,
16,
28,
20,
115]. The KLT relations provide gravity amplitudes only at
tree-level;
D
-dimensional unitarity then provides a means of obtaining quantum
loop amplitudes. In perturbation theory this is tantamount to
quantizing the theory since the complete scattering matrix can,
at least in principle, be systematically constructed this way.
Amusingly, this bypasses the usual formal apparatus [60,
61,
5,
79] associated with quantizing constrained systems. More generally,
the unitarity method provides a way to systematically obtain the
complete set of quantum loop corrections order-by-order in the
perturbative expansion whenever the full analytic behavior of
tree amplitudes as a function of
D
is known. It always works when the particles in the theory are
all massless. The method is well tested in explicit calculations
and has, for example, recently been applied to state-of-the-art
perturbative QCD loop computations [21,
12].
In quantum field theory the
S
-matrix links initial and final states. A basic physical property
is that the
S
matrix must be unitary [95,
91,
96,
35]:
. In perturbation theory the Feynman diagrams describe a
transition matrix
T
defined by
, so that the unitarity condition reads
where
i
and
f
are initial and final states, and the ``sum'' is over
intermediate states
j
(and includes an integral over intermediate on-mass-shell
momenta). Perturbative unitarity consists of expanding both sides
of Eq. (42
) in terms of coupling constants,
g
for gauge theory and
for gravity, and collecting terms of the same order. For
example, the imaginary (or absorptive) parts of one-loop
four-point amplitudes, which is order
in gravity, are given in terms of the product of two four-point
tree amplitudes, each carrying a power of
. This is then summed over all two-particle states that can
appear and integrated over the intermediate phase space (see
Fig.
9).
. The blobs represent tree amplitudes.
This provides a means of obtaining loop amplitudes from tree
amplitudes. However, if one were to directly apply Eq. (42) in integer dimensions one would encounter a difficulty with
fully reconstructing the loop scattering amplitudes. Since
Eq. (42) gives only the imaginary part one then needs to reconstruct the
real part. This is traditionally done via dispersion relations,
which are based on the analytic properties of the
S
matrix [95,
91,
96,
35]. However, the dispersion integrals do not generally converge.
This leads to a set of subtraction ambiguities in the real part.
These ambiguities are related to the appearance of rational
functions with vanishing imaginary parts
, where the
are the kinematic variables for the amplitude.
A convenient way to deal with this problem [15,
16,
28,
20,
115] is to consider unitarity in the context of dimensional
regularization [131,
137]. By considering the amplitudes as an analytic function of
dimension, at least for a massless theory, these ambiguities are
not present, and the only remaining ambiguities are the usual
ones associated with renormalization in quantum field theory. The
reason there can be no ambiguity relative to Feynman diagrams
follows from simple dimensional analysis for amplitudes in
dimension
. With dimensional regularization, amplitudes for massless
particles necessarily acquire a factor of
for each loop, from the measure
. For small
,
, so every term has an imaginary part (for some
), though not necessarily in terms which survive as
. Thus, the unitarity cuts evaluated to
provide sufficient information for the complete reconstruction
of an amplitude. Furthermore, by adjusting the specific rules for
the analytic continuation of the tree amplitudes to
D
-dimensions one can obtain results in the different varieties of
dimensional regularization, such as the conventional one [34], the t' Hooft-Veltman scheme [131], dimensional reduction [122], and the four-dimensional helicity scheme [27,
13].
It is useful to view the unitarity-based technique as an alternate way of evaluating sets of ordinary Feynman diagrams by collecting together gauge-invariant sets of terms containing residues of poles in the integrands corresponding to those of the propagators of the cut lines. This gives a region of loop-momentum integration where the cut loop momenta go on shell and the corresponding internal lines become intermediate states in a unitarity relation. From this point of view, even more restricted regions of loop momentum integration may be considered, where additional internal lines go on mass shell. This amounts to imposing cut conditions on additional internal lines. In constructing the full amplitude from the cuts it is convenient to use unrestricted integrations over loop momenta, instead of phase space integrals, because in this way one can obtain simultaneously both the real and imaginary parts. The generalized cuts then allow one to obtain multi-loop amplitudes directly from combinations of tree amplitudes.
As a first example, the generalized cut for a one-loop
four-point amplitude in the channel carrying momentum
, as shown in Fig.
9, is given by
where
, and the sum runs over all physical states of the theory
crossing the cut. In this generalized cut, the on-shell
conditions
are applied even though the loop momentum is unrestricted. In
addition, any physical state conditions on the intermediate
particles should also be included. The real and imaginary parts
of the integral functions that do have cuts in this channel are
reliably computed in this way. However, the use of the on-shell
conditions inside the unrestricted loop momentum integrals does
introduce an arbitrariness in functions that do not have cuts in
this channel. Such integral functions should instead be obtained
from cuts in the other two channels.
A less trivial two-loop example of a generalized ``double'' two particle cut is illustrated in panel (a) of Fig. 10 . The product of tree amplitudes appearing in this cut is
where the loop integrals and cut propagators have been
suppressed for convenience. In this expression the on-shell
conditions
are imposed on the
,
i
= 1, 2, 3, 4 appearing on the right-hand side. This double cut
may seem a bit odd from the traditional viewpoint in which each
cut can be interpreted as the imaginary part of the integral. It
should instead be understood as a means to obtain part of the
information on the structure of the integrand of the two-loop
amplitude. Namely, it contains the information on all integral
functions where the cut propagators are not cancelled. There are,
of course, other generalized cuts at two loops. For example, in
panel (b) of Fig.
10, a different arrangement of the cut trees is shown.
Complete amplitudes are found by combining the various cuts
into a single function with the correct cuts in all channels.
This method works for any theory where the particles can be taken
to be massless and where the tree amplitudes are known as an
analytic function of dimension. The restriction to massless
amplitudes is irrelevant for the application of studying the
ultra-violet divergences of gravity theories. In any case,
gravitons and their associated superpartners in a supersymmetric
theory are massless. (For the case with masses present the extra
technical complication has to do with the appearance of functions
such as
which have no cuts in any channel. See Ref. [28] for a description and partial solution of this problem.) This
method has been extensively applied to the case of one- and
two-loop gauge theory amplitudes [15,
16,
20,
21,
12] and has been carefully cross-checked with Feynman diagram
calculations. Here, the method is used to obtain loop amplitudes
directly from the gravity tree amplitudes given by the KLT
equations. In the next section an example of how the method works
in practice for the case of gravity is provided.
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Perturbative Quantum Gravity and its Relation to Gauge
Theory
Zvi Bern http://www.livingreviews.org/lrr-2002-5 © Max-Planck-Gesellschaft. ISSN 1433-8351 Problems/Comments to livrev@aei-potsdam.mpg.de |
