Lattice Gauge Theory: A Non-Perturbative Approach to QCD

Transcription

1 Lattice Gauge Theory: A Non-Perturbative Approach to QCD Michael Dine Department of Physics University of California, Santa Cruz May 2011

2 Non-Perturbative Tools in Quantum Field Theory Limited: 1 Semi-classical methods. E.g. finite action solutions of the classical, Euclidean, equations of motion, Instantons.. 2 Lattice gauge theory.

3 Instantons in Non-Abelian Gauge Theories Won t review here. Require A µ µ g g (1) as x, so that the contribution to the action tends to. For gauge group SU(2), a guess ( Ansatz"): A µ = f (x 2 ) µ g g (2) where g = x 0 + i x σ x 2 (3)

4 Plugging in the Yang-Mills equations gives an equation for f (x 2 ). But the problem is simplified by noting that the Yang-Mills action can be written in terms of F µν and F µν = 1 2 ɛ µνρσf ρσ. (4) Start with d 4 x(f ± F) 2 = d 4 x ( F 2 + F 2 ± 2F F ) 0 (5) allowing one to write an inequality: S = 1 4g 2 d 4 xf 2 1 4g 2 So we can minimize the action if we can solve d 4 xf F. (6) F = ± F; S = 1 4g 2 (7) Such configurations are said to be (anti-) self-dual. d 4 xf F has a topological significance.

5 Now plugging in the self-dual equations gives a first order equation for the function f, with solution: Applications:. 1 U(1) problem 2 Strong CP problem. f (x) = x 2 x 2 + ρ 2. (8)

6 Lattice Gauge Theory The path integral allows us to give a definition of a quantum field theory which is not simply based in perturbation theory. If we take space-time to be made up of discrete points, and to have finite volume, the path integral reduces to an ordinary integral (perhaps of very high dimension). The problem becomes one of doing the integral, and perhaps of proving the existence of various limits (small lattice spacing, large volume). This approach has proven particularly useful in QCD, as summarized in review article by De Tar and in the February, 2004 issue of Physics Today. An excellent introduction to lattice gauge theory is provided by the original paper by Ken Wilson, Confinement of Quarks", Phys.Rev.D10: ,1974. I will put a link to the article on my website. In addition, there are various texts, e.g. Creutz, Michael, Quarks, gluons, and lattices" which I will put on reserve.

7 Crucial to the construction of a gauge-invariant action is the Wilson line: U = Pe i dx A where P denotes ordering of matrices along the path (this is the analog of time ordering), and we have written the gauge field in a matrix form. This object has a simple transformation property under gauge transformations: (9) U(x 2, x 1 ) g(x 2 )U(x 2, x 1 )g (x 1 ). (10)

8 It is best to understand this first in the context of a U(1) gauge theory. In this case, the A s are not matrices and path ordering is trivial. Then U(x 1, x 2 ) = e i x x dx µ A µ e iω(x 2) U(x 1, x 2 )e iω(x 1). (11)

9 We can form a gauge invariant object (the Wilson line) by integrating around a closed loop. If we take the closed loop to be a small square (plaquette) on the lattice, and if we approximate: U ɛ = x+aɛ x dx µ A µ = 1 a(a(x) ɛ + A(x + aɛ) ɛ) (12) 2 then U µν e ia2 F µν. (13)

10 To see that, in the non-abelian case, the Wilson line transforms properly under gauge transforms, one can proceed as follows. 1 Assume that U(x 2, x 1 ) transforms properly. 2 Write the transformation law for U(x + ɛ, y) for infinitesimal ɛ. 3 From this result, write a differential equation for U g. 4 Note that that U(x + ɛ, y) = U(x + ɛ, y) + ia(x) ɛu and verify that this transforms as above. 5 Formulate these statements as differential equations satisfied by U.

11 Using this, one can one can again construct plaquette operators as for the Abelian theory For a square lattice, the plaquette operator U P is just the product of U s around a plaquette of the lattice. (See Peskin and Schroeder, fig. 15.1) In order to construct a lattice action, it is helpful to return to the abelian case. The formal expressions in the non-abelian case are virtually identical. For small a, 1 4g 2 (1 U ) a 4 ( 1 4g 2 ) F 2 µν (14) This is known as the Wilson action. It generalizes immediately to the non-abelian case.

12 Adding fermions It is a simple matter to generalize the Dirac lagrangian. One can use the Wilson line to define a generalized covariant derivative: q(x)d µ q(x) m qq 1 2a ( q(x + aˆx µ )U(x + aˆx µ, x)q(x) q(x)q(x a (15) which is gauge invariant. Subtleties arise, however, when we consider the dispersion relation for fermions. Consider, for example, the propagator. Writing q(x) = e ikx, we see that we can restrict k < 2πa.

13 Calling s µ = sin(k µ ), we have G(k) = i k + m s 2 + m 2. (16) This is perfectly fine for k a 1. But for k near π, there is another pole in the dispersion relation. This problem leads to a doubling of the fermion spectrum. In fact, it turns out to be a theorem that one can t obtain chiral representations of the gauge group in this way. To avoid this problem, it is necessary to either explicitly break the chiral symmetry ( Wilson fermions"), or to somehow massage the path integral so as to obtain fewer fermions (staggered, or Kogut-Susskind fermions).

14 Further fermion complications Computers are not wired with Grassmann numbers. To deal with these, the usual procedure is to note that the fermion functional integral, for any fixed values of the gauge fields (the U, or link", variables), is Gaussian. So for any fixed value of the gauge fields, one can do the integral over fermions, leaving a determinant. In principle, this is straightforward, but determinants are computationally expensive, so the problem of rapidly evaluating determinants is one of the great challenges of numerical lattice gauge theories.

15 What is required of a successful lattice computation? 1 Work with a large enough lattice that one can take g(a) to be small, and still a hadron can fit comfortably. The physical scale associated with a particular g is given from the knowledge of the gauge coupling as a function of q 2 or distance: 8π 2 g 2 (a) = b 0 log(aλ QCD ) (17) where Λ QCD 300 MeV. So if, say, a = 3 GeV 1, one would like the lattice to have of order 20 spacings in each dimension. With four dimensions, this is a huge number. 2 Demonstrate that physical masses behave as by varying g; this requires that g be small. m phys = a 1 e 8π 2 b 0 g 2 (18) 3 Insure that rotational invariance is restored in the small a limit. 4 Insure that chiral symmetry is respected (while spontaneously broken) in the chiral limit.

16 After many years, lattice gauge theorists have achieved these goals. E.g. QCD potential:

17

18

19 Quark masses: