The analytic properties of gravity amplitudes as momenta
become either soft
or collinear (
parallel to
) are especially interesting because they supply a simple
demonstration of the tight link between the two theories.
Moreover, these analytic properties are crucial for constructing
and checking gravity amplitudes with an arbitrary number of
external legs. The properties as gravitons become soft have been
known for a long time [139
,
10] but the collinear properties were first obtained using the
known collinear properties of gauge theories together with the
KLT relations.
Helicity amplitudes in quantum chromodynamics have a
well-known behavior as momenta of external legs become collinear
or soft [99,
20]. For the collinear case, at tree-level in quantum
chromodynamics when two nearest neighboring legs in the
color-stripped amplitudes become collinear,
e.g.,
,
, and
, the amplitude behaves as [99]:
The function
is a splitting amplitude, and
is the helicity of the intermediate state
P
. (The other helicity labels are implicit.) The contribution
given in Eq. (21
) is singular for
parallel to
; other terms in the amplitude are suppressed by a power of
, which vanishes in the collinear limit, compared to the ones in
Eq. (21). For the pure glue case, one such splitting amplitude is
where
are spinor inner products, and
is a momentum-dependent phase that may be found in, for example,
Ref. [99]. In general, it is convenient to express splitting amplitudes
in terms of these spinor inner products. The `+' and `-' labels
refer to the helicity of the outgoing gluons. Since the spinor
inner products behave as
, the splitting amplitudes develop square-root singularities in
the collinear limits. If the two collinear legs are not next to
each other in the color ordering, then there is no singular
contribution,
e.g.
no singularity develops in
for
collinear to
.
From the structure of the KLT relations it is clear that a
universal collinear behavior similar to Eq. (21) must hold for gravity since gravity amplitudes can be obtained
from gauge theory ones. The KLT relations give a simple way to
determine the gravity splitting amplitudes,
. The value of the splitting amplitude may be obtained by taking
the collinear limit of two of the legs in, for example, the
five-point amplitude. Taking
parallel to
in the five-point relation (11) and using Eq. (13) yields:
where
More explicitly, using Eq. (22) then gives:
Using the KLT relations at n -points, it is not difficult to verify that the splitting behavior is universal for an arbitrary number of external legs, i.e. :
(Since the KLT relations are not manifestly crossing-symmetric, it is simpler to check this formula for some legs being collinear rather than others; at the end all possible combinations of legs must give the same results, though.) The general structure holds for any particle content of the theory because of the general applicability of the KLT relations.
In contrast to the gauge theory splitting amplitude (22), the gravity splitting amplitude (26) is not singular in the collinear limit. The
factor in Eq. (25) has canceled the pole. However, a phase singularity remains
from the form of the spinor inner products given in Eq. (23), which distinguishes terms with the splitting amplitude from
any others. In Eq. (23), the phase factor
rotates by
as
and
rotate once around their sum
as shown in Fig.
8
. The ratio of spinors in Eq. (26) then undergoes a
rotation accounting for the angular-momentum mismatch of 2
between the graviton
and the pair of gravitons
and
. In the gauge theory case, the terms proportional to the
splitting amplitudes (21) dominate the collinear limit. In the gravitational formula
(27), there are other terms of the same magnitude as
as
. However, these non-universal terms do not acquire any
additional phase as the collinear vectors
and
are rotated around each other. Thus, they can be separated from
the universal terms. The collinear limit of any gravity tree
amplitude must contain the universal terms given in Eq. (27) thereby putting a severe restriction on the analytic structure
of the amplitudes.
Even for the well-studied case of momenta becoming soft one
may again use the KLT relation to extract the behavior and to
rewrite it in terms of the soft behavior of gauge theory
amplitudes. Gravity tree amplitudes have the well known
behavior [139]
as the momentum of graviton
n
becomes soft. In Eq. (28) the soft graviton is taken to carry positive helicity; parity
can be used to obtain the other helicity case.
One can obtain the explicit form of the soft factors directly
from the KLT relations, but a more symmetric looking soft factor
can be obtained by first expressing the three-graviton vertex in
terms of a Yang-Mills three-vertex [26] (see Eq. (40)). This three-vertex can then be used to directly construct the
soft factor. The result is a simple formula expressing the
universal function describing soft gravitons in terms of the
universal functions describing soft gluons [26]:
where
is the eikonal factor for a positive helicity soft gluon in
QCD labeled by
n, and
a
and
b
are labels for legs neighboring the soft gluon. In Eq. (29) the momenta
and
are arbitrary null ``reference'' momenta. Although not manifest,
the soft factor (29) is independent of the choices of these reference momenta. By
choosing
and
the form of the soft graviton factor for
used in, for example, Refs. [10,
22,
23] is recovered. The important point is that in the form (29), the graviton soft factor is expressed directly in terms of the
QCD gluon soft factor. Since the soft amplitudes for gravity are
expressed in terms of gauge theory ones, the probability of
emitting a soft graviton can also be expressed in terms of the
probability of emitting a soft gluon.
One interesting feature of the gravitational soft and
collinear functions is that, unlike the gauge theory case, they
do not suffer any quantum corrections [23]. This is due to the dimensionful nature of the gravity coupling
, which causes any quantum corrections to be suppressed by powers
of a vanishing kinematic invariant. The dimensions of the
coupling constant must be absorbed by additional powers of the
kinematic invariants appearing in the problem, which all vanish
in the collinear or soft limits. This observation is helpful
because it can be used to put severe constraints on the analytic
structure of gravity amplitudes at any loop order.
|
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 |
