace back its origin, we find that it arises from the path integral measure so in a sense this is a non-perturbative effect, even though a" the analysis is (semi-)classical. Claim that this is the most important effect -- a"ows passage to the true non-perturbative vacuu) φ(x) e ipax aφ R e ip ax a 2. take large t Hooft coupling limit : p 2 i m 2 r 1 the Hamiltonian density reduces to the matrix model H = 1 T r ( π 2 + m 2 2 rφ ) 1 T r 2 ) ( δ2 δφ 2 + m2 rφ

20 Holomorphic Invariance now recall the crucial fact of holomorphic invariance: the master field H transforms homogeneously under such a transformation H hhh in particular, the vacuum transformations H = 1 is preserved by constant unitary Therefore in the planar limit the matrix φ transforms as φ hφh will also be an -matrix that since we must require invariance under this action, we must restrict to th* singlet sector of the matrix model thus, the degrees of freedom are reduced from o(n 2 ) to o(n) 20

21 A Flowchart for 3d QCD QCD3 Ham. gauge, large N 3d Collective Field Theory low momentum 2d Collective Field Theory Das-Jevicki (M! "(#)) singlet sector of 1d matrix model holo. inv. M&P oscillators QCD3 string 21

22 The Planar Spectrum of 2+1 YM we can then study the spectrum in the planar limit by looking at the singlet sector of the one-matrix model it is well known that the density of eigenvalues (λ) of the matrix φ is given by the semi-circular Wigner-Dyson distribution ρ W D (λ) = 1 π 2α m 2 r λ 2 at least in the approximation where the potential is quadratic the master field can be described in terms of non-commutative probability theory, and the spectrum of excitations determined by considering linearized perturbations 22

23 Spectrum for a quadratic potential, an analysis of small fluctuations gives solutions η(q) cos(nπ q q(λ) ) ω = nω c n = 1, 2,... ( ) ω c = π Λ dλ [2(α V (λ))] 1 2 = mr q(λ) = λ 0 dx [2(α V (x))] 1 2 Λ = 2α m 2 r so we have a spectrum consisting of a tower of equally spaced states, with fundamental frequency set by ω c so there is a mass gap, which emerges because of the finiteness of the cut of th* semi-circle distribution. 23

24 Spectrum, comments Lorentz covariance may be restored to these expressions states (glueballs) assemble into Regge trajectory although we have lost information about spins of the states although we have taken the low-momentum limit, the result does seem to make sense more generally can reconstruct the vacuum wavefunctional has smooth m 0 limit -- recover Coulomb behaviour can recast matrix model result as a gap equatio+ ( ) M = π Λ(mr (M)) 1 2 dλ [2 (α V (λ, m 0 r (M)))] 1 2 because of the positive definite nature of the potential, a solution to this equation should exist. The lowest numerical value, in the units of the t Hooft coupling should correspond to the physical gap. 24

25 QCD String Field Theory one can interpret the formalism as providing a (second quantized) string field theory however, a worldsheet theory is not obvious (presumably interacting,...) one can construct oscillators, and think of the spectrum in that way. QCD3 Ham. gauge, large N 3d Collective Field Theory low momentum 2d Collective Field Theory Das-Jevicki (M! "(#)) singlet sector of 1d matrix model holo. inv. M&P oscillators QCD3 string 25

26 Outlook one can also consider adding fermions to the theory a rough sketch is as follows: if one introduces fermions in the fundamental representation, one can define gauge-invariant fermions (``constituent quarks? ) P = M ψ these are not holomorphic invariant and so do not create physical states however, one can construct physical meson states... presumably, this makes simple modifications (e.g. a shift of the level of WZW Jacobian) there are lots of correlation functions that could be calculated, etc. for example, four dimensions? trf 2 trf 2 deconstruction?? similar reparameterization? finite temperature? study confinement/deconfinement transition Hagedorn behavior? 26