The Hamiltonian Approach to Yang-Mills (2+1)

Transcription

1 The Hamiltonian Approach to Yang-Mills (2+1) Part I: Basics and Update V. P. NAIR City College of the CUNY New Frontiers in Large N Gauge Theories Institute for Nuclear Theory, Seattle February 4, 2009 INT-Seattle p. 1/40

2 Why is YM(2+1) interesting? Interesting in its own right YM(1+1) is exactly solvable, but has no propagating degrees of freedom YM(3+1) is highly nontrivial and difficult YM(2+1) has propagating degrees of freedom, it is nontrivial. Can be amenable to a Hamiltonian analysis. It has a dimensional coupling constant and is super-renormalizable. This helps to simplify it. A real physical context for YM(2+1) Mass gap of YM(2+1) Magnetic screening mass of YM(3+1) at high temperatures We will use a Hamiltonian approach because some exact calculations are possible Collaborators: Dimitra Karabali, Chanju Kim, Alexander Yelnikov, Abhishek Agarwal INT-Seattle p. 2/40

3 Hamiltonian Analysis As in any theory, Hamiltonian analysis requires 3 basic ingredients 1. Inner product Matrix variables, calculation of gauge-invariant volume Proper gauge-invariant variables (CFT argument) 2. Hamiltonian H in the new variables Propagator mass, comparison with resummation techniques, Solve the Schrödinger equation HΨ = EΨ Ψ 0, the vacuum wave function Once we have these ingredients, we can discuss 1. String tension, corrections, comparison with lattice estimates 2. Magnetic screening mass 3. Screening of some representations 4. Glueball masses INT-Seattle p. 3/40

4 Matrix variables, volume element Choose A 0 = 0, this leaves A i, i = 1, 2. Gauge transformations act as A g i = g A i g 1 i g g 1 Wave functions are gauge-invariant ( Gauss law on Ψ) Choose complex coordinates, z = x 1 ix 2, z = x 1 + ix 2 Parametrize A as A A z = 1 2 (A 1 + ia 2 ), Ā = 1 2 (A 1 ia 2 ) A = M M 1 Ā = M 1 M G = SU(N) = M SL(N, C) = SU(N) C (Generally G G C ) This parametrization is well-known in 2-d YM context, WZW models, etc. INT-Seattle p. 4/40

5 Matrix variables, volume element (cont d.) There is an ambiguity, M and MV ( z) = same A (We will come back to this later) The basic advantage of this parametrization is the behaviour under gauge transformation, A A g i = g A i g 1 i g g 1 = M M g = g M H = M M is gauge-invariant Variation of the potentials is δa = (δmm 1 ) + [ MM 1, δmm 1 ] = D(δMM 1 ) δā = D(M 1 δm ) INT-Seattle p. 5/40

6 Matrix variables, volume element (cont d.) Calculation of volume element of the configuration space ds 2 A = = ds 2 SL(N,C) = d 2 x Tr(δAδĀ) Tr [ (M 1 δm )( DD)(δMM 1 ) ] Tr(M 1 δm δmm 1 ) dµ A = det( DD) dµ(m, M ) }{{} Haar measure for SL(N, C) We can split the SL(N,C) volume element as dµ(m, M ) = dµ(h) }{{} Haar for SL(N,C)/ SU(N) dµ(u) }{{} Haar for SU(N) INT-Seattle p. 6/40

7 Matrix variables, volume element (cont d.) The volume element is now dµ A = det( DD) dµ(h) dµ(u) For the gauge-invariant configuration space dµ(c) = det( DD) dµ(h) = dµ(h) exp[2 c A S wzw (H)] S wzw (H) is the Wess-Zumino-Witten (WZW) action, S wzw (H) = 1 2π Tr( H H 1 ) c A δ ab = f amn f bmn = N δ ab for SU(N). i 12π Tr(H 1 dh) 3 INT-Seattle p. 7/40

8 The inner product and current The inner product is now given as 1 2 = dµ(h) exp[2 c A S wzw (H)] Ψ 1 Ψ 2 The Wilson loop operator is given by ( W(C) = Tr Pe H π A = Tr P exp J = c A π H H 1 c A ) J All gauge-invariant quantities can be made from J. Also we see that Matrix elements in YM(2+1) = Correlators of a hermitian WZW model The integral of Ψ = 1 is the hermitian WZW partition function. INT-Seattle p. 8/40

9 Remarks on the volume element Parametrize H = M M as H = e ta ϕ a, so that H 1 δh = δϕ a R ab (ϕ)t b dµ(h) = [dϕ] detr We start with Γ = log det( DD) = Trlog( DD). Varying δγ [ ] δa a (x) = Tr D 1 Reg (x, y)( ita ) y x = Tr [ G(x, x)( it a ) ] = 1 π Tr [ MM 1 ( it a )] We re-integrate this to get the determinant. YM (2+1) has Gribov problem. But inner product formula has no difficulty due to this, it is exact INT-Seattle p. 9/40

10 The Hermitian WZW model The hermitian model can be analyzed by comparison with the unitary model Unitary model Level k exp[ks wzw (U)] Hermitian model Level k + 2 c A exp[(k + 2 c A )S wzw (H)] κ = k + c A κ = (k + c A ) Integrable rep s spin k Compare using (κ κ) Nonintegrable... = 0 Nonintegrable... = Finite norm = Ψ s made of integrable representations k = 0 for us, = Ψ s are functions of the current J INT-Seattle p. 10/40

11 Construction of H The Hamiltonian is given by H = e2 E a E a 2 }{{} + 1 B a B a 2e } 2 {{} T + V The kinetic term is simplified via the chain rule T Ψ = e2 2 = e2 2 δ 2 x δa(x)δā(x) Ψ [ δj(u) δj(v) δa(x) δā(x) }{{} δ 2 Ψ δj(u)δj(v) + δ 2 J(u) δa(x)δā(x) }{{} ] δψ δj(u) = Ω ab (u, v) Ω δ 2 Ψ δj a (u)δj b (v) + ω a (u) δψ δj a (u) ω INT-Seattle p. 11/40

12 Construction of H (cont d.) ω a (u) needs regularization ω a = e2 2 x δ 2 J a (u) δa b (x)δāb (x) = ( e 2 c A /2π ) M am(x) Tr [ t m D 1 reg(y, x) ] y x = m J a m = e 2 c A /2π (This is the basic mass scale of the theory.) The kinetic energy is thus given by [ T = m Ω ab (u, v) = c A π 2 J a δ δj a + δ ab (u v) 2 if abcj c (v) u v Ω ab (u, v) δ 2 ] δj a (u)δj b (v) + O(ǫ) Can be rechecked, J δ δj essential for self-adjointness of T. INT-Seattle p. 12/40

13 Regularization We start with a regularization of the δ-function This is equivalent to δ (2) (u, w) = σ( u, w, ǫ) = 1 ( ) πǫ exp u w 2 ǫ Ḡ( x, y) = = Ḡ( x, y) = 1 π(x y) Ḡ( x, u) σ( u, y; ǫ)h(u, ȳ)h 1 (y, ȳ) u This simplifies as G ma (x, y) = 1 π(x y) [ δ ma e (x y)2 ǫ [H(x, ȳ)h 1 (y, ȳ)] ma ] All results checked using regularization. INT-Seattle p. 13/40

14 Back to the Hamiltonian H T can be written as 1 T 2 = e2 4 [ dµ(h)e 2c AS wzw (H) (Gp) a ψ 1 H ab (Gp) b ψ 2 + (Ḡ p) a ψ 1 H ba (Ḡ p) b ψ 2 ] [p a ( x), M( y)] = M( y)( it a ) δ( y x) [ p a ( x), M ( y)] = ( it a )M ( y) δ( y x) Self-adjointness is manifest, 1 T 2 = T 1 2. We use this form for most calculations. The potential energy is easy to simplify V = 1 2e 2 B a B a = π mc A : J J : INT-Seattle p. 14/40

15 Back to the Hamiltonian H (cont d.) The regularized form is V = π mc A [ x,y σ( x, y; λ) J a ( x) [H(x, ȳ)h 1 (y, ȳ))] ab Jb ( y) c ] AdimG π 2 λ 2 The regularization for T and for V have to agree (in the choice of the parameter λ) so that H transforms correctly under Lorentz boosts. Once this choice is made, it can be shown, by a long calculation, T V = 2m V This is a very crucial result, allowing for the solution of the Schrödinger equation. INT-Seattle p. 15/40

16 Vacuum wave function Ignore V for the moment. Then we can take Ψ 0 = 1, this is okay since T Ψ 0 = 0, and since it is normalizable. dµ(h)e 2c AS wzw (H) Ψ 0Ψ 0 < Include V perturbatively (in 1/e 2 ). Writing Ψ 0 = e P, P = π : m 2 c J J : A ( ) 2 π m 2 : c JD J : + 1 : A 3 J[J, 2 J] : + + }{{}}{{} sum sum INT-Seattle p. 16/40

17 Vacuum wave function (cont d.) The summed-up result is P = 2 e 2 [ π 2 c 2 A J a (x) K(x, y) J a (y) + f abc J a (x)j b (y)j c (z)f(x, y, z) +... ] K(x, y) = [ ] 1 m + m 2 2 x,y f(x, y, z) = e ikx+ipy+iqz (2π) 2 δ(k + p + q) f(k, p, q) [ ] 3 π (E k m)(e p m) f(k, p, q) = 2c A E k + E p + E q E p = p 2 + m 2 k p kp INT-Seattle p. 17/40

18 Vacuum wave function (cont d.) The vacuum wave function is thus [ Ψ 0 = e P exp 1 B 1 ] 2e B 2 2 [ exp 1 ] B 2 4e 2 m k m 1 k m 1 O(J 3, J 4 ) terms are small at k e 2 and at k e 2. The high k limit agrees with perturbation theory. There are a couple of independent checks of this wave function. One is as follows. INT-Seattle p. 18/40

19 Vacuum wave function: A different argument Absorb exp(2c A S wzw ) from the inner product into the wave function by Ψ = e c AS wzw (H) Φ. The Hamiltonian acting on Φ is H e c AS wzw (H) H e c AS wzw (H) Consider H = e ta ϕ a 1 + t a ϕ a +, a small ϕ limit appropriate for a (resummed) perturbation theory. The new Hamiltonian is H = 1 2 where φ a ( k) = c A k k/(2πm) ϕ a ( k). ] [ δ2 δφ 2 + φ( 2 + m 2 )φ + The vacuum wave function is [ Φ 0 exp 1 2 φ a m 2 2 φ a ] INT-Seattle p. 19/40

20 Vacuum wave function: A different argument (cont d.) Transforming back to Ψ, Ψ 0 exp [ c A πm [ ] ] ( ϕ a 1 ) m + ( ϕ a ) + m 2 2 The full wave function must be a functional of J. The only form consistent with the above is [ [ ] ] Ψ 0 = exp 2π2 J a 1 e 2 c 2 (x) A m + J a (y) + m 2 2 since J (c A /π) ϕ + O(ϕ 2 ). This indicates the robustness of the Gaussian term in Ψ 0, since this argument only presumes 1. Existence of a regulator, so that the transformation Ψ Φ can be carried out 2. The two-dimensional anomaly calculation x,y INT-Seattle p. 20/40

21 String tension The expectation value of the Wilson loop can be calculated with Ψ 0 W R (C) = Tr R P e π c A H J The string tension σ is given as (constant) exp( σa C ) σ = e 4 c Ac R 4π c R = Casimir for representation R c A = Casimir for the adjoint representation, equal to N for SU(N) Consistent with large N expectations, even though we did not use large N analysis INT-Seattle p. 21/40

22 Comparison with lattice calculations Group SU(2) SU(3) Representations k=1 k=2 k=3 k=2 k=3 k=3 Fund. antisym antisym sym sym mixed SU(4) SU(5) SU(6) SU(N) N N N Comparison of σ/e 2 with lattice estimates (lower entry, in red) from Lucini &Teper, Bringholtz & Teper. k is the rank of the representation. INT-Seattle p. 22/40

23 Comment on higher corrections The differences are < 3%. The large N limit differs from lattice value by only 1.2%. Nevertheless, the difference is statistically significant. (Bringoltz & Teper) A different method of lattice calculation gives the value N at large N corresponding to a deviation of about 1.55%. (Kiskis & Narayanan) How can such corrections arise? Higher terms in recursive solution of Schrödinger equation For string tension, corrections can also arise from O(J 3 ) and higher terms in Ψ 0. To get parametric control on the calculation with higher terms, define an extension of the theory with e 2 and m as independent parameters. We set m = e 2 c A /2π at the end. The smallness of the corrections have to come from numerical factors. INT-Seattle p. 23/40

24 Comment on higher corrections Two types of corrections possible Corrections to coupling, purely numerical. Because it is R-independent, the ratios σ R /σ F are unaffected. For this, we will calculate the correction to the coefficient of J J in Ψ 0 by developing a well-defined expansion scheme Corrections via new diagrams to Wilson line expectation value (under investigation) An important R-dependent question, in the context of the second type of corrections, is: Can we see screening of the adjoint and other screenable representations? INT-Seattle p. 24/40

25 A systematic expansion Rescale J ej and write the Hamiltonian as H = H 0 + H 1, H 0 = m H 1 = e ic A 2π δ c 2 δj + A a 2π 3 + 2π2 c 2 : J J : A f abc J c (v) u v J a δ 2 δj a (u)δj b (v) δ ab δ 2 (u v) 2 δj a (u)δj b (v) At this stage, m is considered as an independent parameter. Solve the Schrödinger equation for Ψ 0 recursively in the parameter e. This gives a result of the form Ψ 0 = exp( 1 2S), with S = f (2) (x, y) J(x) J(y) + f (3) (x, y, z)j(x)j(y)j(z) +f (4) (w, x, y, z)j(w)j(x)j(y)j(z) + INT-Seattle p. 25/40

26 A systematic expansion (cont d.) The kernels have the expansion [ ] f (2) (x, y) = 4π2 1 e 2 c 2 A m + + f (2) m (x, y) x,y }{{} + }{{} previous result O(e 2 ) + f (3) (x, y, z) = f (3) 1 (x, y, z) }{{} + O(e) + f (4) (w, x, y, z) = f (4) 1 (w, x, y, z) }{{} + O(e 2 ) + Expectation values require O = Ψ 0Ψ 0 O = e S O INT-Seattle p. 26/40

27 A systematic expansion (cont d.) Corrections to a given order: Write Direct corrections to f s from recursion procedure, e.g., f (2) 1 (x, y) Loop corrections from J 3, J 4 type interaction terms e S(J) O(J) = This gives a two-dimensional field theory S(ϕ) = [dϕ] e S(ϕ) O(ϕ) [ Z 2 ϕ ϕ + Z 1 ϕt a ϕ(x) F (2) (x, y) Z ] 1 ϕt a ϕ(y) + Z 2 Z 2 In this case, we find ϕt a ϕ(x) ϕt b ϕ(y) = c Aδ ab π where E k = k 2 + m 2. d 2 k k m (2π) 2 k eik(x y) E k INT-Seattle p. 27/40

28 A systematic expansion (cont d.) Strategy for computing corrections Get corrections to f s from recursive solution Obtain all loop corrections to the given order from the interaction terms Numerical values of loop corrections decrease with increasing powers of m/e k, calculate to a given order in powers of m/e k. Set m = e 2 c A /2π at the end Ψ 0Ψ 0 = e Wilsonian reduction 2π 2 me 2 c 2 A R J J+ = e 2π2 g 2 c2 A R J J+ Computation to first order gives the corrected string tension Calculate this σr = e 2 ca c R 4π [ 1 + δ ] 3.1% Will be discussed in more detail in talk by A. Yelnikov INT-Seattle p. 28/40

29 Adjoint string breaking We now turn to the screening of the adjoint (and other screenable representations) This is due to the breaking of the string to glue-lump states. The glue-lump is made of the external charge and a gluon. There are two issues here: 1. The matrix element for String = Glue-lumps 2. The energy at which the string breaks The bound state can arise from diagrams like this which are in the integral over Ψ 0Ψ 0 INT-Seattle p. 29/40

30 Adjoint string breaking (cont d.) We do not have a calculation of the matrix element, but the energy E at which the string breaks can be estimated(agarwal, Karabali & VPN) A glue-lump state is given by Ψ G = d 2 xd 2 y f( x, y) J a ( x)[h(x, ȳ)h 1 (y, ȳ)] ab χ b ( y) Ψ 0 d 2 xd 2 y f( x, y) J a ( x)[per x y HH 1 ] ab χ b ( y) Ψ 0 χ = M φm 1 = A heavy scalar field of mass µ We obtain an eigenstate of the Hamiltonian if f(x, y) obeys the equation [ ] µ + m 2 x 2m + σ A x y f( x, y) E f( x, y) (This equation is valid when e 2 is large, µ and N.) INT-Seattle p. 30/40

31 Adjoint string breaking (cont d.) A variational estimate gives E 7.92 m. The lattice estimate is 8.68 m for SU(2) (de Forcrand & Kratochvila). The difference is 8.8%. It will be extremely useful to do a lattice calculation of the string-breaking energy for other groups. INT-Seattle p. 31/40

32 Magnetic mass, resummed perturbation theory Since T = m [ J δ δj + Ω δ δj δ δj ], Including the potential energy, T J a = m J a (T + V ) J a Ψ 0 = k 2 + m 2 J a Ψ 0 + J a is a gauge-invariant definition of a gluon. This is brought out more clearly by Ψ = e c AS wzw (H) Φ H = 1 2 ] [ δ2 δφ + 2 φ( 2 + m 2 )φ + This Lorentz-covariant form requires the Gaussian kernel to be just what we have, viz., K(x, y) = [ m + m 2 2] 1 x,y INT-Seattle p. 32/40

33 Magnetic mass (cont d.) We get a massive propagator in resummed perturbation theory 1 k0 2 k 2 m = 1 2 k k 2 m2 k k 2 m2 k 2 m2 k 2 + Propagator mass for gauge particles (magnetic mass) m = e2 c A 2π 0.32 e2 for SU(2) m/e 2 = 0.35 Common factor for glueball masses (lattice, Philipsen) 0.51 Max. Abelian gauge (lattice, Karsch et al) 0.52 Landau gauge ( " ) 0.44 λ 3 = 2 gauge ( " ) 0.38 Resummation of P.T. (Alexanian & VPN) 0.28 Resummation of P.T. (Buchmuller & Philipsen, Jackiw & Pi) 0.37 Gauge-invariant lattice definition (Philipsen) INT-Seattle p. 33/40

34 Glueballs We showed T J a = m J a, but J a is not a good state. M and MV ( z) give the same A via A = MM 1. We need invariance (holomorphic invariance) under J V JV 1 V V 1, J V J V 1 A 2J- state with holomorphic invariance is given by Ψ 2 = f(x, y) : J a (x) [H(x, ȳ)h 1 (y, ȳ)] ab J b (y) (This can give 0 ++ states.) If we use the approximate equation for f, { [ ] 2m 2 x + 2 y /2m + σa x y + }f E f M 0 ++ = 5.73 m compared to 5.17 m for the lattice estimate. INT-Seattle p. 34/40

35 Glueballs (cont d.) Leigh, Minic & Yelnikov analyze vacuum correlators (with a modified kernel) to obtain estimates of glueball masses The vacuum wave function is taken to be Ψ 0 = exp [ π 2m 2 c A K[L] = J 2(4 L) L J1 (4 L) J K[L] J ] Here L = D /m 2, and J 1, J 2 are Bessel functions of orders 1 and 2 respectively. This is motivated by equations like T ( J L n 2 J) = m n ( J L n 2 J) The case n = 2 is well-established (our earlier result), the others are not entirely clear. INT-Seattle p. 35/40

36 Glueballs (cont d.) The kernel, despite appearances, is very close to the KKN kernel LMY KN p m The two-point function, ignoring another class of interaction terms, is then : J a J a : x : J a J a : y 1 x y (M n M m ) 3 2 e (M n +M m ) x y M n = 1 2 m j 2,n zeros of the Bessel function J 2 INT-Seattle p. 36/40 n,m

37 Glueballs (cont d.) A number of glueball masses are obtained in this way State LMY Calculation Lattice ± ± ± ± ± ± ± ± ± ± ± ± ± ± 0.46 Lattice values are from Meyer & Teper and Teper INT-Seattle p. 37/40

38 A comment on the wave function The difference in kernels can perhaps be understood as follows. [ Ψ 0Ψ 0 = exp 1 ] [ e S 2 2d = Ψ 0Ψ 0 = exp 1 ] e S }{{} 2 2d,eff }{{} KKN form LMY form Wilsonian reduction? INT-Seattle p. 38/40

39 Status Report A gauge-invariant Hamiltonian formulation Vacuum wave function is of the form " Z» Ψ 0 = exp 2π2 J a 1 e 2 c 2 (x) A m + m 2 2 x,y J a (y) + # A systematic expansion scheme for which this is the lowest order result String tensions: lowest order and corrections of order a few percent Possibility of screening of W R (C) via string-breaking in some representations Magnetic screening mass Some results on glueballs Yang-Mills-Chern-Simons Theory Formulation on R S 2 INT-Seattle p. 39/40

40 Unclear issues, more questions More accurate calculation of higher order corrections to string tension Better handle on glueballs Calculations on the torus can help understand the theory at finite temperature Connecting the formulation on R S 2 to the duality-matrix model approach by changing the radius of S 2? Fermions, supersymmetric cases Geometrical properties of the configuration space A/G INT-Seattle p. 40/40