Gauge Field Strength Tensor from the Overlap Dirac Operator

We derive the classical continuum limit of the operator tr$_s \sigma_{\mu\nu} D^{ov}(x,x)$ with $D^{ov}$ being the overlap Dirac operator and show that it corresponds to the gauge field strength tensor $F_{\mu\nu}(x)$.


Introduction
Lattice gauge operators are usually constructed explicitly from link variables. For example, the Wilson gauge action uses the product of gauge links at the boundary of the square plaquette. Similarly, the gauge field strength tensor can be defined through suitable combinations of such elementary plaquettes. To improve scaling behavior of the action and other gauge operators, rectangular and more sophisticated loops have been included [1,2,3,4]. Furthermore, it was shown that operators with smeared gauge links, being less ultralocal, are effective in filtering out ultraviolet fluctuations and improving the efficiency of numerical calculations [5,6,7,8].
A different approach is realized in constructing topological charge densities via Ginsparg-Wilson Dirac operators [9]. It uses matrix elements of the Dirac operator as a starting point. By virtue of the fact that this type of Dirac operators are inevitably non-ultralocal [10], they entail a sum of all gauge loops and are thus automatically smeared. While technically involved, this approach has the advantage that Ginsparg-Wilson operators incorporate an exact lattice chiral symmetry [11]. This leads to an index theorem on the lattice [9].
The overlap Dirac operator [12] offers a concrete example which satisfies Ginsparg-Wilson relation [13]. Its compact form makes it amenable to analytic as well as numerical calculations. The associated topological charge density was shown to have the correct classical continuum limit via weak-coupling expansion [14] and by direct calculations [15,16,17]. It is with this topological density operator that the subdimensional long-range structure has been discovered [18,19,20] and confirmed [21] in QCD, as well as in 2-D CP(N-1) models [22]. It is also with this operator that the required negativity of the topological density correlator is borne out clearly with only a handful of configurations [23]. Whereas, detecting this negativity to the same precision using conventional operators, such as those used in the glueball calculation [24], would require much larger statistics. Based on these observations, it was suggested by one of the authors that the condition of chiral symmetry plays a relevant role in efficient suppression of the ultraviolet noise [25], and that all gauge operators can be constructed from the chirally symmetric Dirac operator [26]. This way, one can also have a formulation of lattice QCD where the gauge action, the θ term and the fermion action are all expressed in terms of the lattice Dirac operator [26].
In the present work, we will concentrate on the gauge field strength tensor. It was suggested that the classical limit of the tensor component of the overlap operator, tr s σ µν D ov (x, x), is proportional to the gauge field strength tensor F µν (x) [26,27]. We shall explicitly calculate it and show that this is indeed the case. In view of the fact the chirally symmetric Dirac operators are non-ultralocal [10], the derivation is somewhat non-trivial. Since the overlap Dirac operator is expected to be local on gauge configurations of interest [28,29], our result implies that the quantum operator constructed this way represents a valid definition of gauge field strength tensor in lattice QCD. This could provide a practical tool for evaluating gluonic observables. A preliminary version of the calculation was given in Ref. [30].

Formulation
Our goal in this paper is to show that if we discretize a classical gauge field A µ (x) on a hypercubic lattice with (classical) lattice spacing a, and consider the overlap Dirac operator D ov on such background, then where "tr s " denotes the spinor trace. 1 By classical SU(3) gauge configuration we mean any configuration of the gauge field that is smooth (differentiable arbitrarily many times) almost everywhere. In the above equation we have implicitly assumed that x is not a singular point of A µ (x). It was also implicitly assumed that point x is a lattice grid point of a superimposed lattice for arbitrary lattice spacing a. This is technically most easily (and without loss of generality) achieved if we focus on the point x = 0 with the origin of the lattice coordinate system aligned with the Cartesian coordinate system of R 4 . Also, to avoid an extensive discussion of technical issues associated with the transcription of the field with singularities onto the hypercubic lattice, we will implicitly assume in the following that the field is smooth everywhere. This is sensible since due to the locality of the overlap operator, the result is expected to be valid for an arbitrary non-singular space-time point x. For a discussion relevant to this point, the reader is referred to Ref. [15].
In the convention that we will use, the continuum gauge potential A µ (x) is the vector field of traceless Hermitian matrices 2 and the corresponding field-strength tensor is With the covariant derivative defined as one has 1 Note that we use the convention that the real-valued arguments of lattice quantities (such as D(x, x) on LHS of Eq. (1)) are given in parenthesis, while the integer-valued lattice coordinates are written as subscripts (such as D n,n ).
2 Note that this differs from conventions of Ref. [26], where anti-Hermitian gauge potentials were used instead. The equations below can be obtained from equations of Ref. [26] via substitutions A µ (x) → iA µ (x), F µν (x) → iF µν (x). The value of constant c T in Eq.(9) is the same in both conventions.
The transcription of A µ (x) to the hypercubic lattice with integer coordinates n ≡ (n 1 , n 2 , n 3 , n 4 ) is accomplished in a standard manner. If a is the classical lattice spacing, we associate the lattice site n with the space-time point x = an, and the lattice link variable U n,µ is defined as The overlap Dirac operator D ov is given by [12] where −ρ, ρ ∈ (0, 2r), is the negative mass parameter and with We shall take the Euclidean γ-matrices to be Hermitian, i.e. γ † µ = γ µ and {γ µ , γ ν } = 2δ µ,ν .
With the above defining relations, we shall proceed to show the following in an explicit calculation: If A µ (x) is a smooth SU(3) gauge potential on R 4 , and U(a) is the transcription of this field to the hypercubic lattice with classical lattice spacing a, then where D ov 0,0 is the matrix element of the overlap operator at (m, n) = (0, 0). The

Calculation
To proceed with the calculation, we shall assume that tr s σ µν D ov 0,0 has a Taylor expansion in a and we will compute the leading contributions.
To evaluate the diagonal element D ov n,n we introduce the momentum variable in the following way [15,17] This will allow us to evaluate the inverse square root in Eq. (6). Next, we define the diagonal matrices K(k) as (K(k)) n,m ≡ e ikn δ n,m .
These matrices are unitary: If we now introduce the vector 1 such that 1 n = 1, a vector with all entries set to 1, we can rewrite the above expression as To calculate K −1 (k)D ov K(k), we assume that we can express 1 √ X † X as a power series in X † X. Then where where s µ = sin k µ , c µ = cos k µ .

Computational strategy
As we mentioned before, we assume that tr s σ µν D ov 0,0 has a Taylor expansion in a. We will compute the leading contributions by taking derivatives with respect to a and then evaluating the limit a → 0. We assume that all the matrices and matrix products are well defined and that we can take derivatives in the usual fashion. The non-trivial part of the calculation is taking the derivative of 1 √ Y † Y with respect to a. For this we will write under the condition that Y † Y does not have zero eigenvalues, which is satisfied for the case of classical gauge fields sufficiently close to the continuum limit [28]. Noting We see that the problem is reduced to computing derivatives of Y with respect to a. The only matrices that depend on a are the link matrices U µ . Since we are only interested in the limit a → 0 and since we will only calculate the contributions up to order a 2 , we only need the following limits where with n α being the component of the 4-vector n. We see that in the limit a → 0 both U µ and d da U µ reduce to an identity matrix in the space-time coordinates, but they are not necessarily diagonal in color space. However, the second derivative N∇A µ (0) has a term that is different: this matrix is still diagonal in the space-time index (since it is the derivative of a diagonal matrix) but the diagonal elements are not equal.
To compute (Y 1 √ Y † Y 1) 0 we will need to justify several relations. • Relation 1: where and is a number.
To show Eq. (22), we write The commutator is zero since [S µ , S ν ] = 0 and thus we have It is easy to see thatR 0 1 = 0 andD / 0 1 = 0 since the "derivative" like term, S µ − 1, vanishes when acting on 1. This "derivative" term comes from the continuum limit of (U µ S µ − 1) which becomes (S µ − 1) as a → 0. We thus have • Relation 2: This can be straight-forwardly shown if we expand as a power series in Y † 0 Y 0 and apply Relation 1 in Eq. (22) successively.
• Relation 3: where To compute the second order derivative in Eq. (19) we will need this relation when the matrix acts on non-constant vectors of the form N∇A µ (0)1 in Eq. (20).
As we mentioned before, we assume that our function admits a Taylor expansion. In this case, we can write We will now proceed to carry out our calculation order by order.

Calculation details 3.2.1 Order 0
We need to compute where the relevant term is tr s σ µν Y 0 due to the factR 0 1 =D / 0 1 = 0. Since Y 0 1 has only scalar and vector spinor components and no tensor component, it leads to tr s σ µν Y 0 1 = 0 and the zeroth order contribution is thus zero, i.e:

Order 1
We need to compute The first term contribution vanishes since and then we are left again with only scalar and vector spinor components. Thus To evaluate the second term we need to compute (Y † Y ) ′ 0 1 = (2MR ′ 0 −2i α s αD ′ α0 )1 using the following identities We have thenR ′ a constant vector. We have then We see that we only have scalar and vector spinor components and this term vanishes too after taking the spinor trace. Thus the first order contribution vanishes.

Order 2
The derivation of the second order contribution is somewhat involved, but employs the same steps as above. The details of the calculations are presented in Appendix A for the perusal of interested readers. The main result is that the second order contribution is not zero and is given by Together with the (null) results from the zeroth and first orders in Eqs. (39) and (46), the final result is Comparing with Eq. (9), we find With r = 1 and ρ = 1.368 (which corresponds to κ = 0.19 in the Wilson Dirac operator), we find c T = 0.11157. 3

Conclusions
We have shown in an explicit calculation that, for the overlap Dirac operator, the classical continuum limit of tr s σ µν D ov (x, x) is proportional to the gauge field strength tensor F µν (x).
Based on the experience of studying the QCD vacuum structure with the topological charge density defined from the overlap operator, it is found that one can obtain clear signals with only a handful of gauge configurations [18,19,20,21,22,23]. This 3 We should remark that D µ and R in Eq. (7) can be written as , such as defined in Ref. [17]. Upon taking derivatives with respect to a in e ±a∂µ and U µ in Eq. (36), the results in Eqs. (39), (46), (47), and (48) were obtained [30]. However, due to the fact that σ µν in Ref. [30] is defined with an opposite sign from the one used here which is σ µν ≡ 1 2i [γ µ , γ ν ], the result in Ref. [30] is negative of that in Eq. (48).
is presumably due to the non-ultralocal nature of the overlap operator which serves as an efficient filter of the ultraviolet fluctuations [25,26]. It is worthwhile then to study whether other operators defined in a similar fashion share this property. For example, it would be interesting to see if the calculation of glueball masses, glue momentum and angular momentum in the nucleon, etc. can benefit from employing the overlap-based definition of the field strength tensor which is properly normalized. We should point out that the value of the constant c T = c T (ρ) depends on the mass parameter ρ used to define the overlap operator. We will study this ρ-dependence in detail elsewhere [31].
Finally, we wish to mention that, for the purposes of studying QCD vacuum structure it is useful to be able to expand gauge observables in low-lying Dirac eigenmodes. Indeed, such expansions in the case of overlap-based topological density proved to be useful in studying the low-energy behavior of the topological vacuum structure [32,19]. Thus, one rationale for defining all gauge operators in terms of Dirac kernels is the fact that it allows such an expansion for an arbitrary operator [26]. We note that for the purpose of eigenmode expansion, the expression for the field strength tensor in terms of the squared lattice Dirac operators was also considered in Ref. [33].

Appendix A
We need to compute lim a→0 tr s σ µν The first term in Eq. (51) turns out to be zero, since the expression only has scalar and vector spinor components and, as a result, it vanishes upon taking the the spinor trace in Eq. (51). The second term in Eq. (51) also vanishes, because also has only scalar and vector components. The third term in Eq. (51) vanishes for the same reason, since (54) has only spinor and vector components.
The only non-zero contributions come from the fourth term in Eq. (51). To compute its contribution, we need to evaluate We will compute each term separately. For the first term, we have where U ′′ µ0 = −A µ (0) 2 + 2iN∇A µ (0). In this case, Y ′′ 0 1 will have a constant vector part and a non-constant part In considering the first term in Eq. (55), we note that the derivative terms in Y † 0 , i.e.R † 0 +D / † 0 , vanish when acting on constant vectors as shown in Eq. (38). Furthermore, the derivatives acting on the non-constant part produce constant vectors as in Eq. (33). As a result, we get a constant term and a non-constant term From Eq. (56), we have Consequently, the second term, Y ′′ The last term to evaluate is 2Y ′ 0 † Y ′ 0 1. We note that since this term only involves first derivatives it will only produce a constant vector. Using we get Putting all the contributions from Eqs. (59), (61), and (63) together, we obtain and We now return to evaluating the last term of the second order contribution in Eq. (51) From Eq. (30), we find for the non-constant term contribution (67) It is clear from Eq. (65) that the non-constant term ((Y † Y ) ′′ 0 1) nc is a scalar. Furthermore, from Eq. (31), we see that ∆ is a scalar too. Since Y 0 has only scalar and vector components, the terms in Eq. (67) above have the same spinor structure. Thus, after taking the spinor trace, all these terms vanish.
To finish our calculation, we will use the fact that z and M are even functions of k µ and thus any integral over k µ that involves an odd power of s µ will vanish. For the spinor trace we use the relation tr s σ µν γ α γ β = 4i(δ µα δ νβ − δ µβ δ να ) and we finally obtain