Chapter II: Recursion Relations in Four Dimensions
2.5 Examples
In this section, we illustrate the power of our recursion relations in various theories. The calculation is straightforward once the largezbehavior is known.
YM + ψ + φ. Consider a gauge theory with fermion and scalar matter in the adjoint representation. In addition to the gauge interactions, there are Yukawa interactions of the form Tr(φ{ψ, ψ}). Here we construct the color- ordered amplitudeM(ψ−, ψ−, φ, φ, φ) via a 3-line shift [{2},{3,4}i. The seed amplitudes for the recursion relation are
M(ψ−, ψ−, φ)= yh12i M(ψ−, ψ+, A−)= gh31i2/h12i
M(φ, φ, A−)= gh31ih23i/h12i M(φ, φ, φ, φ)= g2 1+ [13]2[24]2
[12][23][34][41]
!
M(ψ−, ψ+, φ, φ)= g2[23][24]
[12][34] −y2[24]
[41],
(2.39)
where y and g are the Yukawa and gauge coupling constants, respectively.
There are only two non-vanishing factorization channels. Based on these seeds, it’s straightforward to write down
M(ψ−, ψ−, φ, φ, φ)=yg2 1
[12] + [14]2[35]2
[13][12][34][45][51] − [35][34]
[13][23][45]
!
+y3 [35]
[23][51].
(2.40)
Note that the spurious pole [13] cancels between terms. From the final answer, we see that neither the BCFW shifts, like [{2},{3}i and [{1},{2}i, nor the Risager shift on [{2,3,4},0ican construct the amplitude. Thus, a 3-line shift such as [{2},{3,4}i is necessary to construct theories with both gauge and Yukawa interactions.
N = 1 SUSY. We have shown all massless supersymmetric theories are 3- line constructible. Consider an N = 1 supersymmetric gauge theory with an SU(3) flavor multiplet of adjoint chiral multiplets Φa. We assume a superpo- tential
W = iλTr(Φa[Φb,Φc]), (2.41) where a, b, c are fixed SU(3) flavor indices, no summation implied. We ap- ply our recursion relations on the (color-ordered) 6-point scalar amplitude M(φ−a, φ−b , φ−c, φ+c, φ+b, φ+a), where the superscripts and subscripts denote R- symmetry and flavor indices, respectively. In the massless limit, all scalars in the chiral multiplets carry equal R-charge. Therefore we can shift the three holomorphic scalars, namely, [{1,2,3},0i. The relevant lower point amplitudes for recursion are
M(A−, φ±a, φ∓a) = h31ih12i h23i M(φ−a, φ−b, φ+b, φ+a) = h13ih42i
h41ih23i +(1−λ2) M(A+, φ−a, φ−b, φ+b, φ+a) = h24ih53i
h51ih12ih34i+ (1−λ2) h52i h51ih12i.
(2.42)
Crucially, all of them are holomorphic in spinors. Under [{1,2,3},0i shift, it is straightforward to obtain the result by an MHV expansion from the above amplitudes [39, 47]
M(φ−a, φ−b, φ−c , φ+c, φ+b, φ+a)
= [6η][η1]
[61]h5P/61η]h2P/61η]
h24ih53i
h34i +(1−λ2)h52i
!
+ [3η][η4]
[34]h2P/34η]h5P/34η]
h51ih26i
h61i +(1−λ2)h25i
!
+ 1 P6122
h1P/612η]h62i
h2P/612η]h61i +(1−λ2)
! h4P/612η]h35i
h5P/612η]h34i +(1−λ2)
!
+ 1 P5612
h3P/561η]h24i
h2P/561η]h34i +(1−λ2)
! h6P/561η]h51i
h5P/561η]h61i +(1−λ2)
!
,
(2.43)
where η is the reference spinor and PF denotes the total momentum of the states in the factorization channel F. We have verified numerically that the answer is, as expected, independent of referenceη. Since the scalar amplitude is independent of the fermions, this result applies to any theory with the same bosonic sector. When λ = 1, the SU(3) flavor symmetry together with the U(1) R-symmetry combine to form the SU(4) R-symmetry of N = 4 SYM.
Our expression agrees with known answer in this limit.
(a) general scalar.
+
− +
− +
−
(b) U(1) charged scalar.
Figure 2.2: Factorization channels in the 6-point scalar amplitude inφ4theory.
The left and right diagrams show the factorization channels for the general case and the case of a U(1) charged scalar, respectively.
φ4 Theory. Next, consider amplitudes in a theory of interacting scalars. We have shown that a 5-line shift is sufficient to construct all amplitudes, while a 3-line shift suffices if every scalar has equal charge under a U(1) symmetry.
It is straightforward to see how these apply to the 6-point scalar amplitude in φ4 theory. Applying a 5-line shift, the factorization channel is depicted in Fig. 2.2 where we sum over all non-trivial permutations of external particles.
If the scalar is complex and carriesU(1) charge, namely|φ|4 theory, then only channels satisfying charge conservation can appear. Thus, three plus charged scalars never appear on one side of factorization. Consequently, shifting three plus charge scalars will construct the amplitude by exposing all physical poles.
ψ4 Theory. From our previous discussion, we know four fermion theory can be constructed by aQ2, 0 5-line shift. Consider a 6ptM(ψ+, ψ−, ψ+, ψ−, ψ+, ψ−) amplitude. Using a [{2,4},{1,3,5}i shift, we find
M(ψ+, ψ−, ψ+, ψ−, ψ+, ψ−)
= X
P(1,3,5),P(2,4,6)
(−1)σ[13]h46i 4P4562
z+,456h2|Pˆ456|5]|z−,456−(z+,456 ↔z−,456) z+,456−z−,456
= X
P(1,3,5),P(4,5,6)
(−1)σ[13]h46ih2|P456|5]
4P4562 , (2.44)
where hatted variable is evaluated at factorization limit andz±,456 are the two solutions of ˆP4562 = 0. The result is summed over permutation of (1,3,5) and (2,4,6) withσbeing the number of total permutations. In the last line, we use
the fact that h2|Pˆ456|5] is linear in z and only the non-deformed part survives after exchangingz±,456. We see the final answer has no square root as claimed before.
Maxwell-Einstein Theory. We discuss the theory where a U(1) photon minimally couples to gravity. The coupling constant has the same dimension as in GR (see Table 2.2). But as a photon has less spin than a graviton, the large z behavior is worse. We focus on the amplitudes with only external photons given that any amplitude with a graviton can be recursed by BCFW shift[43]. Using a m-line Q2 , 0 shift, we find Mf . zn+2−2m at large z;
thus, it’s always possible to construct such an amplitude when m > (n+2)/2.
Together with BCFW shift on gravitons, the theory is fully constructible!
Using Eqs. (2.29) and (2.32), the results for Q2 =0 m-line shifts are γ ≤
1+n−3m/2, for [{−,−, ...},0i n−3m/2, for [{−,−, ...},{+}i
. (2.45)
For the 4pt M(A−, A−, A+, A+) amplitude, we choose a [{1,2}; 4i shift so γ < 0. The inputs for recursions are 3pt functions obtained from consistency relation [35],M(A−, A+, G−) =h31i4/h12i2andM(A−, A+, G+) =[23]4/[12]2. The amplitude then follows
M(A−, A−, A+, A+) = h13ih1 ˆ3P[13][24]24|4]4 2
z24 + h23ih2 ˆ3P[23][14]14|4]4 2 z14
= h12i2[34]2
1 P242 + P12
14
. (2.46)
FvOperators. Consider amplitudes with a single insertion of aFvoperator.
Applying an [m,0i-line shift on minus helicity gluons and [m−1,1i m-line shift on all-but-one minus helicity gluons, Eq. (2.29) and Eq. (2.32) predict
γ ≤
v−m, for [{−,−, ...},0i v−1−m, for [{−,−, ...},{+}i
. (2.47)
We conclude [v+1,0i- and [v−1,1i-line shifts suffice to construct the amplitude with the given helicity configuration.
The case of the F3 operator has been studied extensively in Ref. [48]. Given the large z behavior above, the general MHV-like expression in Ref. [49] can be proven inductively by a [{−,−},{+}i shift. In addition, the vanishing of
the boundary term in the [{−,−, ...},0i shift directly proves the validity of CSW-expansion in Ref. [48]. We demonstrate it with the MHV-like amplitude M(A−, A−, A−, A+) where a single F3 operator is inserted. Note that the all- minus amplitude M(1−,2−,3−) = h12ih23ih31i is induced by anF3 operator.
Taking this as an input for the [{2,3},4ishift, we find M(A−, A−, A−, A+) =h12ih23ih31i h23i
h34ih2ˆ4i|z
12 − h12i h41ih2ˆ4i|z
23
!
= h12i2h23i2h31i2 h12ih23ih34ih41i.
(2.48)
This agrees with the result in Ref. [49, 50].
The case of the φtr(F F) operator, which is popular for the study of Higgs phenomenology, is very similar to F3 operator. The MHV-like formula and CSW expansion in Ref. [49] can also be proved analogously.
Rv Operators. Such operators often arise in effective theories from string action. Consider amplitudes with a single insertion of an Rv operator. The amplitude scales as z2v+n−3m under an m-line Q2 , 0 shift. For a given Rr operator, any(n > v)-pt amplitude can be constructed under an all-lineQ2 ,0 shift. If we use Q2 =0 shifts, Eq. (2.29) and Eq. (2.32) give
γ ≤
n+v −2m, for [{−,−, ...},0i n+v −2−2m, for [{−,−, ...},{+}i
. (2.49)
So if the helicity configuration is available, the amplitude is constructible under the [m,0i- and [m−1,1i-line shifts for m > (n+v)/2 andm > (n+v)/2−1, respectively.
Consider the 4pt M(G−, G−, G−, G+) amplitude with oneR3 operator inser- tion. We adopt the [{2,3}; 4i shift to construct it. The amplitude factorizes into the anti-MHV amplitude in GR and M(G−, G−, G−) = h12ih23ih31i in- duced by one insertion of R3 operator. We find
M(G−, G−, G−, G+)
=(h12ih23ih31i)2×
"
h12i2[41]
hˆ42i2h41i
z41 +(cyclic in (1,2,3))
#
,
=(h12ih23ih31i)2
"
[41]hξ1i2
h41ihξ4i2 + [42]hξ2i2
h42ihξ4i2 + [43]hξ3i2 h43ihξ4i2
#
=P122 M(1−A,2−A,4+A,3−A)M(1−A,2−A,3−A,4+A),
(2.50)
where |ξi is a reference spinor in 3-line shift. The result in the second line manifest the leading soft factor of particle 4. After canceling the reference spinor, the result in the last line is expressed in a KLT-relation form, where M(1−A,2−A,3−A,4+A) is the corresponding amplitude in gauge theory with the F3 operator given in Eq. (2.48). It agrees with Ref. [48]. It obvious from the answer that any [m,0i shift cannot construct the amplitude.