
where τ = τ
1
+ iτ
2
is the modular parameter, and z = x + iy is the displacement between
the two vertices. In the above the exponent is what is left of the Green’s function at large
τ
2
, while s = k
1
· k
2
is the kinematic invariant.
1
We will ignore the accompanying pre-
factors because these would be the same as in the effective particle theory. In this large
τ
2
region of the fundamental domain the y and τ
1
integrals become “inert” with the latter
simply enforcing the level-matching of the physical spectrum. The whole integral is then
“projected” to a worldline integral over x (the usual Feynman parameter) and πα
0
τ
2
≡ t
(the usual Schwinger parameter). In other words the particle limit yields a result directly
in the worldline formalism,
A ∼
X
i=physical
Z
1
0
dx
Z
∞
∼α
0
dt
t
e
−(sx(1−x)+m
2
i
)t+...
, (1.2)
where the m
2
i
term in the exponent drops out of the partition function Z(τ ) and where
√
α
0
is the string-length.
This encapsulates the effective particle theory contribution to the string amplitude.
But note that invariance of the whole amplitude under the τ → −1/τ modular transfor-
mation means that one could equally write the integral in the domain where it approaches
the cusp at τ → 0:
A ∼
X
i=physical
Z
1
0
dy
Z
∼α
0
0
dt
t
e
−(sy(1−y)+m
2
i
)
1
M
4
t
+...
, (1.3)
where we define M
2
= 1/πα
0
. In this limit it is y rather than x that drops out of the
worldsheet Green’s function to end up playing the role of the Feynman parameter. But
since the other variable is inert, the integral as t → 0 — which is a copy of (1.2) — can just
as well be interpreted as continuing the t-integral into the deep UV, but with t → 1/(M
4
t).
In [81] this was used to argued that one can capture the behaviour of the entire amplitude
by writing a nonlocal theory with t replaced by T (t) = t + 1/M
4
t and integrating over all
t. This approximation reproduces the asymptotic behaviour at the IR and UV cusps. It
is reminiscent of string theory in the sense that the deep UV is identified as just another
IR, with the difference being that in the full string theory there are an infinite number
of fundamental domains not just two. In summary the gross UV/IR mixing behaviour
of strings (and modular invariance) can be mimicked in the particle context by suitably
modifying the Klein-Gordon propagator so that it exhibits a worldline inversion symmetry,
t → 1/(M
4
t) .
In this paper we will explore such string theory inspired nonlocal field theories in the
gravitational context. Our aim is to formulate a gravitational theory whose propagator
around Minkowski space exhibits the above worldline inversion symmetry, and to inves-
tigate some of its consquences. This is possible despite the technicalities of writing the
higher spin components of the theory in the worldline formalism [86]. The only price to
pay is the introduction of nonlocality.
1
Having the correct conformal weight for the vertices requires k
2
1
= k
2
2
= 0 and being on shell would
imply s = 0, so we are implicitly employing the usual trick of slightly violating Lorentz invariance to retain
explicit dependence on s.
– 3 –