Lattice worldline representation of correlators in a background field

We use a discrete worldline representation in order to study the continuum limit of the one-loop expectation value of dimension two and four local operators in a background field. We illustrate this technique in the case of a scalar field coupled to a non-Abelian background gauge field. The first two coefficients of the expansion in powers of the lattice spacing can be expressed as sums over random walks on a d-dimensional cubic lattice. Using combinatorial identities for the distribution of the areas of closed random walks on a lattice, these coefficients can be turned into simple integrals. Our results are valid for an anisotropic lattice, with arbitrary lattice spacings in each direction.


Introduction
In a recent work [1], we have studied the short distance behavior of the one-loop expectation value of local operators in lattice scalar quantum chromodynamics. For this study, we used a discrete version of the worldline formalism [2,3,4,5], in which propagators are represented as a sum over all the random walks on the lattice, that connect the two endpoints of the propagator. In this approach, the coefficients of the expansion in powers of the lattice spacing are related to combinatorial properties of these random walks. The leading coefficient merely counts the random walks of a certain length, but the following coefficients are related to areas enclosed by the random walk (for local operators, the two endpoints of the propagator are at the same point, and therefore its worldline representation is a sum over closed random walks). In fact, the successive terms of this expansion can be related to the moments of the distribution of the areas. In the case of an isotropic lattice (i.e. with identical lattice spacings in all directions), the results we needed could be found in ref. [6], in which the authors obtain the general 1 structure of the moments of the area distribution over the set of closed random walks of fixed length 2n. Moreover, this paper contains explicit formulas for the moments of order 2 and 4, that were sufficient for our purposes in the isotropic case.
However, the generalization of this expansion to the case of an anisotropic lattice (i.e. with different lattice spacings in each direction) requires the moments of the area over the set of closed random walks that make n 1 steps in the direction +1 and n 2 steps in the direction +2 (and also n 1 , n 2 steps in the directions −1, −2). Some numerical explorations and guesswork suggested a very simple form for these moments, namely the product of the number of such closed walks times a polynomial in n 1 , n 2 . Then, by taking this structure for granted, we obtained the expressions of these polynomials for the moments 2, 4, 6, 8 and 10 by making an exhaustive list of all the closed random walks up to the length 2n = 20. The purpose of the present paper is to prove these formulas, by exploiting the relationship between the distribution of areas and Quantum Electrodynamics (QED) on a 2-dimensional lattice.
Let us consider an infinite planar square lattice, and denote by Γ n1,n2 the set of the closed loops (with a fixed base point to avoid considering loops that are identical up to a translation) drawn on the edges of this lattice, that make 2n 1 hops in the directions ±1 and 2n 2 hops in the directions ±2. We are interested in the distribution of the algebraic areas enclosed by the loops in Γ n1,n2 . In this paper, we prove that γ∈Γn 1 ,n 2 (Area (γ)) 2l = (2(n 1 +n 2 ))! n 1 ! 2 n 2 ! 2 P 2l (n 1 , n 2 ) , where P 2l is a symmetric polynomial in n 1 , n 2 of degree 2l. Note that the combinatorial factor (2(n 1 +n 2 ))!/(n 1 ! 2 n 2 ! 2 ) is nothing but the cardinal of the set Γ n1,n2 . Therefore, the polynomial P 2l (n 1 , n 2 ) can be interpreted as the mean value of (Area (γ)) 2l over this set. Our approach provides an algorithm for calculating these polynomials explicitly. The first two non-zero moments involve the following polynomials 1 , P 2 (n 1 , n 2 ) = n 1 n 2 3 , (The polynomials that enter in eq. (1) up to 2l ≤ 12 are listed at the end of the section 3.) 1 By summing these expressions over all n 1 + n 2 = n, we recover the results of ref. [6], n 1 +n 2 =n (2(n 1 +n 2 ))! n 1 ! 2 n 2 ! 2 n 1 n 2 3 = 2n n 2 n 2 (n − 1) 6(2n − 1) , n 1 +n 2 =n (2(n 1 +n 2 ))! n 1 ! 2 n 2 ! 2 n 1 n 2 7n 1 n 2 −(n 1 +n 2 ) 15 = 2n n The rest of this paper is devoted to a proof of eqs. (1) and (2). Our approach is based on the formulation of lattice scalar quantum electrodynamics in the worldline formalism. When this formulation is used in two dimensions, in the presence of a transverse magnetic field, the propagator of the scalar particle can be viewed as a generating function for the distribution of the areas of random loops on a square lattice. In the section 2, we define more precisely the model and recall this correspondence. We also explain how to perform an expansion in powers of the interactions with the magnetic field. In the section 3, we prove the main result, namely eq. (1). Our proof leads to an algorithm to obtain explicit expressions for the polynomial P 2l , that we have implemented in order to obtain the formulas (2) and several higher moments. A few intermediate calculations are explained in the appendix A.
2 Lattice scalar QED in two dimensions 2.1 Area distribution of random walks from lattice QED Consider the following operator, acting on complex valued functions on 2 : ( In Quantum Electrodynamics, this operator is the square of the discrete 2dimensional covariant derivative 2 in two dimensions, acting on a scalar field. Here, we choose an anisotropic lattice, with lattice spacings a 1 and a 2 in the two directions. The coefficients U 1,ij and U 2,ij are complex phases called "link variables". Their name comes from the fact that they live on the edges of the lattice, while the function f ij lives on the nodes of the lattice. The link variables have a direction and an orientation: in the notation U 1,ij , the index 1 is the direction of the link, and the coordinates ij denote the starting point of the oriented link. Thus, U 1,ij lives on the edge that connects the points (i, j) and (i + 1, j). The complex conjugate of a link variable lives on the same link but has the opposite orientation. An interesting object is the inverse G ij,kl of this operator, which in QED is nothing but the propagator of a particle in a given external electromagnetic field. It is defined by In the worldline formalism, this inverse can be written as a sum over closed random walks (see ref. [1]). When evaluated with identical endpoints (we take the point of coordinates (0, 0), but this choice is irrelevant), its expression reads where ℓ∈γ U ℓ denotes the product of the link variables encountered along the closed path γ. We have also defined 2 By using an anisotropic lattice, we can isolate closed random walks with a given number of steps in each direction by selecting a given term in the Taylor series of the left hand side in powers of h 1 and h 2 . In other words, the order in h 1 and h 2 counts the numbers of steps made by the random walk in the directions ±1 and ±2. In order to obtain an explicit connection to the area enclosed by the paths, we need to specialize the link variables in such a way that they represent a magnetic field transverse to the plane of the lattice. By Gauss's law, the product ℓ∈γ U ℓ will then become the exponential of the magnetic flux through the loop γ, and will give direct access to the area. For a given transverse magnetic field B, there are in fact infinitely many ways to represent it in terms of link variables, due to the gauge invariance of electrodynamics. The choice that we adopt in the rest of this paper is the following : It is easy to check that where Area (γ) is the algebraic area enclosed by the path γ, measured as a number of elementary lattice plaquettes. Note that this equation would be true with any other gauge equivalent choice for the link variables 3 , since the left hand side is a gauge invariant quantity. In the rest of the paper, we denote φ ≡ Ba 1 a 2 the magnetic flux through one elementary plaquette of the lattice.
Knowing the left hand side of eq. (5) as a function of φ, h 1,2 would therefore give access to the full distribution of areas, since we can write Only the even moments are non-zero, since one gets the opposite algebraic area by reversing the path γ. Note that this correspondence is equivalent to the well-known relationship between the distribution of areas and the traces of the powers of the Harper Hamiltonian [7,8,9].

Rules for the expansion in powers of the magnetic field
In terms of the rescaled lattice spacings defined in eq. (6), the operator D reads where we have now specialized to the gauge choice of eq. (7). This operator can be separated into a vacuum part and a term due to the interactions with the magnetic field, with Let us first consider the vacuum case, in order to establish the notations. We consider an infinite lattice. In this case, the propagator can be conveniently represented in Fourier space by the following formula, .
By evaluating this expression with equal endpoints, i = k, j = l, and expanding in powers of h 1,2 , we can easily recover eq. (5) for a vanishing flux φ = 0. For this, we only need to recall the number of random loops that make n 1 hops in the direction +1 and n 2 hops in the direction +2 : In order to go beyond this trivial result, we must include the effect of the interaction term ∆. The inverse of eq. (11) can be written as a formal series Note that this is not exactly an expansion in powers of φ, since ∆ is itself an infinite series in φ. However, since ∆ starts at the order φ, we need only to calculate the terms up to the order ∆ 2l if we are interested in the order φ 2l . Note that, although the free propagator depends on a single momentum p ≡ (p 1 , p 2 ), the full propagator carries different momenta at its two endpoints, because of the interaction with the magnetic field. Thus, eq. (13) is replaced by where G(p, p ′ ) is the full propagator in momentum space. Here, we define the momenta in such a way that p enters at one endpoint and p ′ exits at the other endpoint of the propagator. The off-diagonal momentum components of the propagator are inherited from those of the interaction term, With the gauge choice that we have adopted, the interactions with the magnetic field always conserve the component p 2 of the momentum. Therefore, the full propagator G(p, p ′ ) is itself proportional to δ(p ′ 2 −p 2 ). In contrast, the magnetic field can change the component p 1 , but only in discrete increments ±φ (or 0 if we pick the third term in eq. (17)). For bookkeeping purposes, it is useful to write ∆ as a sum of three terms, The index ǫ denotes the increment (in units of φ) of the momentum p 1 caused by the interaction. Note that which implies that each insertion of the interaction term increases the order in φ by at least one unit.

Term of order ∆ k
We can organize the expression of G (k) as follows, where G (0) is the Fourier transform of the free propagator, that can be read off from eq. (13). Tracking explicitly the momentum flow, this can be written as where we denote 6 The contribution of this term to the left hand side of eq. (5) reads G (k;ǫ1···ǫ k ) 00,00 = a 2 4 dp 1 dp 2 (2π) 2 In order to make the connection with eq. (5), let us expand the integrand in powers of h 1,2 : where we denote (Note that n − + n 0 + n + = k.) The integral over p 2 is of the form 2π 0 dp 2 2π The final expression is valid only if |n| ≤ m and m, n have the same parity, otherwise the integral is zero. Therefore, m + n and m − n should both be even. Since m = n 0 + b i and n = n + − n − , this implies that k + b i should be even. This was of course expected for a closed random walk, since k + b i is the number of steps in the directions ±2 (see the order in h 2 in eq. (24)). For the integral over p 1 , we need 2π 0 dp 1 2π Wrapping everything together, we obtain where we have introduced The sum over the b i 's can be performed easily by using the following identity (see the appendix A.1) where A ≡ a 0 + · · · + a k . Therefore, eq. (28) can be rewritten as with F k;ǫ1···ǫ k (n 1 , n 2 ; φ) ≡ (−1) n0 n 1 ! 2 (2n 1 +k)!
In this formula, the sum over the α i 's and β i 's, as well as the combinatorial factors under this sum, are symmetric under the exchange α i ↔ β i , while the argument of the exponential is antisymmetric. Therefore, F k;ǫ1···ǫ k (n 1 , n 2 ; φ) is even in φ.

Moments of the distribution of areas
3.1 Terms of order φ 0 As a check, one can first evaluate eq. (32) for φ = 0. For this, we need the following sum (see the appendix A.2) α 0 +···+α k =n 1 β 0 +···+β k =n 1 8 which leads to When summing over all the possible assignments of ǫ i in {−1, 0, +1}, we obtain (see the appendix A.3) that vanishes as expected since the interaction terms do not contribute to the φ-independent part of the result.

Terms of order φ 2
In order to obtain the terms that are quadratic in the magnetic flux, we need to calculate the sum α 0 +···+α k =n 1 β 0 +···+β k =n 1 The terms that appear in the right hand side involve sums of one of the following types (see the appendices A.4 and A.5), α 0 +···+α k =n 1 β 0 +···+β k =n 1 α 0 +···+α k =n 1 β 0 +···+β k =n 1 (i = j in the second equation) and we obtain α 0 +···+α k =n 1 β 0 +···+β k =n 1 Since the expansion in ∆ k is in fact an expansion in powers of e ±iφ − 1, it is a priori obvious that the terms of order φ 2 can only appear in the orders k = 1 and k = 2. This corresponds to a fairly small number of terms (3 for k = 1 and 3 2 for k = 2) that can be summed by hand to obtain so that the total up to the order φ 2 reads From this formula, we can read out the second moment of the distribution of areas, which ends the proof of the first of eqs. (2).

General structure
Let us now consider the general case of the terms of arbitrary (but even) order φ 2l . Now, the sum we must calculate is α 0 +···+α k =n 1 β 0 +···+β k =n 1 In order to evaluate the sum underlined in the last line, we proceed as in the appendix A.4. First, we define a generating function The desired quantity is the coefficient of (xy) n1 in the Taylor expansion of this function. Following the appendices A.4 and A.5, this function is also equal to After evaluating the derivatives, the generating function takes the form where the A n (x, y) are polynomials defined iteratively by The degree of A n is thus equal to n, and the product in the numerator of eq. (46) can be expanded as Each term in this sum gives the following contribution to the coefficient of (xy) n1 in the Taylor expansion of the generating function : Let us combine C l1···l k (n 1 ) with the other factors that depend on n 1 in eq. (32), at the exception of the factor (2(n 1 + n 2 ))!/(n 1 ! 2 n 2 ! 2 ), The three underlined factors are polynomials in n 1 , of respective degrees m, n and 2l − m − n, and therefore P (k) l1···l k (n 1 ) is also a polynomial in n 1 whose degree is bounded by 2l (at this level of the discussion, it is not possible to see the cancellations that may reduce the final degree when we sum on m, n).
Let us now focus on the dependence on n 2 , and combine all the n 2 -dependent factors into the following quantity, except the factor (2(n 1 + n 2 ))!/(n 1 ! 2 n 2 ! 2 ), which is a sum of products of three polynomials in n 2 , of respective degrees n + , n − and n 0 (the total degree is thus k). Therefore, Q (k) l1···l k (n 2 ) is a polynomial in n 2 whose degree is bounded by k (again, cancellations that may decrease the degree when summing over the ǫ i 's are beyond the reach of this argument -see the subsection 3.3.2).
In order to obtain the moment 2l of the distribution of areas, we just need to sum 4 on 1 ≤ k ≤ 2l and for each k on the partitions l 1 + · · · + l k = 2l. This leads to the following expression for the 2l-th moment : where P 2l is a polynomial in n 1 , n 2 defined as : The above counting only provides an upper bound 4l for the total degree of this polynomial. However, the asymptotic behavior of these moments for large random loops is known, since their area grows like their perimeter: this implies that the degree should in fact be exactly 2l. In addition, this polynomial should obviously be symmetric in n 1,2 . This last property is non trivial in our approach, since the choice of the gauge in which we have represented the magnetic field breaks the manifest symmetry between the two directions of space.

Degree of Q
Starting from the expression (51) and expanding the factors σ li i , we see that Q (k) l1···l k (n 2 ) is a linear combination of the following polynomials : where the λ i are integers ≥ 0, such that (This follows from the fact that the right hand side of eq. (51) is homogeneous in the ǫ i 's.) Firstly, one can prove that S (k) λ1···λ k (n 2 ) = 0 if any of the λ i 's is zero. Let us assume for instance that λ k = 0, and introduce the following notations : so that Then, the sum over the three values of ǫ k contains a factor (Notice that n − + n 0 + n + = k − 1.) From now on, we need only to consider the situation where all the λ i 's are strictly positive. First, note that since the λ i 's are non-zero, we can exclude ǫ i = 0 from the sum in eq. (54). Let us assume that k e of the λ i 's are even (≥ 2) and that k o of them are odd (k e + k o = k). Without loss of generality, we can assume that λ 1 , · · · , λ ko are odd and λ ko+1 , · · · , λ k are even. If λ k is even, it is easy to check that by performing explicitly the sum over ǫ k = ±1. By iterating this formula, we can eliminate all the even λ i 's to obtain We need now to examine eq. (54) in the case where all the λ i 's are odd, in order to evaluate the last factor in the right hand side of the previous equation. When λ is odd and ǫ = ±1, we have ǫ λ = ǫ. Therefore, where we denote Thus, we have 13 Since the right hand side depends on the ǫ i 's only via n o± , we can rewrite this as Note that since k i=1 λ i = 2l is even, the number k o of odd λ i 's is also even, and we are in the second case. By counting the powers of n 2 in eqs. (60) and (64), we see that Given our assumption that none of the λ i 's is zero, this leads to 5 Since the polynomial Q λ1···λ k (n 2 ), its degree is at most l. By symmetry, the maximum degree in the variable n 1 in the moment of order 2l is also l.
The first one is a consequence of the fact that if n 1 or n 2 is zero, then all the closed paths one can construct have a vanishing area. The second one follows from the fact that for n 1 = n 2 = 1, all the closed paths have area −1, 0 or +1, and therefore contribute equally to all the even moments. The practical limitations of this algorithm are the growth of the number of terms that need to be summed, and the dramatic cancellations that occur among these terms: the intermediate calculations contain polynomials with rational coefficients whose representation involves very large integers, while the final result has a rather moderate complexity. This suggests that there may be a better way to organize the calculation, that would help avoiding these cancellations.

Conclusions
In this paper, we have used the worldline representation of scalar QED lattice propagators in two dimensions in order to obtain the general structure of the moments of the distribution of the areas of closed random walks that make fixed number of steps in the two directions. We find that these moments are the product of the number of such random walks, times a polynomial in the number of steps made in each direction. The derivation of this formula also provides an algorithm to obtain this polynomial explicitly (although this is practical only for moments of low order).
In this approach, one must choose a "gauge" to represent the transverse magnetic field whose flux measures the areas, and there are infinitely many ways of doing this. Although the final result -in particular the polynomials P 2l -is gauge invariant, each gauge choice may lead to an alternative to eqs. (50), (51) and (53) for representing this polynomial, thereby potentially providing a more efficient way of computing it.