From Trees to Loops

5 From Trees to Loops

In this section, the above discussion is extended to quantum loops through use of $D$ -dimensional unitarity [16 , 17 , 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 , 80 ] 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 [96 , 92 , 97 , 35 ]: $S†S = 1$ . In perturbation theory the Feynman diagrams describe a transition matrix $T$ defined by $S ≡ 1 + iT$ , 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 $κ$ 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 $κ4$ in gravity, are given in terms of the product of two four-point tree amplitudes, each carrying a power of $κ2$ . This is then summed over all two-particle states that can appear and integrated over the intermediate phase space (see Figure 9

Figure 9: The two-particle cut at one loop in the channel carrying momentum $k1 + k2$ . 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 [96 , 92 , 97 , 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 $R (si)$ , where the $si$ are the kinematic variables for the amplitude.

A convenient way to deal with this problem [16 , 17 , 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 $D = 4 − 2𝜖$ . With dimensional regularization, amplitudes for massless particles necessarily acquire a factor of $(− si)− 𝜖$ for each loop, from the measure $∫ dDL$ . For small $𝜖$ , $(− si)− 𝜖R (si) = R(si) − 𝜖ln(− si)R (si) + 𝒪 (𝜖2)$ , so every term has an imaginary part (for some $si > 0$ ), though not necessarily in terms which survive as $𝜖 → 0$ . Thus, the unitarity cuts evaluated to $𝒪 (𝜖)$ 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 $k + k 1 2$ , as shown in Figure 9 , is given by

where $L2 = L1 − k1 − k2$ , and the sum runs over all physical states of the theory crossing the cut. In this generalized cut, the on-shell conditions $2 2 L 1 = L 2 = 0$ 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.

Figure 10: Two examples of generalized cuts. Double two-particle cuts of a two-loop amplitude are shown. This separates the amplitude into a product of three tree amplitudes, integrated over loop momenta. The dashed lines represent the cuts.

A less trivial two-loop example of a generalized “double” two particle cut is illustrated in panel (a) of Figure 10 . The product of tree amplitudes appearing in this cut is

∑ 𝒜tr4ee(1,2,− L2,− L1) × 𝒜tr4ee(L1,L2,− L3, − L4 ) states | × 𝒜tree(L4,L3,3, 4)|| , (44 ) 4 2×2−cut

where the loop integrals and cut propagators have been suppressed for convenience. In this expression the on-shell conditions $L2 = 0 i$ are imposed on the $L i$ , $i = 1, 2,3,4$ 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 Figure 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 $m − 2𝜖$ 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 [16 , 17 , 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.