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 th...
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| Cite this: | New Techniques for Worldline Integration. James P. Edwards, C. Moctezuma Mata, Uwe Müller and Christian Schubert. SIGMA 17 (2021), 065, 19 pages |
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| description | 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.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 17 (2021), 065, 19 pages
New Techniques for Worldline Integration
James P. EDWARDS a, C. Moctezuma MATA a, Uwe MÜLLER b and Christian SCHUBERT a
a) Instituto de F́ısica y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo,
Apdo. Postal 2-82, C.P. 58040, Morelia, Michoacan, Mexico
E-mail: james.p.edwards@umich.mx, cesarfismat@gmail.com, christianschubert137@gmail.com
b) Brandenburg an der Havel, Brandenburg, Germany
E-mail: uwe.mueller.math.phys@web.de
Received March 01, 2021, in final form June 23, 2021; Published online July 03, 2021
https://doi.org/10.3842/SIGMA.2021.065
Abstract. 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.
Key words: worldline formalism; Bernoulli numbers; Bernoulli polynomials; Feynman dia-
gram
2020 Mathematics Subject Classification: 11B68; 33C65; 81Q30
1 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 per-
turbative QFT, which later was to become his hallmark. At the time, our interest in common lay
in the “γ5-problem”, i.e., the difficulty of giving a consistent and practicable definition for the γ5
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 Feyn-
man 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 coeffi-
cients to fifth order [54] and of the two-loop effective Lagrangian for a constant SU(2) back-
This paper is a contribution to the Special Issue on Algebraic Structures in Perturbative Quan-
tum Field Theory in honor of Dirk Kreimer for his 60th birthday. The full collection is available at
https://www.emis.de/journals/SIGMA/Kreimer.html
mailto:james.p.edwards@umich.mx
mailto:cesarfismat@gmail.com
mailto:christianschubert137@gmail.com
mailto:uwe.mueller.math.phys@web.de
https://doi.org/10.3842/SIGMA.2021.065
https://www.emis.de/journals/SIGMA/Kreimer.html
2 J.P. Edwards, C.M. Mata, U. Müller and C. Schubert
ground field [76], as well as for obtaining tensor decompositions of off-shell gluon amplitudes [8].
Very recently it has also been shown how to apply it to the computation of deeply inelastic
structure functions [68, 77]. Yukawa and axial couplings have been included using dimensional
reduction [41, 42, 66, 67] as well as direct approaches [65].
Extending the formalism to curved space turned out to be mathematically subtle [14, 23, 24,
29, 40] but eventually led to the development of new efficient methods for the calculation of
amplitudes involving gravitons [2, 15, 18, 19, 30], gravitational corrections to the QED effective
Lagrangian [26], curved-space refractive indices [62], and the first one-loop calculation of the
photon-graviton conversion amplitude in a magnetic field [27, 28].
A well-known application of worldline path integrals is to the calculation of anomalies and
index densities [9, 10, 14, 29, 35, 40, 57]. In particular, a number of special cases of the Atiyah–
Singer index theorem can be reproduced in an elementary way by rewriting supertraces of heat
kernels in terms of supersymmetric particle path integrals [9, 10, 57].
Beyond standard QFT, the worldline approach has shown to be a natural tool for the con-
struction of higher-spin field interactions [16, 20], and it lends itself to generalization to non-
commutative spaces [4, 5, 33, 63] as well as spaces with boundary [21, 22, 36].
Although most applications of the formalism are based on an analytical calculation of the path
integral, efficient algorithms have also been developed for the direct numerical computation of
worldline path integrals [47, 60]. Applications include scalar bound states [70], Casimir energies
with Dirichlet boundary conditions [61], Schwinger pair-creation rates [59] and the curved-space
heat kernel [37].
Many mathematical aspects of spacetime field theory reappear in the worldline formalism
in a one-dimensional setting, such as worldline supersymmetry [9, 10, 57, 75], worldline instan-
tons [1, 45], regularisation dependence of UV counterterms (summarized in [29]), and Fadeev–
Popov determinants induced by gauge fixing [25].
Probably the most interesting string-related feature of the worldline formalism is that, for
many cases of interest, it allows one to derive parameter integral representations summing up
whole classes of Feynman diagrams (examples will be shown in Sections 4 and 5 below). How-
ever, making this fact useful for actual calculations poses a mathematical challenge, since it leads
to a highly non-standard integration problem. We will summarize the progress that has been
achieved along these lines, including the solution of the problem at the polynomial level, and
explain a recent algorithm that makes it possible to analytically compute the worldline integral
representing the standard one-loop scalar N -point diagram together with all the “crossed” dia-
grams. A central role in worldline integration is played by the Bernoulli polynomials, which over
the years has also led to some number-theoretical spin-off, some of which will be discussed, too.
2 Feynman’s worldline formalism
We start with a short review of Feynman’s worldline formalism. Consider the Green’s function
for the covariantized Klein–Gordon operator (∂ + ieA)2 +m2,
Dxx′ [A] ≡
〈
x′
∣∣∣∣
1
−(∂ + ieA)2 +m2
∣∣∣∣x
〉
.
We work with euclidean conventions, and as usual set ~ = c = 1. We exponentiate the denomi-
nator using a Schwinger proper-time parameter T :
Dxx′ [A] =
〈
x′
∣∣∣∣
∫ ∞
0
dT exp
[
−T
(
−(∂ + ieA)2 +m2
)]∣∣∣∣x
〉
.
New Techniques for Worldline Integration 3
By a standard discretization procedure, the x-space matrix element can be transformed into
a path integral,
Dxx′ [A] =
∫ ∞
0
dT e−m
2T
∫ x(T )=x′
x(0)=x
Dx(τ)e−
∫ T
0 dτ( 1
4
ẋ2+ieẋ·A(x(τ))).
Choosing the background field as a sum of N plane waves,
Aµ(x(τ)) =
N∑
i=1
εµi eiki·x(τ) (2.1)
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.
8
Substituting this vertex operator in Eq. (??), and applying the split in Eq. (??), one gets
Γ[x, x�; k1, ε1; · · · ; kN , εN ] = (−ie)N
� ∞
0
dT e−m2T e−
1
4T (x−x�)2
�
q(0)=q(T )=0
Dq(τ) e−
1
4
� T
0
dτ q̇2
×
� T
0
N�
i=1
dτi e
�N
i=1
�
εi· (x−x�)
T +εi·q̇(τi)+iki·(x−x�) τi
T +iki·x�+iki·q(τi)
����
lin(ε1ε2···εN )
.
(4.3)
After completing the square in the exponential, we obtain the following tree-level “Bern-Kosower-type formula” in
configuration space:
Γ[x, x�; k1, ε1; · · · ; kN , εN ] = (−ie)N
� ∞
0
dT e−m2T e−
1
4T (x−x�)2
�
4πT
�−D
2
×
� T
0
N�
i=1
dτi e
�N
i=1
�
εi· (x−x�)
T +iki·(x−x�) τi
T +iki·x�
�
e
�N
i,j=1
�
∆ijki·kj−2i•∆ijεi·kj−•∆•
ijεi·εj
����
lin(ε1ε2···εN )
.
(4.4)
Now, we also Fourier transform the scalar legs of the master formula in Eq. (??) to momentum space,
Γ[p; p�; k1, ε1; · · · ; kN , εN ] =
�
dDx
�
dDx� eip·x+ip�·x�
Γ[x, x�; k1, ε1; · · · ; kN , εN ] . (4.5)
This gives a representation of the multi-photon Compton scattering diagram as depicted in FIG. ?? (together with
all the permuted and “seagulled” ones).
8
After completing the square in the exponential, we obtain the following tree-level “Bern-Kosower-type formula” in
configuration space for FIG. ??
Γ[x, x�; k1, ε1; · · · ; kN , εN ] = (−ie)N
� ∞
0
dT e−m2T e−
1
4T (x−x�)2
�
4πT
�−D
2
×
� T
0
N�
i=1
dτi e
�N
i=1
�
εi· (x−x�)
T +iki·(x−x�) τi
T +iki·x�
�
e
�N
i,j=1
�
∆ijki·kj−2i•∆ijεi·kj−•∆•
ijεi·εj
����
lin(ε1ε2···εN )
, (4.4)
Now, we Fourier transform to momentum space also the scalar legs of the master formula Eq. (4.4),
Γ[p; p�; k1, ε1; · · · ; kN , εN ] =
�
dDx
�
dDx� eip·x+ip�·x�
Γ[x, x�; k1, ε1; · · · ; kN , εN ] . (4.5)
This gives a representation of the multi-photon Compton scattering diagram as depicted in FIG. 2 (together with
all the permuted and seagulled ones).
p
k1 k2 k3
· · ·
kN
··· p�
FIG. 2: Multiphoton diagram in momentum space.
Changing the integral variables to
x − x� = x− and x + x� = 2x+ ,
the integral over x+ just produces the usual energy-momentum conservation factor:
Γ[p; p�; k1, ε1; · · · ; kN , εN ] = (−ie)N (2π)DδD
�
p + p� +
�
i
ki
��
dDx−
� ∞
0
dT e−m2T e−
1
4T x2
−(4πT )−
D
2
×
� T
0
�
i
dτi eix−·(p+
�
i
kiτi
T )e
�
i
εi·x−
T e
�N
i,j=1
�
∆ijki·kj−2i•∆ijεi·kj−•∆•
ijεi·εj
����
lin(ε1ε2···εN )
. (4.6)
After performing also the x− integral, and some rearrangements, one arrives at
Γ[p; p�; k1, ε1; · · · ; kN , εN ] = (−ie)N (2π)DδD
�
p + p� +
�
i
ki
�� ∞
0
dT e−T (m2+p2)
×
� T
0
N�
i=1
dτi e
�N
i=1(−2ki·pτi+2iεi·p)+
�N
i,j=1
�
(
|τi−τj |
2 − τi+τj
2 )ki·kj−i(sign(τi−τj)−1)εi·kj+δ(τi−τj)εi·εj
����
lin(ε1ε2···εN )
.
(4.7)
This is our final representation of the N - propagator in momentum space. On-shell it corresponds to multi-photon
Compton scattering, while off-shell it can be used for constructing higher-loop amplitudes by sewing. Since this
momentum space version involves the integration variables only linearly in the exponent, for any given ordering of
the photon legs it is straightforward to do the integrals and verify, that they correspond to the usual sum of Feynman
diagrams. The main point of the formula (4.7) is its ability to combine all the N ! orderings. This may not appear very
relevant at tree level, but when used as a building block for higher-loop amplitudes leads to integral representations
for nontrivial sums of diagrams. For example, taking two copies of the N - propagator, pairing off the photons on
each side, and connecting them by free photon propagators, we can construct an integral representation of the sum
of ladder plus crossed-ladder diagrams, important for the study of scalar bound states in Scalar QED. For the case of
scalar field theory this construction was already carried out in [? ], and the resulting integral representation used to
first asymptotically sum over N and then apply a saddle-point approximation for obtaining the lowest bound state
mass in the ladder plus crossed-ladder approximation.
FIG. 3: Multi-photon Compton-scattering diagram.
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:
Γ[p; p�; k1, ε1; · · · ; kN , εN ] = (−ie)N (2π)DδD
�
p + p� +
N�
i=1
ki
�� ∞
0
dT e−m2T (4πT )−
D
2
�
dDx− e−
1
4T x2
−
×
� T
0
N�
i=1
dτi eix−·
�
p+
�N
i=1
kiτi
T
�
e
�N
i=1
εi·x−
T e
�N
i,j=1
�
∆ijki·kj−2i•∆ijεi·kj−•∆•
ijεi·εj
����
lin(ε1ε2···εN )
.
(4.6)
After performing also the x− integral, and some rearrangements, one arrives at
Γ[p; p�; k1, ε1; · · · ; kN , εN ] = (−ie)N (2π)DδD
�
p + p� +
N�
i=1
ki
�� ∞
0
dT e−T (m2+p2)
×
� T
0
N�
i=1
dτi e
�N
i=1(−2ki·pτi+2iεi·p)+
�N
i,j=1
�
(
|τi−τj |
2 − τi+τj
2 )ki·kj−i(sign(τi−τj)−1)εi·kj+δ(τi−τj)εi·εj
����
lin(ε1ε2···εN )
.
(4.7)
Figure 1. Scalar propagator dressed with N -photons.
For the one-loop effective action, the same procedure gives
Γ[A] = −Tr ln
[
−(∂ + ieA)2 +m2
]
=
∫ ∞
0
dT
T
Tr exp
[
−T
(
−(∂ + ieA)2 +m2
)]
=
∫ ∞
0
dT
T
e−m
2T
∫
x(0)=x(T )
Dx(τ)e−
∫ T
0 dτ( 1
4
ẋ2+ieẋ·A(x(τ))). (2.2)
Upon specialization to the plane-wave background (2.1) and an expansion to Nth order in the
coupling, one obtains the one-loop N , photon amplitudes in the form
Γ[{ki, εi}] = (−ie)N
∫
dT
T
e−m
2T
∫
DxV γ
scal[k1,ε1] · · ·V γ
scal[kN ,εN ]e−
∫ T
0 dτ ẋ
2
4 . (2.3)
Here V γ
scal denotes the same photon vertex operator as is used in string perturbation theory,
V γ
scal[k, ε] =
∫ T
0
dτε · ẋ(τ)eikx(τ).
From these two building blocks, arbitrary scalar QED amplitudes can be constructed by sewing
pairs of photons. See, e.g., Figure 2.
Figure 2. A typical multiloop diagram in scalar QED.
4 J.P. Edwards, C.M. Mata, U. Müller and C. Schubert
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,
Spin[x,A] = trΓP exp
[
i
4
e[γµ, γν ]
∫ T
0
dτFµν(x(τ))
]
. (2.4)
Here P indicates that the exponential function in general will not be an ordinary but path-
ordered one, since the exponents at different proper times will not commute in general.
3 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]:
Spin[x,A]→
∫
Dψ(τ) exp
[
−
∫ T
0
dτ
(
1
2
ψ · ψ̇ − ieψµFµνψ
ν
)]
.
Here ψµ(τ) is a Lorentz-vector valued anticommuting function of the proper-time,
ψ(τ1)ψ(τ2) = −ψ(τ2)ψ(τ1)
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 dis-
cussed 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 demon-
strated 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(τ1, τ2), GF (τ1, τ2), obeying
〈xµ(τ1)xν(τ2)〉 = −G(τ1, τ2)δµν , G(τ1, τ2) = |τ1 − τ2| −
1
T
(
τ1 − τ2
)2
,
〈ψµ(τ1)ψν(τ2)〉 =
1
2
GF (τ1, τ2)δµν , GF (τ1, τ2) = sgn(τ1 − τ2) (3.1)
(note that in the worldline formalism sgn(0) = 0 in general). For example, to calculate the path-
integral (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
xµ0 ≡
1
T
∫ T
0
dτxµ(τ),
whose integral will just produce the global energy-momentum conservation factor (2π)Dδ(
∑
ki).
It then requires only a formal exponentiation ε · ẋ(τ)eikx(τ) = eikx+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”,
Γ[{ki, εi}] = (−ie)N
∫ ∞
0
dT
T
(4πT )−
D
2 e−m
2T
N∏
i=1
∫ T
0
dτi
New Techniques for Worldline Integration 5
× exp
{ N∑
i,j=1
[
1
2
Gijki · kj + iĠijki · εj +
1
2
G̈ijεi · εj
]}∣∣∣∣
lin(ε1,...,εN )
, (3.2)
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,
Ġ(τ1, τ2) = sgn(τ1 − τ2)− 2
(τ1 − τ2)
T
,
G̈(τ1, τ2) = 2δ(τ1 − τ2)− 2
T
, (3.3)
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 corre-
sponding 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 back-
ground 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(τi − τj) contained
in G̈ij , equation (3.3).
5. All the G̈ij can be removed by a systematic integration-by-parts procedure, which homoge-
nizes the integrand and at the same time leads to the appearance of photon field strength
tensors fµνi ≡ k
µ
i ε
ν
i − εµi kνi , as was noted by Strassler [74].
4 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.
1 2
34
+
1 2
43
+
1 3
24
+
1 3
42
+
1 4
23
+
1 4
32
Figure 3. Feynman diagrams for photon-photon scattering.
At the four-point level, the requirement of removing all the G̈ij 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 all G̈ij , it also reduces the
number of independent tensor structures arising to its minimum possible, which is five. It leads
to the following decomposition:
Γ̂4 = Γ̂
(1)
4 + Γ̂
(2)
4 + Γ̂
(3)
4 + Γ̂
(4)
4 + Γ̂
(5)
4 ,
Γ̂
(1)
4 = Γ̂
(1)
(1234)T
(1)
(1234) + Γ̂
(1)
(1243)T
(1)
(1243) + Γ̂
(1)
(1324)T
(1)
(1324),
Γ̂
(2)
4 = Γ̂
(2)
(12)(34)T
(2)
(12)(34) + Γ̂
(2)
(13)(24)T
(2)
(13)(24) + Γ̂
(2)
(14)(23)T
(2)
(14)(23),
Γ̂
(3)
4 =
∑
i=1,2,3
Γ̂
(3)
(123)iT
(3)r4
(123)i +
∑
i=2,3,4
Γ̂
(3)
(234)iT
(3)r1
(234)i +
∑
i=3,4,1
Γ̂
(3)
(341)iT
(3)r2
(341)i +
∑
i=4,1,2
Γ̂
(3)
(412)iT
(3)r3
(412)i,
Γ̂
(4)
4 =
∑
i<j
Γ̂
(4)
(ij)iiT
(4)
(ij)ii +
∑
i<j
Γ̂
(4)
(ij)jjT
(4)
(ij)jj ,
Γ̂
(5)
4 =
∑
i<j
Γ̂
(5)
(ij)ijT
(5)
(ij)ij +
∑
i<j
Γ̂
(5)
(ij)jiT
(5)
(ij)ji
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,
T
(1)
(1234) ≡ Z4(1234),
T
(2)
(12)(34) ≡ Z2(12)Z2(34),
T
(3)r4
(123)i ≡ Z3(123)
r4 · f4 · ki
r4 · k4
, i = 1, 2, 3,
T
(4)
(12)ii ≡ Z2(12)
ki · f3 · f4 · ki
k3 · k4
, i = 1, 2,
T
(5)
(12)ij ≡ Z2(12)
ki · f3 · f4 · kj
k3 · k4
, (i, j) = (1, 2), (2, 1),
where we abbreviated
Z2(ij) ≡ 1
2
tr
(
fifj
)
= εi · kjεj · ki − εi · εjki · kj ,
Zn(i1i2 · · · in) ≡ tr
( n∏
j=1
fij
)
, n ≥ 3.
Note that the tensor T
(3)r4
(123)i still depends on a “reference momentum” r4, which is arbitrary
except for the condition r4 · k4 6= 0. The coefficient functions are given by [6]
Γ̂
(k)
··· =
∫ ∞
0
dT
T
T 4−D
2 e−m
2T
∫ 1
0
4∏
i=1
duiγ̂
(k)
...
(
Ġij
)
eT
∑4
i<j=1Gijki·kj ,
where, for spinor QED
γ̂
(1)
(1234) = Ġ12Ġ23Ġ34Ġ41 −GF12GF23GF34GF41,
γ̂
(2)
(12)(34) =
(
Ġ12Ġ21 −GF12GF21
)(
Ġ34Ġ43 −GF34GF43
)
,
γ̂
(3)
(123)1 =
(
Ġ12Ġ23Ġ31 −GF12GF23GF31
)
Ġ41,
γ̂
(4)
(12)11 =
(
Ġ12Ġ21 −GF12GF21
)
Ġ13Ġ41,
γ̂
(5)
(12)12 =
(
Ġ12Ġ21 −GF12GF21
)
Ġ13Ġ42
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 GFij . 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 off-
shell. 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.
5 On to multiloop
For example, from the four-photon amplitude we can, by sewing, construct the two-loop photon
propagator, Figure 4
Figure 4. The two-loop photon propagator.
and from the one-loop six-photon amplitude we get the three-loop quenched propagator, Fi-
gure 5,
Figure 5. The three-loop quenched photon propagator.
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, K. Chetyrkin, A. Davydychev and other experts in multiloop
integration were present. The coefficients known at the time (up to four loops in spinor QED,
up to three loops in scalar QED) were all rational [34]:
βspin
3 = −2
after a cancellation of ζ(3) between diagrams,
βspin
4 = −46
after a cancellation of ζ(3)s and ζ(5)s. And in scalar QED, too, the ζ(3) contributions cancelled
at three loops,
βscal
3 =
29
2
.
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 βspin
5 [11]. The Lord sometimes can be malicious. Still,
although incomplete the cancellations are remarkable and ought to find an explanation!
8 J.P. Edwards, C.M. Mata, U. Müller and C. Schubert
6 The fundamental problem of worldline integration
Returning to the one-loop level, using that GFijs always appear in closed chains and can be
eliminated by
GFijGFjkGFki = −
(
Ġij + Ġjk + Ġki
)
the most general integral that one will ever have to compute in the worldline approach to QED
is of the form
∫ 1
0
du1du2 · · · duNPol
(
Ġij
)
e
∑N
i<j=1 λ
2
ijGij
with arbitrary N and polynomial Pol
(
Ġij
)
, where
Gij = |ui − uj | − (ui − uj)2, Ġij = sgn(ui − uj)− 2(ui − uj).
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 in Ġij “full circle” in any partic-
ular variable ui yields again a polynomial expression in the remaining Ġij . To give just one
particularly neat example,
∫ 1
0
du Ġ(u, u1)Ġ(u, u2)Ġ(u, u3) = −1
6
(
Ġ12 − Ġ23
)(
Ġ23 − Ġ31
)(
Ġ31 − Ġ12
)
.
However, the result can generally be written in many different equivalent ways due to identities
such as
(
Ġij + Ġjk + Ġki
)2
= 1.
Eventually, using the master integral
∫ 1
0
du e
∑n
i=1 ciĠ(u,ui) =
∑n
i=1 sinh(ci)e
∑n
j=1 cjĠij
∑n
i=1 ci
with constants c1, . . . , cn the following formula was found which allows one to integrate an
arbitrary monomial in Ġij in one of the variables, and gives the result as a polynomial in the
remaining Ġij :
∫ 1
0
du Ġ(u, u1)k1Ġ(u, u2)k2 · · · Ġ(u, un)kn
=
1
2n
n∑
i=1
∏
j 6=i
kj∑
lj=0
(
kj
lj
)
Ġ
kj−lj
ij
ki∑
li=0
(
ki
li
)
(−1)
∑n
j=1 lj
(1 +
∑n
j=1 lj)n
∑n
j=1 lj
×
{(∑
j 6=i
Ġij + 1
)1+
∑n
j=1 lj
− (−1)ki−li
(∑
j 6=i
Ġij − 1
)1+
∑n
j=1 lj
}
.
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 ∂nP is invertible, and
the integral kernel of the inverse is given essentially by the nth Bernoulli polynomial Bn(x):
〈u | ∂−nP | u′〉 = − 1
n!
Bn(|u− u′|)sgnn(u− u′), n ≥ 1, (7.1)
〈u|∂0|u′〉 = δ(u− u′)− 1. (7.2)
This fact is related to the well-known Fourier series
Bn(x) = − n!
(2πi)n
∑
k 6=0
e2πikx
kn
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 the Ġij as a “chain
integral”,
〈
u1 | ∂−nP | un+1
〉
=
1
2n
∫ 1
0
du2 · · · dunĠ12Ġ23 · · · Ġn(n+1)
since the integral just constructs a multiple inverse derivative by iteration of a single one.
Thus the ubiquitous appearance of the Bernoulli numbers Bn (= Bn(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)(LE)
spin (all+) = −απm
4
8π2
(
2e
m2
)N
c
(1,2)
spin
(
N
2
)
χ+
N ,
Γ
(1,2)(LE)
scal (all+) =
απm4
16π2
(
2e
m2
)N
c
(1,2)
scal
(
N
2
)
χ+
N .
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,
c
(1)
spin(n) =
(−1)n+1B2n
2n(2n− 2)
= c
(1)
scal (n)
however this degeneracy is lifted at the two-loop level,
c
(2)
spin(n) =
1
(2π)2
{
2n− 3
2n− 2
B2n−2 + 3
n−1∑
k=1
B2k
2k
B2n−2k
2n− 2k
}
,
c
(2)
scal(n) =
1
(2π)2
{
2n− 3
2n− 2
B2n−2 +
3
2
n−1∑
k=1
B2k
2k
B2n−2k
2n− 2k
}
.
(The two-loop result is actually the quenched contribution only, there is also a non-quenched
one [48, 58].)
10 J.P. Edwards, C.M. Mata, U. Müller and C. Schubert
8 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 back-
ground 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 for-
malism, we obtained there two different formulas in terms of Bernoulli numbers for the same
weak-field expansion coefficients,
c
(2)
scal(n) =
1
(2π)2
{
2n− 3
2n− 2
B2n−2 +
3
2
n−1∑
k=1
B2k
2k
B2n−2k
2n− 2k
}
(8.1)
vs.
c
(2)
scal(n) =
1
(2π)2
{
2n− 3
2n− 2
B2n−2 + 3
[
ψ(2n+ 1)− 2
2n− 1
+ γ − 1
]
B2n
2n
+ 3
n−1∑
k=1
(
2n− 2
2k − 2
)
B2k
2k
B2n−2k
2n− 2k
}
. (8.2)
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 com-
binatorics, and indeed he showed us how to demonstrate the equivalence by combining the
probably best-known of Bernoulli number identities, Euler’s identity
n−1∑
k=1
(
2n
2k
)
B2kB2n−2k = −(2n+ 1)B2n
with a much deeper identity due to Miki (1977): for integer n ≥ 2,
n−1∑
k=1
B2kB2n−2k
2k(2n− 2k)
=
n−1∑
k=1
B2kB2n−2k
2k(2n− 2k)
(
2n
2k
)
+
B2n
n
H2n,
where Hi 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,
n−1∑
k=1
B̄2kB̄2n−2k
2k(2n− 2k)
=
1
n
n∑
k=1
B2kB̄2n−2k
2k
(
2n
2k
)
+
B̄2n
n
H2n−1,
where we have introduced the further abbreviation
B̄n ≡
(
1− 2n−1
2n−1
)
Bn.
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 e2πikx with positive integers k. The inverse
derivative 〈u | ∂−nP | u′〉 then acquires also an imaginary part, which up to normalization is the
nth Clausen function Cln(2π(u− u′)).
In [69] two of the authors along these lines constructed a worldline QFT geared towards the
derivation of identities between multiple zeta values, defined by
ζ(k1, . . . , km) =
∑
n1>n2>···>nm>0
1
nk11 · · ·nkmm
.
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 φ3 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 φ3-theory in D dimensions. For this case the worldline master formula reads [72],
IN (p1, . . . , pN ) =
1
2
(4π)−D/2(2π)Dδ
( N∑
i=1
pi
)
ÎN (p1, . . . , pN ),
ÎN (p1, . . . , pN ) =
∫ ∞
0
dT
T
TN−D/2e−m
2T
∫
12...N
eT
∑N
i<j=1Gijpij ,
where we abbreviated
∫
12...N ≡
∫ 1
0 du1 · · ·
∫ 1
0 duN and pij ≡ pi · pj .
The ordered integrals lead off-shell to hypergeometric functions such as 2F1 (2-point), F1
(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
IN (n12, n13, . . . , n(N−1)N ) ≡
∫
12...N
N∏
i<j=1
G
nij
ij .
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.
11 Expansion in inverse derivatives
Thus, we expand each exponential using the identity (whose proof is given in the appendix)
epabGab = 1 + 2
∞∑
n=1
p
n−1/2
ab H2n−1
(√
pab
2
)
〈ua|∂−2n|ub〉. (11.1)
12 J.P. Edwards, C.M. Mata, U. Müller and C. Schubert
Here the Hn(x) are Hermite polynomials, and we have abbreviated
〈ua|∂−2n|ub〉 ≡ 〈ua|∂−2n|ub〉 − 〈ua|∂−2n|ua〉.
By integration, this also gives
√
π
2x
erf(x)ex
2
= 1 +
∞∑
n=1
22nB̂2nx
2n−1H2n−1(x)
(
B̂n ≡ 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
ep12G12+p13G13+p23G23 =
{
1 + 2
∞∑
i=1
p
i− 1
2
12 H2i−1
(√
p12
2
)[
〈u1|∂−2i|u2〉+ B̂2i
]}
×
{
1 + 2
∞∑
j=1
p
j− 1
2
13 H2j−1
(√
p13
2
)[
〈u1|∂−2j |u3〉+ B̂2j
]}
×
{
1 + 2
∞∑
k=1
p
k− 1
2
23 H2k−1
(√
p23
2
)[
〈u2|∂−2k|u3〉+ B̂2k
]}
.
Since
∫ 1
0 dui〈ui|∂−2n|uj〉 =
∫ 1
0 duj〈ui|∂−2n|uj〉 = 0, the three 〈ui|∂−2n|uj〉 must go together, and
then we can apply the completeness relation
∫ 1
0 du|u〉〈u| = 1l to get
∫
123
〈u1|∂−2i|u2〉〈u2|∂−2k|u3〉〈u3|∂−2j |u1〉 = Tr
(
∂−2(i+j+k)
)
= −B̂2(i+j+k),
where we have used (7.1) in the last step. In this way we get a closed form-expression for the
N = 3 coefficients,
I3(a, b, c) ≡
∫
123
Ga12G
b
13G
c
23
= a!b!c!
a∑
i=b1+a/2c
b∑
j=b1+b/2c
c∑
k=b1+c/2c
hai h
b
jh
c
k
(
B̂2iB̂2jB̂2k − B̂2(i+j+k)
)
.
In writing this identity we have assumed that a, b, c are all different from zero. The coefficients hai
can be found from the rearrangement
2
∞∑
i=1
λi−
1
2H2i−1
(√
λ
2
)
B̂2i =
∞∑
a=1
λa
a∑
i=b1+a/2c
hai B̂2i.
From the explicit formula for the Hermite polynomials
Hn(x) = n!
bn
2
c∑
m=0
(−1)m
(2x)n−2m
m!(n− 2m)!
we find
hai = (−1)a+1 2(2i− 1)!
(2i− a− 1)!(2a− 2i+ 1)!
.
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”
V ijk
3 ≡
∫ 1
0
du〈u|∂−i|u1〉〈u|∂−j |u2〉〈u|∂−k|u3〉.
This vertex can be computed by partial integration as follows:
V ijk
3 ≡
∫ 1
0
du〈u|∂−i|u1〉〈u|∂−j |u2〉〈u|∂−k|u3〉
=
i+j−1∑
a=i
(−1)a
(
a− 1
i− 1
)∫ 1
0
du〈u|∂0|u1〉〈u|∂−(i+j−a)|u2〉〈u|∂−(k+a)|u3〉+{i↔ j, u1 ↔ u2}
=
i+j−1∑
a=i
(−1)a
(
a− 1
i− 1
)[
〈u1|∂−(i+j−a)|u2〉〈u1|∂−(k+a)|u3〉−(−1)i+j−a〈u2|∂−(i+j+k)|u3〉
]
+
i+j−1∑
a=j
(−1)a
(
a− 1
j − 1
)[
〈u2|∂−(i+j−a)|u1〉〈u2|∂−(k+a)|u3〉−(−1)i+j−a〈u1|∂−(i+j+k)|u3〉
]
.
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 i1i2···im
m . In the calcula-
tion of the N -point function, the worldline vertices V3, V4, . . . , VN−1 will be needed.
12 Outlook
Although the results that we have presented here on scalar N -point functions are still prelimi-
nary, 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 IN
(
n12, n13, . . . , 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,
B2n ∼ (−1)n+12
(2n)!
(2π)2n
might become useful for approximations.
A Proof of identity (11.1)
The following identity shall be proved.
Identity A.1.
epGab = 1 + 2
∞∑
n=1
pn−
1
2H2n−1
(√
p
2
)
〈ua|∂−2n|ub〉,
where ua and ub are 1-periodic coordinates on a circle with length 1, Gab = |ua−ub|− (ua−ub)2
(|ua|, |ub| < 1), Hn(x) Hermite polynomials, p some parameter, ∂−1 the inverse derivative on
the circle with zero integration constant and
〈ua|∂−2n|ub〉 =
〈
ua|∂−2n|ub
〉
−
〈
ua|∂−2n|ua
〉
.
14 J.P. Edwards, C.M. Mata, U. Müller and C. Schubert
Using translation invariance on the circle, we can express the identity in terms of the difference
x = ua − ub. The inverse derivatives can then be expressed by periodically continued Bernoulli
polynomials
〈
ua|∂−n|ub
〉
= δn,0
∞∑
k=−∞
δ(x− k)− 1
n!
B̄n(x), n ≥ 0,
where B̄n(x) = Bn(x− [x]) ([x] denotes the largest integer smaller or equal to x) coincides with
the Bernoulli polynomial Bn(x) on the interval x ∈ [0, 1) and is periodically continued to x ∈ R
by B̄n(x+ 1) = B̄n(x) (hence, B̄n(x) is no polynomial for n > 0, when it is regarded as defined
on R). B̄n(x) is continuous at x ∈ Z for all n 6= 1, whereas B̄1(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 B1, which is −1
2 by convention.
Using these definitions, and restricting x = ua− ub 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:
Identity A.2.
epx(1−x) = 1 + 2
∞∑
n=1
pn−
1
2H2n−1
(√
p
2
)
(B2n(x)−B2n(0)).
The proof given below requires the following lemma:
Lemma A.3. Let r be any odd positive integer, x an indeterminate. Then the following poly-
nomial identity holds
r∑
n=0
(
r
n
)
(2x− r + n)n(x)r−nBn = 0,
where Bn = Bn(0) denotes the nth Bernoulli number and (a)0 = 1, (a)n = (a)n−1(a−n+ 1) the
Pochhammer symbol denoting here the falling factorial (which is a polynomial if a is one).
Proof. For any odd r, the Bernoulli polynomial Br(x) has the property Br(1 − x) = −Br(x).
Hence, given any integrable function f : [0, 1]→ C with f(1−x) = f(x), we have by substituting
x 7→ 1− x and using the transformation properties
∫ 1
0
f(x)Br(x) dx =
∫ 1
0
f(1− x)Br(1− x) dx = −
∫ 1
0
f(x)Br(x) dx = 0.
Applying this to the explicit form of the Bernoulli polynomials,
Br(x) =
r∑
n=0
(
r
n
)
Bnx
r−n, (A.1)
we get
r∑
n=0
(
r
n
)
Bn
∫ 1
0
f(x)xr−ndx = 0, f(x) = f(1− x), r > 0 odd.
Choosing f(x) = xu−1(1− x)u−1 for any u ∈ C, Reu > 0, we can express the integral explicitly
by Γ-functions
∫ 1
0
xu−1(1− x)u−1xr−ndx =
Γ(u+ r − n)Γ(u)
Γ(2u+ r − n)
New Techniques for Worldline Integration 15
and obtain an identity of meromorphic functions in u
r∑
n=0
(
r
n
)
Bn
Γ(u+ r − n)Γ(u)
Γ(2u+ r − n)
= 0, r > 0 odd.
Using the Γ-function identity Γ(x + 1) = xΓ(x) and assuming r, n ∈ Z, 0 ≤ n ≤ r, we can
introduce Pochhammer symbols1 for falling factorials
Γ(u+ r − n) = (u+ r − n− 1)r−nΓ(u) = (−1)r−n(−u)r−nΓ(u),
Γ(2u+ r − n) =
Γ(2u+ r)
(2u+ r − 1)n
= (−1)n
Γ(2u+ r)
(−2u− r + n)n
and obtain after dividing out the factors independent of n
r∑
n=0
(
r
n
)
Bn(−u)r−n(−2u− r + n)n = 0, r > 0 odd.
Now we have an equality between analytic functions in u, which are in fact polynomials, such
that we can replace the complex variable −u with the indeterminate x and obtain the polynomial
identity of the lemma. �
Corollary A.4.
2r+1∑
n=0
(
2r + 1
n
)
Bn
(
n+ 2m− 2r − 1
m
)
= 0, r,m ∈ Z, 0 ≤ r < m.
Proof. Writing the second binomial coefficient as
(
n+ 2m− 2r − 1
m
)
=
1
m!
(n+ 2m− 2r − 1)m
=
1
m!
(n+ 2m− 2r − 1)n(2m− 2r − 1)m−2r−1(m)2r+1−n,
the sum of the corollary becomes
1
m!
(2m− 2r − 1)m−2r−1
2r+1∑
n=0
(
2r + 1
n
)
Bn(n+ 2m− 2r − 1)n(m)2r+1−n,
which is zero by Lemma A.3 setting x = m. �
We come now to the main proof.
Proof of Identity A.2. Using the explicit expression for the Hermite polynomials
Hn(x) =
[n2 ]∑
k=0
n!
(n− 2k)!k!
(−1)k(2x)n−2k,
and comparing equal powers pm (m ≥ 1), the identity reduces to the equivalent form
1
m!
xm(1− x)m =
∑
m+1
2
≤n≤m
2
(2n−m− 1)!(2m− 2n+ 1)!
1
2n
(B2n(x)−B2n(0)).
1Other Pochhammer representations using also Γ(x)Γ(1 − x) = π
sin(πx)
do not give other identities, because
they can also be obtained by the identity (a)n = (−1)n(−a+ n− 1)n, where each factor is multiplied with −1.
16 J.P. Edwards, C.M. Mata, U. Müller and C. Schubert
Multiplying by m! 6= 0, inserting the explicit form (A.1) for the Bernoulli polynomials in terms
of Bernoulli numbers, expanding the binomial (1−x)m and comparing equal powers xk (k ≥ 1),
the identity takes the equivalent form
∑
k
2
≤n≤m
1
2n
(
2n
k
)
2B2n−k
(
m
2n−m− 1
)
= (−1)k
(
m
k −m
)
, 1 ≤ k ≤ 2m,
or equivalently, replacing 2n by a new summation variable n restricted to even values
2m∑
n=k
n even
1
n
(
n
k
)
2Bn−k
(
m
n−m− 1
)
= (−1)k
(
m
k −m
)
, 1 ≤ k ≤ 2m. (A.2)
Now we have two cases:
1. k is odd. Then, because all odd Bernoulli numbers, except B1 = −1
2 , vanish, the sum
in (A.2) consists of the single term for n = k + 1, which gives
1
k + 1
(
k + 1
k
)
2B1
(
m
k −m
)
= −
(
m
k −m
)
,
which is just the right-hand side for odd k. Hence, (A.2) is true in this case.
2. k is even. Then the right-hand side of (A.2), having a positive sign, fits into the sum
as additional term for n = k + 1. Thus, in this case, taking into account that all other odd
Bernoulli numbers vanish, equation (A.2) is equivalent to
2m+1∑
n=k
1
n
(
n
k
)
2Bn−k
(
m
n−m− 1
)
= 0,
where the summation extends now over all, even and odd, integers, and the range had to be
extended to n = 2m+ 1 to cover also the case k = 2m, where the additional term for n = k+ 1
is exactly the term for n = 2m+ 1.
Dividing by 2, shifting n by k, writing the binomial coefficients in factorials and introducing
r ≥ 0 by setting k = 2m− 2r, we obtain
2r+1∑
n=0
(n+ 2m− 2r − 1)!
n!(2m− 2r)!
Bn
m!
(2r + 1− n)!(n+m− 2r − 1)!
= 0.
Multiplying this equation with (2r+1)!(2m−2r)!
(m!)2
6= 0 and recombining the factorials into binomial
coefficients, it can be written equivalently as
2r+1∑
n=0
(
2r + 1
n
)
Bn
(
n+ 2m− 2r − 1
m
)
= 0,
which is true by Corollary A.4. Hence, (A.2) is true also in this case. �
Acknowledgements
We thank Andrei Davydychev and Tord Riemann for sharing with us their expertise on scalar
off-shell N -point functions.
New Techniques for Worldline Integration 17
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1 Introduction
2 Feynman's worldline formalism
3 Modern approach to the worldline formalism
4 The four-photon amplitudes
5 On to multiloop
6 The fundamental problem of worldline integration
7 Bernoulli numbers and polynomials
8 The Miki and Faber–Pandharipande–Zagier identities
9 A toy QFT for multiple zeta-value identities
10 General worldline integral in phi3 theory
11 Expansion in inverse derivatives
12 Outlook
A Proof of identity (11.1)
References
|
| id | nasplib_isofts_kiev_ua-123456789-211358 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T11:03:47Z |
| publishDate | 2021 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Edwards, James P. Mata, C. Moctezuma Müller, Uwe Schubert, Christian 2025-12-30T15:55:16Z 2021 New Techniques for Worldline Integration. James P. Edwards, C. Moctezuma Mata, Uwe Müller and Christian Schubert. SIGMA 17 (2021), 065, 19 pages 1815-0659 2020 Mathematics Subject Classification: 11B68; 33C65; 81Q30 arXiv:2106.12071 https://nasplib.isofts.kiev.ua/handle/123456789/211358 https://doi.org/10.3842/SIGMA.2021.065 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. We thank Andrei Davydychev and Tord Riemann for sharing with us their expertise on scalar off-shell N-point functions. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications New Techniques for Worldline Integration Article published earlier |
| spellingShingle | New Techniques for Worldline Integration Edwards, James P. Mata, C. Moctezuma Müller, Uwe Schubert, Christian |
| title | New Techniques for Worldline Integration |
| title_full | New Techniques for Worldline Integration |
| title_fullStr | New Techniques for Worldline Integration |
| title_full_unstemmed | New Techniques for Worldline Integration |
| title_short | New Techniques for Worldline Integration |
| title_sort | new techniques for worldline integration |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211358 |
| work_keys_str_mv | AT edwardsjamesp newtechniquesforworldlineintegration AT matacmoctezuma newtechniquesforworldlineintegration AT mulleruwe newtechniquesforworldlineintegration AT schubertchristian newtechniquesforworldlineintegration |