PHY646 - Quantum Field Theory and the Standard Model

Even Term 2020 Dr. Anosh Joseph, IISER Mohali

LECTURE 43

Tuesday, April 7, 2020 (Note: This is an online lecture due to COVID-19 interruption.)

Topic: Lattice Gauge Theories.

Lattice Gauge Theories

The QCD coupling constant is a scale-dependent quantity. At short distances the coupling becomes small, and it is possible to compare results from experiments with theoretical predictions based on perturbation theory. However, at long distances, the coupling is strong and the strong interactions are truly strong, leading to a break down of perturbation theory. In these situations some non- perturbative technique is necessary to perform calculations of quantities that are sensitive to the long-distance behavior of QCD. One of these techniques is called Lattice QCD. The approach of lattice QCD involves regularizing QCD by introducing a spacetime lattice with an ultraviolet cutoff (the lattice spacing a) and an infrared cutoff (the lattice volume V ). The degrees of freedom of QCD are assigned on the lattice sites, and the links connecting the sites. Matter fields such as quarks are restricted to occupy the sites of the lattice, and force carriers such as gluons live on the links joining the neighboring sites. Introducing the lattice spacing converts the Feynman path integral for QCD into an ordinary integral of very large dimensionality. The path integral expectation value of any observable can be computed by evaluating the integral on a computer, typically using Monte Carlo methods. Of course, the lattice spacing itself is an unphysical quantity; it was introduced in order to perform the calculation. Physical predictions require an extrapolation from nonzero lattice spacing to zero lattice spacing. The particular replacement of a continuum gauge field theory by a lattice field theory dates back to Wilson’s seminal work in 1974. Wilson showed that a lattice formulation of QCD exhibited confinement in the strong coupling limit. His work led to an explosive growth of research on the properties of strongly-coupled QCD. At first, all of this research was analytic. The use of Monte Carlo methods for studying statistical systems was already very old, as early as Metropolis et al. (1953); and in 1979 Creutz, Jacobs, PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 and Rebbi (Creutz al., 1979; Creutz 1979) adapted these techniques to the study of lattice gauge theories. The first computer simulations involving quarks began two years later. As techniques have advanced, the lattice have become an ever more accurate and important source of calculations of hadronic spectroscopy and matrix elements used in the studies of QCD, the Standard Model, and beyond. A four-dimensional Euclidean lattice Λ is defined as the hypercube

n o Λ = n = (n1, n2, n3, n4) | n1, n2, n3 = 0, 1, ··· ,NS − 1; n4 = 0, 1, ··· ,NT − 1 , (1) where NS and NT represent the number of sites in the spatial and temporal directions, respectively of the lattice. The vectors n ∈ Λ label points in (Euclidean) spacetime separated by a lattice spacing a. The matter fields ψ(n) and ψ(n) are placed on the lattice sites. We denote µˆ a vector of unit length a in the µ-th direction; so ψ(n +µ ˆ) and ψ(n) are the field values on the nearest-neighbor sites. The partial derivative is discretized with the symmetric expression

1   ∂ ψ(x) → ψ(n +µ ˆ) − ψ(n − µˆ) . (2) µ 2a

Then the lattice version of the free fermion action takes the form

 4  X X ψ(n +µ ˆ) − ψ(n − µˆ) S [ψ, ψ] = a4 ψ(n) γ + mψ(n) . (3) F  µ 2a  n∈Λ µ=1

Let us introduce gauge symmetry on the lattice. Gauge transformations can be implemented as discrete transformations on the lattice. The matter fields are rotated the following way by gauge group elements ψ(n) → g(n)ψ(n), ψ(n) → ψ(n)g†(n), (4) where the group element, g ∈ SU(N), and g(n) = exp(iαk(n)T k) is defined separately on each site. We also need to introduce the gauge field and the covariant derivative on the lattice. The kinetic part of the fermion action given above contains terms that are not gauge invariant. For instance, the fermion bilinear term ψ(n)ψ(n +µ ˆ) breaks gauge invariance on the lattice. If we introduce a link field Uµ(n) directed from site n to site n +µ ˆ transforming as

† Uµ(n) → g(n)Uµ(n)g (n +µ ˆ), (5) we can make the above bilinear term gauge invariant on the lattice. We have

h † ih † ih i ψ(n)Uµ(n)ψ(n +µ ˆ) → ψ(n)g (n) g(n)Uµ(n)g (n +µ ˆ) g(n +µ ˆ)ψ(n +µ ˆ)

= ψ(n)Uµ(n)ψ(n +µ ˆ), (6)

2 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 since g†(n)g(n) = 1. The gauge invariant fermion action takes the form

 4  X X Uµ(n)ψ(n +µ ˆ) − U−µ(n)ψ(n − µˆ) S [ψ, ψ, U] = a4 ψ(n) γ + mψ(n) . (7) F  µ 2a  n∈Λ µ=1

For convenience, we also define

† U−µ(n) = Uµ(n − µˆ), (8) which acts as a link in the opposite direction. In analogy to the continuum case, we also write

Uµ(n) = exp(iaAµ(n)), (9)

k k where Aµ(n) = Aµ(n)T ; and the coupling g has been absorbed into the gauge field. To construct a Yang-Mills action, we need a gauge-invariant object constructed out of the link fields. From the transformation property in Eq. (5), any closed loop of links will be gauge invariant. The simplest nontrivial loop goes in a little square. We call this a plaquette. See Fig. 1. We define a plaquette by

Uµν(n) = U−ν(n +ν ˆ)U−µ(n +µ ˆ +ν ˆ)Uν(n +µ ˆ)Uµ(n) † † = Uν (n)Uµ(n +ν ˆ)Uν(n +µ ˆ)Uµ(n). (10)

n + ν̂ Uμ(n + ν)̂ n + μ̂ + ν̂

Uν(n) Uν(n + μ)̂

n n μ Uμ(n) + ̂

Figure 1: A plaquette is a closed-oriented loop of link fields on the lattice. Continuum Yang-Mills action is recovered from a specific combination of plaquettes.

To connect plaquettes to the continuum field strengths Fµν(x), we can rewrite Uµν with the

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Baker-Campbell-Hausdorff formula

 1  exp(A) exp(B) = exp A + B + [A, B] + ··· . (11) 2

Up to order a2 we find

  ln Uµν = ia Aµ(n) + Aν(n +µ ˆ) − Aµ(n +ν ˆ) − Aν(n) a2 n + [A (n) + A (n +ν ˆ),A (n +µ ˆ) + A (n)] 2 ν µ ν µ o −[Aν(n),Aµ(n +ν ˆ)] − [Aν(n +µ ˆ),Aµ(n)] . (12)

2 To connect the continuum limit, we Taylor expand: Aν(n +µ ˆ) = Aν(n) + a∂µAν(n) + O(a ). This gives,

 2 2 3 Uµν = exp ia (∂µAν(n) − ∂νAµ(n)) + a [Aµ(n),Aν(n)] + O(a )  2 3 = exp ia Fµν(n) + O(a ) , (13) where Fµν ≡ (∂µAν − ∂νAµ) − i[Aµ,Aν], with g = 1. Expanding at small a, we find

a4 U (n) = 1 + ia2F (n) − F 2 (n) + O(a5). (14) µν µν 2 µν

The Wilson Gauge Action

What we are looking for is something that approaches the discretization of the continuum (Eu- 1 clidean) Yang-Mills action. After rescaling Aµ → g Aµ we have

Z  1  ia4 X X S [F ] = i d4x − (F a )2 = − Tr (F 2 ), (15) YM µν 4g2 µν 4Ng2 µν n µν for the gauge group SU(N). We therefore define the Yang-Mills action on the lattice, the Wilson gauge action, as

i X X S [U ] = − Re (Tr [1 − U (n))] . (16) lattice µν 2g2N µν n µν

The lattice action is the sum over all plaquettes, which are in turn defined in terms of link fields

Uµ(n). One can now evaluate the path integral for the lattice (in Euclidean space) numerically by literally summing over values for the links at each site. There are many things one can calculate with the lattice. For a concrete example, the most straightforward physical quantities to calculate are particle masses. These can be extracted from 2-point functions, which are calculated on the lattice by inserting fields at different lattice points into the discretized path integral weighted by the Euclidean action. For example, to calculate the

4 / 5 PHY646 - Quantum Field Theory and the Standard Model Even Term 2020 pion mass in QCD, with one quark flavor, we could calculate the discrete Euclidean version of the two-point function Z −S C(x) = hΩ|O(0)O(x)|Ωi = DAµ Du¯Due O(0)O(x), (17) where the operator O(x) =u ¯(x)γ5u(x). This correlation function should die off at large distances as e−mr, where m is the mass of the lightest particle with the same quantum numbers as O. Thus, by varying the distance between the points, one can extract the asymptotic behavior. The plot given 2 in Fig. 2 indicates that the pion mass scales as the square root of quark masses; mπ ∝ mq. (This result, known as the Gell-Mann-Oakes-Renner relation, can be derived analytically using chiral perturbation theory.) By using one such mass to set the overall units, one can then predict other things, such as other particle masses. Besides masses, the lattice is used to calculate many non- perturbative quantities, such as form factors. The lattice also gives insight into purely theoretical issues, such as spontaneous symmetry breaking in Yang-Mills theories with various number of flavors and colors.

Figure 2: Confirmation of the Gell-Mann-Oakes-Renner relation in lattice QCD. The lattice Monte Carlo simulations were performed on a 163 ×32 lattice at lattice spacing a ≈ 0.15 fm. (Left) Log plot for the pion correlation function; (Right) Effective mass plot (in lattice units). The different sets correspond to different values of the quark mass in the lattice units. The points are connected to guide the eye. This figure is taken from C. Gattringer and C. B. Lang, Quantum Chromodynamics on the Lattice: An Introductory Presentation, Lect. Notes Phys. 788, Springer, Berlin Heidelberg (2010).

References

[1] M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press (2013).

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