Yang-Mills ground-state wavefunctional in 2+1 dimensions

7.5.2008 Seminar, Institut f. Physik, Karl-Franzens-Universität, Graz 1

Yang-Mills ground-state wavefunctional

in 2+1 dimensions

(in collaboration with Jeff Greensite)

J. Greensite, ŠO, Phys. Rev. D 77 (2008) 065003, arXiv:0707.2860

[hep-lat]


Motivation

“QCD field theory with six flavors of quarks with three colors, each represented by a Dirac spinor of four components, and with eight four-vector gluons, is a quantum theory of amplitudes for configurations each of which is 104 numbers at each point in space and time. To visualize all this qualitatively is too difficult. The thing to do is to take some qualitative feature to try to explain, and then to simplify the real situation as much as possible by replacing it by a model which is likely to have the same qualitative feature for analogous physical reasons.

The feature we try to understand is confinement of quarks. We simplify the model in a number of ways.

First, we change from three to two colors as the number of colors does not seem to be essential.

Next we suppose there are no quarks. Our problem of the confinement of quarks when there are no dynamic quarks can be converted, as Wilson has argued, to a question of the expectation of a loop integral. Or again even with no quarks, there is a confinement problem, namely the confinement of gluons.

The next simplification may be more serious. We go from the 3+1 dimensions of the real world to 2+1. There is no good reason to think understanding what goes on in 2+1 can immediately be carried by analogy to 3+1, nor even that the two cases behave similarly at all. There is a serious risk that in working in 2+1 dimensions you are wasting your time, or even that you are getting false impressions of how things work in 3+1. Nevertheless, the ease of visualization is so much greater that I think it worth the risk. So, unfortunately, we describe the situation in 2+1 dimensions, and we shall have to leave it to future work to see what can be carried over to 3+1.”



Introduction

Confinement is the property of the vacuum of quantized non-abelian gauge theories. In the hamiltonian formulation in D=d+1 dimensions and temporal gauge:


At large distance scales one expects:

Halpern (1979), Greensite (1979) Greensite, Iwasaki (1989)

Kawamura, Maeda, Sakamoto (1997) Karabali, Kim, Nair (1998)

Property of dimensional reduction: Computation of a spacelike loop in d+1 dimensions reduces to the calculation of a Wilson loop in Yang-Mills theory in d Euclidean dimensions.


Suggestion for an approximate vacuum wavefunctional


Warm-up example: Abelian ED



Free-field limit (g!0)


Zero-mode, strong-field limit (D=2+1)

D. Diakonov (private communication to JG)

Let’s assume we keep only the zero-mode of the A-field, i.e. fields constant in space, varying in time. The lagrangian is

and the hamiltonian operator

Natural choice - 1/V expansion:


Keeping the leading term in V only:

The equation is solved by:

since


Now the proposed vacuum state coincides with this solution in the strong-field limit, assuming

The covariant laplacian is then

Let’s choose color axes so that both color vectors lie in, say, (12)-plane:


The eigenvalues of M are obtained from

Our wavefunctional becomes

In the strong-field limit


D=3+1



Dimensional reduction and confinement

What about confinement with such a vacuum state? Define “slow” and “fast” components using a mode-number cutoff:

Then:


Effectively for “slow” components

we then get the probability distribution of a 2D YM theory and can compute the string tension analytically (in lattice units):

Non-zero value of m implies non-zero string tension and confinement! Let’s revert the logic: to get with the right scaling behavior ~ 1/2, we need to

choose


Why m02 = -0 + m2 ?

Samuel (1997)


Non-zero m is energetically preferred

Take m as a variational parameter and minimize <H > with respect to m:

Assuming the variation of K with A in the neighborhood of thermalized configurations is small, and neglecting therefore functional derivatives of K w.r.t. A one gets:


Abelian free-field limit: minimum at m2 = 0 → 0.


Non-abelian case: Minimum at non-zero m2 (~ 0.3), though a higher value (~ 0.5) would be required to get the right string tension.

Could (and should) be improved!


Calculation of the mass gap

To extract the mass gap, one would like to compute

in the probability distribution:

Looks hopeless, K[A] is highly non-local, not even known for arbitrary fields.

But if - after choosing a gauge - K[A] does not vary a lot among thermalized configurations … then something can be done.

Numerical simulation


Numerical simulation of |0|2

Define:

Hypothesis:

Iterative procedure:


Practical implementation: choose e.g. axial A1=0 gauge, change variables from A2 to B. Then

Spiral gauge

1. given A2, set A2’=A2,2. the probability P [A;K[A’]] is gaussian in B, diagonalize K[A’] and

generate new B-field (set of Bs) stochastically;3. from B, calculate A2 in axial gauge, and compute everything of interest;4. go back to the first step, repeat as many times as necessary.

All this is done on a lattice. Of interest:

Eigenspectrum of the adjoint covariant laplacian. Connected field-strength correlator, to get the mass gap:

For comparison the same computed on 2D slices of 3D lattices generated by Monte Carlo.


Eigenspectrum of the adjoint covariant laplacian


Mass gap



Summary (of apparent pros)

Our simple approximate form of the confining YM vacuum wavefunctional in 2+1 dimensions has the following properties: It is a solution of the YM Schrödinger equation in the weak-

coupling limit … … and also in the zero-mode, strong-field limit. Dimensional reduction works: There is confinement (non-zero

string tension) if the free mass parameter m is larger than 0. m > 0 seems energetically preferred. If the free parameter m is adjusted to give the correct string

tension at the given coupling, then the correct value of the mass gap is also obtained.


Open questions (or contras?)

Can one improve (systematically) our vacuum wavefunctional Ansatz?

Can one make a more reliable variational estimate of m? Comparison to other proposals?

Karabali, Kim, Nair (1998) Leigh, Minic, Yelnikov (2007)

What about N-ality? Knowing the (approximate) ground state, can one construct an

(approximate) flux-tube state, estimate its energy as a function of separation, and get the right value of the string tension?

How to go to 3+1 dimensions? Much more challenging (Bianchi identity, numerical treatment very

CPU time consuming). The zero-mode, strong-field limit argument valid (in certain

approximation) also in D=3+1. Comparison to KKN

N-alityFlux-tube state


Elements of the KKN approach

Matrix parametrisation:

Jakobian of the transformation leads to appearance of a WZW-like term in the action.


Comparison to KKN

Wavefunctional expressed in terms of still another variable:

It’s argued that the part bilinear in field variables has the form:

The KKN string tension following from the above differs from string tensions obtained by standard MC methods, and the disagreement worsens with increasing .


N-ality

Dimensional reduction form at large distances implies area law for large Wilson loops, but also Casimir scaling of higher-representation Wilson loops.

How does Casimir scaling turn into N-ality dependence, how does color screening enter the game?

A possibility: Necessity to introduce additional term(s), e.g. a gauge-invariant mass term

Cornwall (2007)

… but color screening may be contained!

Strong-coupling:

Greensite (1980)


Guo, Chen, Li (1994)



Flux-tube state

A variational trial state:

The energy of such a state for a given quark-antiquark separation can be computed from:

On a lattice:

Work in progress.


Epilogue/Apologies

“It is normal for the true physicist not to worry too much about mathematical rigor. And why? Because one will have a test at the end of the day which is the confrontation with experiment. This does not mean that sloppiness is admissible: an experimentalist once told me that they check their computations ten times more than the theoreticians! However it’s normal not to be too formalist. This goes with a certain attitude of physicists towards mathematics: loosely speaking, they treat mathematics as a kind of prostitute. They use it in an absolutely free and shameless manner, taking any subject or part of a subject, without having the attitude of the mathematician who will only use something after some real understanding.”

Alain Connes in an interview with C. Goldstein and G. Skandalis, EMS Newsletter, March 2008.

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Yang-Mills ground-state wavefunctional in 2+1 dimensions