New Techniques for Worldline Integration

The worldline formalism provides an alternative to Feynman diagrams in the construction of amplitudes and effective actions that shares some of the superior properties of the organization of amplitudes in string theory. In particular, it allows one to write down integral representations combining the contributions of large classes of Feynman diagrams of different topologies. However, calculating these integrals analytically without splitting them into sectors corresponding to individual diagrams poses a formidable mathematical challenge. We summarize the history and state of the art of this problem, including some natural connections to the theory of Bernoulli numbers and polynomials and multiple zeta values.


Introduction
When I (Christian Schubert) first met Dirk, he was still a PhD student of K. Schilcher at Mainz University, but clearly had already set out on his quest for new mathematical structures in perturbative QFT, which later was to become his hallmark. At the time, our interest in common lay in the "γ 5 -problem", i.e., the difficulty of giving a consistent and practicable definition for the γ 5 matrix in the framework of dimensional renormalization. Later on it shifted to the behaviour of perturbation theory at large orders, to the computation of renormalization group functions, and to the mathematical nature of the objects appearing there. Although we have never actually collaborated, staying in touch with Dirk over all these years has been a fruitful and enjoyable experience.
Our topic here will be the "worldline formalism", which provides an alternative to Feynman diagrams in the construction of the perturbation series in QFT based on first-quantized relativistic particle path integrals. Introduced by Feynman in 1950/1 for QED [51,52], but then largely forgotten, it has since the nineties gained some popularity following developments in string theory [31,32,74,75]. Although here we will be concerned only with examples from QED and scalar field theory, it should be mentioned that worldline path integrals during the last three decades have been applied to a steadily expanding circle of problems in QFT, providing new computational options as well as useful physical intuition (for reviews, see [49,73]).
Their non-abelian generalisation was used for a calculation of the QCD heat-kernel coefficients to fifth order [54] and of the two-loop effective Lagrangian for a constant SU(2) back- We work with euclidean conventions, and as usual set = c = 1. We exponentiate the denominator using a Schwinger proper-time parameter T : New Techniques for Worldline Integration 3 By a standard discretization procedure, the x-space matrix element can be transformed into a path integral, 2 +ieẋ·A(x(τ ))) .
Choosing the background field as a sum of N plane waves, and Fourier transforming the endpoints we get the "photon-dressed propagator", as shown in Figure 1, where for our present purposes it must be emphasized that summation over the N ! permutations of the photons is understood. .

(4.3)
After completing the square in the exponential, we obtain the following tree-level "Bern-Kosower-type formula" in configuration space: .

(4.4)
Now, we also Fourier transform the scalar legs of the master formula in Eq. (??) to momentum space, e Fourier transform to momentum space also the scalar legs of the master formula Eq. (4.4), gives a representation of the multi-photon Compton scattering diagram as depicted in FIG. 2 (togeth permuted and seagulled ones).
nging the integral variables to egral over x + just produces the usual energy-momentum conservation factor: . r performing also the x − integral, and some rearrangements, one arrives at our final representation of the N -propagator in momentum space. On-shell it corresponds to multi on scattering, while off-shell it can be used for constructing higher-loop amplitudes by sewing. Sin tum space version involves the integration variables only linearly in the exponent, for any given ord oton legs it is straightforward to do the integrals and verify, that they correspond to the usual sum of Fe s. The main point of the formula (4.7) is its ability to combine all the N ! orderings. This may not appe Changing the integration variables x, x to x − x = x − and x + x = 2x + , the integral over x + just produces the usual energy-momentum conservation factor: .

(4.6)
After performing also the x − integral, and some rearrangements, one arrives at . For the one-loop effective action, the same procedure gives Upon specialization to the plane-wave background (2.1) and an expansion to N th order in the coupling, one obtains the one-loop N , photon amplitudes in the form Here V γ scal denotes the same photon vertex operator as is used in string perturbation theory, From these two building blocks, arbitrary scalar QED amplitudes can be constructed by sewing pairs of photons. See, e.g., Figure 2. An advantage of this formalism is that scalar QED and spinor QED become more closely related than usual, at least in perturbation theory. For example, to get the effective action for spinor QED from (2.2) requires only a change of the global normalization by a factor of − 1 2 , and the insertion of the following spin factor Spin[x, A] under the path integral, Here P indicates that the exponential function in general will not be an ordinary but pathordered one, since the exponents at different proper times will not commute in general.

Modern approach to the worldline formalism
Nowadays for analytical purposes one usually does not use Feynman's spin factor in the form (2.4), but replaces it by the following Grassmann path integral [56]: Here ψ µ (τ ) is a Lorentz-vector valued anticommuting function of the proper-time, and it has to be functionally integrated using antiperiodic boundary conditions, ψ(T ) = −ψ(0). Apart from removing the path-ordering, which will be essential for the applications to be discussed below, it has the further advantage of exhibiting a supersymmetry between the orbital and spin degrees of freedom of the electron (generalizing the well-known supersymmetry of the Pauli equation in quantum mechanics). A similar approach can be followed in the non-abelian case to eliminate the path ordering due to the presence of colour factors [3,12,13,17]. However, it was only in the nineties, after the development of string theory, which demonstrated the mathematical beauty and computational usefulness of first-quantized path integrals, that Feynman's worldline path-integral formalism was finally taken seriously as a competitor for Feynman diagrams. In this "string-inspired" approach to the worldline formalism [31,32,74,75], a central role is played by gaussian path integration using "worldline Green's functions". In QED, the only worldline correlators required are G(τ 1 , τ 2 ), G F (τ 1 , τ 2 ), obeying (note that in the worldline formalism sgn(0) = 0 in general). For example, to calculate the pathintegral (2.3), first we have to make the kinetic operator invertible, which requires eliminating the constant trajectories. This is best done by separating off the average position of the loop whose integral will just produce the global energy-momentum conservation factor (2π) D δ( k i ).
It then requires only a formal exponentiation ε ·ẋ(τ )e ikx(τ ) = e ikx+ε·ẋ(τ ) | lin(ε) to arrive at a gaussian path integral, which can be performed by a simple completion of the square. This leads to the prototype of a "Bern-Kosower master formula", which is somewhat formal since, after deciding on the number of photons, one still has to expand the exponential factor and to keep only the terms that contain each of the N polarization vectors once. Besides the Green's function (3.1), now also its first two derivatives appear, where it is understood that a "dot" always refers to a derivative with respect to the first variable. The factor (4πT ) − D 2 comes from the free path integral determinant. The master formula (3.2) has many attractive features, some obvious, some less so: 1. It provides a highly compact generating function for the N -photon amplitudes in scalar QED, valid off-shell and on-shell.
2. It represents the sum of the corresponding Feynman diagrams including all non-equivalent orderings of the photons along the loop.
3. Bern and Kosower in their seminal work [31,32] found a set of rules that allows one to obtain from this master formula, by a pure pattern matching procedure, also the corresponding amplitudes with a spinor loop, as well as the N -gluon amplitudes with a scalar, spinor or gluon loop. 4. In this formalism, the worldline Lagrangian contains only a linear coupling to the background field, corresponding to a cubic vertex in field theory. The quartic seagull vertex arises only at the path-integration stage, and is represented by the δ(τ i − τ j ) contained inG ij , equation (3.3).
5. All theG ij can be removed by a systematic integration-by-parts procedure, which homogenizes the integrand and at the same time leads to the appearance of photon field strength tensors f µν i ≡ k µ i ε ν i − ε µ i k ν i , as was noted by Strassler [74].

The four-photon amplitudes
Let us have a closer look at the four-photon case, which is important not only for QED but also serves as the prototypical example for all amplitudes involving four gauge bosons. In terms of Feynman diagrams, in spinor QED it is given by the familiar six diagrams shown in Figure 3, while in scalar QED there are a few more diagrams involving the seagull vertex.  At the four-point level, the requirement of removing all theG ij by no means exhausts the freedom of integration by parts, even if requesting that the full permutation symmetry between the photons be preserved, and it is only very recently [6] that an algorithm was found that is 6 J.P. Edwards, C.M. Mata, U. Müller and C. Schubert optimized in the sense that, apart from achieving the removal of allG ij , it also reduces the number of independent tensor structures arising to its minimum possible, which is five. It leads to the following decomposition: with a set of five tensors T (i) that, remarkably, up to normalization agrees with the one found in 1971 by Costantini, De Tollis and Pistoni [38] using the QED Ward identity, where we abbreviated Note that the tensor T (123)i still depends on a "reference momentum" r 4 , which is arbitrary except for the condition r 4 · k 4 = 0. The coefficient functions are given by [6] Γ (k) where, for spinor QED New Techniques for Worldline Integration 7 (plus permutations thereof), and the coefficient functions for scalar QED are obtained from these simply by deleting all the G F ij . In a forthcoming series of papers [7] this representation is used for a first calculation of the scalar and spinor QED four-photon amplitudes completely offshell. The results should become useful for higher-order calculations in QED with four-photon sub-diagrams, as well as for photonic processes in external fields.   etc. The interesting feature of the worldline formalism is that the sewing results in parameter integrals that represent not some particular Feynman diagram, but the whole set of Feynman diagrams shown in Figures 4 and 5. And it is precisely this type of sums of diagrams that have become notorious in the QED community for the particularly extensive cancellations between diagrams that were discovered as a byproduct of the enormous effort that was invested from the sixties onward in obtaining information on the high-order behaviour of the QED β function. Both physically and mathematically, the most interesting contributions to this function come from the quenched (one fermion-loop) diagrams, and this "quenched QED β function" was a big topic at the Multiloop Workshop that took place at the Aspen Center of Physics in 1995, where besides Dirk also D. Broadhurst Thus at the time everybody believed that the quenched coefficients would stay rational to all loop orders and eventually might be computable in closed form, but then many years later at five loops ζ(3) refused to cancel out in β spin 5 [11]. The Lord sometimes can be malicious. Still, although incomplete the cancellations are remarkable and ought to find an explanation!

The fundamental problem of worldline integration
Returning to the one-loop level, using that G F ij s always appear in closed chains and can be eliminated by the most general integral that one will ever have to compute in the worldline approach to QED is of the form with arbitrary N and polynomial Pol Ġ ij , where For this to become eventually useful in addressing the type of multiloop cancellations that we have just reviewed, one should devise a way of performing such integrals without decomposing the integrand into ordered sectors. At the polynomial level, this turned out to be a quite tractable problem. By Fourier analysis, it is easy to show that integrating a polynomial expression inĠ ij "full circle" in any particular variable u i yields again a polynomial expression in the remainingĠ ij . To give just one particularly neat example, However, the result can generally be written in many different equivalent ways due to identities such as Ġ ij +Ġ jk +Ġ ki 2 = 1.
Eventually, using the master integral with constants c 1 , . . . , c n the following formula was found which allows one to integrate an arbitrary monomial inĠ ij in one of the variables, and gives the result as a polynomial in the remainingĠ ij : Thus by recursion it provides a complete solution to the "circular integration" problem at the polynomial level.
New Techniques for Worldline Integration 9 7 Bernoulli numbers and polynomials Next, let us emphasize that worldline integration naturally relates to the theory of Bernoulli numbers and polynomials. This is because the coordinate path integral was performed in the Hilbert space H P of periodic functions orthogonal to the constant functions (because of the elimination of the zero mode). In this space the ordinary nth derivative ∂ n P is invertible, and the integral kernel of the inverse is given essentially by the nth Bernoulli polynomial B n (x): This fact is related to the well-known Fourier series k n although in the mathematical literature it is usually assumed that 0 < x < 1, which eliminates the sgn and δ functions in (7.1), (7.2), that is, just the fun stuff that makes these inverse derivatives into a well-defined algebra of integral operators in the Hilbert space H P . Note that the left-hand side of (7.1) can also be written in terms of theĠ ij as a "chain integral", since the integral just constructs a multiple inverse derivative by iteration of a single one. Thus the ubiquitous appearance of the Bernoulli numbers B n (= B n (0)) in perturbative QFT, which from the diagrammatic point of view often remains somewhat mysterious, in the worldline formalism finds a natural explanation.
As a nice example for a non-trivial occurrence of the Bernoulli numbers in QED amplitudes, let us show here the "all +" amplitudes in scalar and spinor QED, in the low-energy (LE) limit, at one and two loops [43,64] Γ (1,2) Here χ + N is a spinorial invariant that absorbs all the momenta and polarization vectors, and can be constructed explicitly in the spinor-helicity formalism, see [64]. The one-loop coefficients are equal, however this degeneracy is lifted at the two-loop level, (The two-loop result is actually the quenched contribution only, there is also a non-quenched one [48,58].)

The Miki and Faber-Pandharipande-Zagier identities
The above two-loop result actually led to some mathematical spin-off which merits a digression. It was not obtained by a direct calculation of the N -photon amplitudes, but by a calculation of the scalar and spinor QED effective Lagrangian at two loops in a (Euclidean) self-dual background field by G.V. Dunne and one of the authors in 2002 [43,44]. Let us now focus on the scalar QED case. Doing this calculation in two different ways, both using the worldline formalism, we obtained there two different formulas in terms of Bernoulli numbers for the same weak-field expansion coefficients, Here ψ(x) = Γ (x)/Γ(x), and γ is Euler's constant.
Although there is a rather enormous mathematical literature on identities involving Bernoulli numbers, we could not find anything that would have allowed us to show the equivalence of these two expressions. Dirk suggested to consult Richard Stanley from MIT, a leading expert in combinatorics, and indeed he showed us how to demonstrate the equivalence by combining the probably best-known of Bernoulli number identities, Euler's identity with a much deeper identity due to Miki (1977): for integer n ≥ 2, where H i denotes the ith harmonic number. At about the same time, we learned about a similar identity that was found heuristically by Faber and Pandharipande in 1998 in a string theory calculation, and then proven by Zagier [50]: for integer n ≥ 2, where we have introduced the further abbreviation In [46], we reanalyzed our worldline derivation of (8.1) and (8.2), and used it as a guiding principle to 1. Give a unified proof of the Miki and Faber-Pandharipande-Zagier identities.
2. To generalize them at the quadratic level.
3. Generalize them to the cubic level.
4. Outline the construction of even higher-order identities.
New Techniques for Worldline Integration 11 9 A toy QFT for multiple zeta-value identities As a further mathematical digression, let me mention that the above algebra of inverse derivatives can be mathematically enriched by changing the Hilbert space from H P to one "chiral" half, that is, the Hilbert space H + P generated by the basis e 2πikx with positive integers k. The inverse derivative u | ∂ −n P | u then acquires also an imaginary part, which up to normalization is the nth Clausen function Cl n (2π(u − u )).
In this model, arbitrary such values can be represented as "seashell" Feynman diagrams, and large classes of identities between them can be derived through a systematic integration-by-parts procedure applied to the corresponding Feynman integrals.
10 General worldline integral in φ 3 theory Finally, let us return to the problem of circular integration, and our ongoing work, where we are trying to solve it for the simplest non-polynomial case, the off-shell one-loop N -point amplitude for scalar φ 3 -theory in D dimensions. For this case the worldline master formula reads [72], where we abbreviated 12...N ≡ 1 0 du 1 · · · 1 0 du N and p ij ≡ p i · p j . The ordered integrals lead off-shell to hypergeometric functions such as 2 F 1 (2-point), F 1 (3-point), the Lauricella-Saran function (4-point), see, e.g., [39,53,71].
As a first attempt at calculating the unordered integrals, one might try a brute-force approach expanding all the exponential factors, and computing I N (n 12 , n 13 , . . . , However, although these integrals are individually trivial, it is by no means easy to arrive at a closed-form expression that eventually might allow one to perform a resummation. Instead, we have found it more promising to expand each exponential factor in terms of the inverse derivative operators above.

Expansion in inverse derivatives
Thus, we expand each exponential using the identity (whose proof is given in the appendix) e p ab G ab = 1 + 2 ∞ n=1 p n−1/2 ab Here the H n (x) are Hermite polynomials, and we have abbreviated By integration, this also gives B n ≡ Bn n! . Curiously, we have not been able to find this simple series representation of the error function in the mathematical literature.
Let us have a look at the three-point case. Here using (11.1) in each factor leads to e p 12 G 12 +p 13 G 13 +p 23 Since 1 0 du i u i |∂ −2n |u j = 1 0 du j u i |∂ −2n |u j = 0, the three u i |∂ −2n |u j must go together, and then we can apply the completeness relation where we have used (7.1) in the last step. In this way we get a closed form-expression for the N = 3 coefficients, In writing this identity we have assumed that a, b, c are all different from zero. The coefficients h a i can be found from the rearrangement From the explicit formula for the Hermite polynomials New Techniques for Worldline Integration

13
Differently from the three-point case, for N = 4 we encounter, apart from "chain integrals", the "cubic worldline vertex" This vertex can be computed by partial integration as follows: In the second step (7.2) was used. Here we are, incidentally, just reusing one of the algorithms that was used in the toy model described above to obtain alternative representations for a given multiple zeta value. This algorithm generalizes to the worldline vertex V i 1 i 2 ···im m . In the calculation of the N -point function, the worldline vertices V 3 , V 4 , . . . , V N −1 will be needed.

Outlook
Although the results that we have presented here on scalar N -point functions are still preliminary, what is already clear is that the expansion (11.1) together with integration by parts allows one to perform the unordered integrals, in principle for any N . The challenge then becomes to identify the resulting multiple sums with known hypergeometric functions. As a byproduct, we will get closed formulas for the basic coefficients I N n 12 , n 13 , . . . , n (N −1)N . Those seem less relevant for the momentum-space amplitudes, but fundamental for the associated heat-kernel expansion of the effective action in x-space [55]. Finally, the property of the Bernoulli numbers to converge rather fast towards their asymptotic limit, might become useful for approximations.
A Proof of identity (11.1) The following identity shall be proved.
where u a and u b are 1-periodic coordinates on a circle with length 1, G ab = |u a − u b | − (u a − u b ) 2 (|u a |, |u b | < 1), H n (x) Hermite polynomials, p some parameter, ∂ −1 the inverse derivative on the circle with zero integration constant and Using translation invariance on the circle, we can express the identity in terms of the difference x = u a − u b . The inverse derivatives can then be expressed by periodically continued Bernoulli polynomials denotes the largest integer smaller or equal to x) coincides with the Bernoulli polynomial B n (x) on the interval x ∈ [0, 1) and is periodically continued to x ∈ R byB n (x + 1) =B n (x) (hence,B n (x) is no polynomial for n > 0, when it is regarded as defined on R).B n (x) is continuous at x ∈ Z for all n = 1, whereasB 1 (x) jumps from 1 2 on the left to − 1 2 on the right of integer arguments, and the value at integers is chosen such that it equals the Bernoulli number B 1 , which is − 1 2 by convention. Using these definitions, and restricting x = u a − u b to the interval x ∈ [0, 1), the identity can be written in terms of Bernoulli polynomials, which is the form that we are going to prove: (0)).
The proof given below requires the following lemma: and obtain an identity of meromorphic functions in u r n=0 r n B n Γ(u + r − n)Γ(u) Γ(2u + r − n) = 0, r > 0 odd.
Corollary A.4. We come now to the main proof.