3.4 Soft and collinear properties of gravity amplitudes from gauge theory

The analytic properties of gravity amplitudes as momenta become either soft (ki → 0) or collinear (kj parallel to kj) 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 [139Jump To The Next Citation Point10Jump To The Next Citation Point] 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 [99Jump To The Next Citation Point20Jump To The Next Citation Point]. For the collinear case, at tree-level in quantum chromodynamics when two nearest neighboring legs in the color-stripped amplitudes become collinear, e.g., k1 → zP, k2 → (1 − z)P, and P = k1 + k2, the amplitude behaves as [99Jump To The Next Citation Point]:

tree k1∥k2 ∑ QCD tree tree λ A n (1,2,...,n) − → Split−λ (1,2)A n−1(P ,3,...,n). (21 ) λ= ±
The function SplitQ−CλD tree(1,2) is a splitting amplitude, and λ is the helicity of the intermediate state P. (The other helicity labels are implicit.) The contribution given in Eq. (21View Equation) is singular for k 1 parallel to k2; other terms in the amplitude are suppressed by a power of √ --- s12, which vanishes in the collinear limit, compared to the ones in Eq. (21View Equation). For the pure glue case, one such splitting amplitude is
QCD tree + + 1 1 Split− (1 ,2 ) = ∘--------- ----, (22 ) z(1 − z) ⟨12⟩
where
√ ---iϕjl √ --- −iϕjl ⟨jl⟩ = sij e , [jl] = − sij e , (23 )
are spinor inner products, and ϕjl 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 √--- sij, 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 Atree(1,2,3,...,n ) n for k1 collinear to k3.

From the structure of the KLT relations it is clear that a universal collinear behavior similar to Eq. (21View Equation) 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, Splitgravity tree. 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 k1 parallel to k2 in the five-point relation (11View Equation) and using Eq. (13View Equation) yields:

tree k1∥k2 κ ∑ gravity tree tree λ ℳ 5 (1,2,3,4,5) −→ 2- Split− λ (1,2)ℳ 4 (P ,3,4, 5), (24 ) λ=±
where
Splitgravity tree(1,2 ) = − s × SplitQCD tree(1, 2) × SplitQCD tree(2,1). (25 ) 12
More explicitly, using Eq. (22View Equation) then gives:
Splitgravity tree(1+, 2+) = ---−-1-- [12]. (26 ) − z (1 − z) ⟨12⟩
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.:
tree k1∥k2 κ ∑ gravity tree tree λ ℳ n (1,2,...,n) −→ 2- Split−λ (1,2)ℳ n−1(P ,3,...,n). (27 ) λ=±
(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 (22View Equation), the gravity splitting amplitude (26View Equation) is not singular in the collinear limit. The s12 factor in Eq. (25View Equation) has canceled the pole. However, a phase singularity remains from the form of the spinor inner products given in Eq. (23View Equation), which distinguishes terms with the splitting amplitude from any others. In Eq. (23View Equation), the phase factor ϕ12 rotates by 2π as ⃗k1 and ⃗k2 rotate once around their sum P⃗ as shown in Figure 8View Image. The ratio of spinors in Eq. (26View Equation) then undergoes a 4π rotation accounting for the angular-momentum mismatch of 2¯h between the graviton P+ and the pair of gravitons 1+ and 2+. In the gauge theory case, the terms proportional to the splitting amplitudes (21View Equation) dominate the collinear limit. In the gravitational formula  (27View Equation), there are other terms of the same magnitude as [12]∕⟨12⟩ as s12 → 0. However, these non-universal terms do not acquire any additional phase as the collinear vectors ⃗k1 and ⃗k2 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. (27View Equation) thereby putting a severe restriction on the analytic structure of the amplitudes.

View Image

Figure 8: As two momenta become collinear a gravity amplitude develops a phase singularity that can be detected by rotating the two momenta around the axis formed by their sum.

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 [139Jump To The Next Citation Point]

tree + kn→0 κ gravity + tree ℳ n (1,2,...,n ) −→ --𝒮 (n ) × ℳ n−1(1,2,...,n − 1) (28 ) 2
as the momentum of graviton n becomes soft. In Eq. (28View Equation) 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 [26Jump To The Next Citation Point] (see Eq. (40View Equation)). 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 [26Jump To The Next Citation Point]:

n∑−1 𝒮gravity(n+ ) = sni𝒮QCD (ql,n+, i) × 𝒮QCD (qr,n+, i), (29 ) i=1
where
QCD + ⟨ab ⟩ 𝒮 (a, n ,b) = --------- (30 ) ⟨an ⟩⟨nb ⟩
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. (29View Equation) the momenta ql and qr are arbitrary null “reference” momenta. Although not manifest, the soft factor (29View Equation) is independent of the choices of these reference momenta. By choosing ql = k1 and qr = kn−1 the form of the soft graviton factor for kn → 0 used in, for example, Refs. [10Jump To The Next Citation Point22Jump To The Next Citation Point23Jump To The Next Citation Point] is recovered. The important point is that in the form (29View Equation), 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 [23Jump To The Next Citation Point]. This is due to the dimensionful nature of the gravity coupling κ, 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.


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