Localisation in worldline pair production and lightfront zero-modes

The nonperturbative probability of pair production in electric fields depending on lightfront time is given exactly by the locally constant approximation. We explain this by showing that the worldline path integral defining the effective action contains a constraint, which localises contributing paths on hypersurfaces of constant lightfront time. These paths are lightfront zero-modes and there can be no pair production without them; the effective action vanishes if they are projected out.


I. INTRODUCTION
Schwinger's calculation of the pair production probability in a constant electric field is one of the most well known nonperturbative results in quantum field theory [1][2][3].Going beyond constant fields leads to richer mathematics and physics, the most well-studied case being time-dependent electric fields E(x 0 ) [4][5][6][7], though there are only a few cases which can be solved exactly.To instead estimate the pair production probability in a given field (describing e.g.focussed laser pulses [8][9][10][11][12][13]), the locally constant approximation (LCA) is often used, in which one replaces constant E with E(x), and a volume factor in Schwinger's expression with an integral over x µ .Exactly solvable cases show, though, that the LCA is not exact in general [14][15][16].
pair production in longitudinal electric fields E 3 (x + ), depending on lightfront time x + = x 0 + x 3 [17], was investigated in [18][19][20][21] using lightfront quantisation, i.e. quantisation on hypersurfaces of constant x + [22,23].In this approach, a prescription is required for treating ubiquitous 1/p + terms in order to deal with zeromodes, states for which the momentum p + vanishes.It was shown in [18,19] that standard regularisations of zero-modes would imply that there is no pair production in fields E 3 (x + ), contradicting e.g.Schwinger's result.This problem was solved in [18,19] by quantising on two lightlike hypersurfaces, one of fixed x + and one of fixed x − = x 0 − x 3 [24,25].Although effectively abandoning the usual lightfront approach, this allowed control of the zero-modes and recovered a nonzero pair production probability.Surprisingly, this was found to be given exactly by the LCA, and for arbitrarily field dependence on x + .This result was recovered from an independent functional computation of the effective action in [26].
Explaining why this remarkably simple result should hold is the goal of this paper.We reconsider pair production in fields E 3 (x + ) using the worldline formalism, reviewed in [27], which has found particular success in applications to pair production in external fields [28,29].The paper is organised as follows.In Sect.II we com- * anton.ilderton@chalmers.sepute the effective action using the worldline path integral, giving a simple derivation of the result in [26].We will see explicitly why the effective action localises, and what role the zero-modes play in this.In Sect.III we give extensions of our results, compare with pair production in fields E(x 0 ), for which the LCA does not hold, and relate the zero-mode contributions to the triviality of the lightfront vacuum.We conclude in Sect.IV.

II. THE EFFECTIVE ACTION
The effective action is given by the functional integral where δ is a UV cutoff and the worldline particle action S is a function of the closed path x µ and the intrinsic length of the path, T , (For clarity we consider scalar particles; the extension to spinors is given below.)A closed path x µ can be decomposed into a constant, average, piece x µ c and an orthogonal oscillatory piece y µ .Details are given in Appendix A, here we need only that the measure on these variables is, for each coordinate [30], We will ignore constant prefactors, which can be recovered by comparison with Schwinger's result.We define lightfront coordinates x ± = x 0 ± x 3 for the time and longitudinal directions, and x ⊥ = {x 1 , x 2 } the transverse directions.Following [18,19] we take A − ≡ A − (x + ) and all other components zero (Γ is trivially gauge invariant).
Writing a := −eA − , the longitudinal electric field is and taking a constant gives Schwinger's constant field case [3].Extensions will be discussed below.

arXiv:1406.1513v1 [hep-th] 5 Jun 2014
A. Localisation and zero-modes Lightfront zero-modes correspond to vanishing momentum p + , i.e. dynamics within a lightlike hypersurface x + = constant.We are therefore particularly interested in contributions to (1) from paths which lie in such hypersurfaces.
In lightfront quantisation, some degrees of freedom are typically constrained, and it is in solving these constraints that the issue of zero-modes must be confronted, for example in defining the operator 1/p + , see.Something similar will happen here: we will first integrate out the longitudinal degrees of freedom x − , which will generate a constraint leading us to the zero-modes.
The part of the action depending on x − is Since x − c does not appear, the dx − c integral contributes only a volume, V − .The boundary conditions on the paths allow us to integrate by parts in ( 5) and remove all derivatives from y − .After doing so, the Dy − integral can be performed and gives a delta functional on the space of paths y + : Thus, the paths in {x + , x ⊥ }-space which can contribute to the effective action are constrained by (6), which we must solve in order to proceed.Although the constraint looks highly nonlinear our key result is that, because of boundary conditions, the delta functional has support only on y + ≡ 0, as we will now show.Integrating the equation O[y + ] = 0, implied by ( 6), we obtain confirming that ẏ is continuous (since y and a are) and obeys the same boundary conditions as y.Hence Rolle's theorem can be applied [31]: there exists ξ 0 with ẏ(ξ 0 ) = 0. Now, if we instead expand ȧ = ẏ+ a in ( 6) and use an integrating factor, we obtain the implicit relation which shows immediately that ẏ(ξ) ≡ 0. The purely oscillatory y(ξ) is therefore constant, but the only allowed constant is y(ξ) ≡ 0. Thus the delta functional in (6) has support only on y + = 0, and we have x + x 2 x 1 x + = const.
x + = const.where, from here on dropping the c-subscript from x + c , (10) Having solved the constraint (6), we see that the effective action (in the considered class of fields) is supported entirely on loops for which y + = 0 ⇐⇒ ẋ+ = 0.In other words, the effective action is supported entirely on loops lying in hyperplanes of constant x + , which are the lightfront zero-modes, see Fig. 1.It is this which localises the effective action, as the constraint turns the functional integral Dx + into an ordinary sum dx + over contributions Det −1 O from hypersufraces of fixed lightfront time x + : Since the field has only a one-dimensional homogeneity, and because contributing paths are localised in this dimension, (1) will become equal to that obtained by applying the LCA to the constant field result.We now make this explicit by evaluating the determinant in (9) and hence the effective action.

B. Evaluation
The remaining transverse integrals in (1) are those of the free theory, and contribute the standard factor V ⊥ /T .The effective action then takes the form The determinant can be evaluated by returning to (5), replacing a → a (x + c ) ẏ+ , expanding in Fourier modes as in Appendix A and performing the resulting elementary integrals.The result is using zeta-function regularisation, in the second step, to define the infinite product.Inserting the inverse of ( 13) into ( 12) we arrive at The integral is convergent in the IR but divergent in the UV, T → 0. Subtracting the two problematic terms [3], and changing variables T → 2T /m 2 , leaves where ε = eE/m 2 = E/E S .This exact result agrees with that given by the LCA applied to the constant field case; E is replaced by E(x + ) and a nontrivial integral dx + appears instead of an infinite volume V + .Thus we neatly recover (the scalar version of) the results in [18,19,26] from the worldline formalism.
As a function of T , the integrand in (15) has no UV or IR divergences, or poles on the real line.It has an imaginary part only because of the imaginary exponent.To evaluate Im Γ we rotate the T -contour counterclockwise to, but not past, the imaginary axis, where the function does have poles.The contour then runs up the imaginary axis, with semicircular deviations into the upperright quadrant [32].The on-axis segments contribute to the real part, whereas the semicircles contribute to the imaginary part, the final result for which is The extension to QED proper is direct.The spin factor to be inserted into the integrand of (1) is For our choice of field, the x − -integral can be performed as above, which localises the rest of the integrand, including the above spin factor.The fermion integrals can then be performed as in the constant field case, and the final result is still given by the LCA.

III. DISCUSSION
A. Comparison with E(x 0 ) Our results provide another example of the fact that, even in essentially one-dimensional systems, pair produc-tion is sensitive to the spacetime support of the external field; temporal field inhomogeneities tend to enhance pair production, for example, while spatial field inhomogeneities tend to suppress it [28].Here we discuss the similarities and differences between pair production in fields E 3 (x + ), for which the LCA holds, and E 3 (x 0 ), for which it does not.
An elegant way to see the similarities and differences between these two cases is to follow the phase-space analysis in [33].For the constant field, there are two approaches which lead to the same result.For the first, take A 3 = Ex 0 , then one finds that three canonical momenta, k ⊥ and k 3 , are conserved [34].Integrating out the remaining component k 0 (τ ) then gives an effective classical action depending on x 0 (τ ) and constant k 3 , k ⊥ .The effectively 1-D problem can be analysed by e.g.going to Euclidean space and studying the instanton structure [33].For the second approach, take instead A − = − 1 2 Ex + (same field, different gauge).One now finds that the three components k + and k − are conserved.However, integrating out the remaining k + (τ ) gives a delta functional as above, the support of which contains all the physics which would be obtained from e.g. the instantons.Because the effective action is gauge invariant, even though k µ is not, these two approaches are equivalent (see also [35,36]).Now, if the electric field is allowed to depend nontrivially on x + , the second option remains available, as we saw above in equations ( 5) and ( 6).If, however, the field depends nontrivially on x 0 = (x + + x − )/2, only the first option is available, since beyond the constant field case the action cannot be made linear in any single variable, as in (5), and there is no way to generate a 'localising' delta functional as in (6).

B. Minkowski vs. Euclidean
Worldine integrals are often computed in Euclidean space, where one can take advantage of the fact that instanton contributions dominate [28,29], as well as employ numerical methods [37,38].We have worked instead in Minksowski space, and it is interesting to compare the two approaches.
Beginning again with a constant field, we have seen that ( 8) has no non-trivial solutions in Minkowski space, which is what forces contributing paths to be localised on hypersurfaces x + = constant [39].Rotating to Euclidean space, though, the boundary conditions can be satisfied if the exponent satisfies This is both the periodicity condition for instanton solutions, which give the dominant contribution to the effective action, and the locations of the poles in (15) after rotating to Euclidean T -space [40].For fields which depend nontrivially on x + , the same periodicity would seem to hold, although it is not at all apparent how to relate this to the instanton or pole structure [41].This rough analysis suggests that the Minkowski and Euclidean calculations must look very different; the delta-functional (6), for example, will clearly not arise in Euclidean space.This will be investigated elsewhere.

C. Extension to other fields
There is a wider class of fields for which the effective action is supported only on zero-modes, and in some cases the remaining x ⊥ -integrals can still be performed.Had we included a potential A + (x + , x ⊥ ), the corresponding term ẋ+ A + in the action would have vanished after obtaining the delta-functional.Hence the presence of fields leads to the same effective action as found above; this is natural since all such fields (a class which includes plane waves) have vanishing Schwinger invariants F F = F F = 0 and are orthogonal to the longitudinal electric field above.Including instead a potential A ⊥ (x + , x ⊥ ) we can describe, in addition to (19), a longitudinal magnetic field The x − -integrals can be performed as before, leading to localisation in x + .If both E 3 and B 3 depend only on x + ; the effective action is again given exactly by the LCA applied to the known expressions for parallel, constant, E and B [42][43][44].

D. Zero-modes and the trivial vacuum
In order to underline the importance of zero-modes for pair production, we can explicitly project them out, from the beginning, and investigate what happens.Following [45] we write 1 = P + Q in which P is the zero-mode (or vacuum) projector, and Q = 1 − P is the non-zero-mode (or particle) projector.These are, in ordinary momentum space, distributions which can be constructed from nascent delta functions, with a standard representation being, see Fig. 2, Since zero-modes correspond to vanishing kinematic longitudinal momentum, the natural functional equivalent of the non-zero-mode projector Q is (See also Appendix A.) Inserting this projector into the effective action (1) and repeating our earlier calculation, we see that the x − integral can still be performed to obtain the delta functional (6).However, this now multiples the projector (23), which vanishes when y + = 0, killing the whole path integral.So, when we omit the zero-modes, not only does the imaginary part of the effective action vanish (no pair production) but the whole effective action vanishes and Γ → 0. This is the statement that the vacuum is trivial [46].In particular, the vacuum to vacuum transition amplitude then becomes One of the great simplifications encountered in lightfront quantisation is that the front form vacuum is trivial up to zero-modes [22,23,45,47], and our calculation provides a rather explicit example of this.Without the zero-modes, the effective action vanishes, which is indeed consistent with a trivial vacuum, but then there is no Schwinger pair production.This is recovered, though, once the zeromodes are included.

IV. CONCLUSIONS
The worldline approach to pair production shows that the effective action in longitudinal electric fields E 3 (x + ) is a sum over loops constrained to lie in hypersurfaces of constant x + .The localisation of these loops is the reason why the locally constant approximation is exact for the chosen class of fields.If the electric field instead depends nontrivially on x 0 , no such localisation occurs, and indeed it is known that the locally constant approximation is not exact in that case.
It would be interesting to compare in detail our Minkowski space calculation with the corresponding calculation in Euclidean space, and to investigate the relation to real and complex worldline instantons [33], as well as to the instanton calculations of [48,49].
The entire effective action, not just the imaginary part responsible for pair production, is supported on lightfront zero-modes.Without these, the effective action vanishes, and the vacuum persistence amplitude becomes unity.This is consistent with the vacuum being trivial up to zero-modes.
It seems to remain unclear as to how it is possible to recover, within canonical lightfront quantisation, the results of [18][19][20][21].As suggested in those papers, it may be possible to integrate out the degrees of freedom on x − = 0 to obtain an effective theory which includes pair production.(We have done something similar here, albeit in a worldline approach.)Discretised LightCone Quantisation, or DLCQ [50,51], seems well suited for such an investigation.However, boundary conditions may be subtle, since the zero-mode lies at vanishing physical momentum p + , rather than vanishing canonical momentum k + conjugate to the compactified direction x − .

FIG. 1 :
FIG.1: Contributing paths (after integrating out x − ): the blue loops, which lie within hypersurfaces of constant x + , contribute to the effective action.The red loops, which cut those hypersurfaces, do not contribute.