Introduction
Recursion Relations in Four Dimensions
Introduction
For our analysis, we define the "covering space" of the recursion relations shown in Eq. 2.2), which includes natural generalizations of the BCFW [4] and Risager [39] recursion relations. 2.4 we present our main result, which is a classification of the minimal recursion relations required for the construction of various renormalizable and non-renormalizable theories.
Covering Space of Recursion Relations
Second, there is a very significant difference in the structure of the recursion relation depending on whether Q2 vanishes in all channels. In this case the recursion coincides with the form of displacements of the line [1, m−1i and [m−1,1i of the line in Eq.
Large z Behavior of Amplitudes
Nesting the large z scaling of the external wavefunctions in Eq. 2.21) with the numerator and denominator of the Feynman diagram in Eq. To determine the large z behavior of the [1, m−1i line shift, we start with our previous result on the [0, mi line shift.
On-Shell Constructible Theories
The standard model can also be built with 3 lines simply because it has a single scalar – the Higgs boson – that carries the hypercharge. Substituting the above equation and the number of vertices u= (m+nf −2)/(v−2) into the behavior of large z for the general displacement of the m-line in Eq.
![Table 2.2: The dimensionality of the coupling constant, [g ], for an n-point am- am-plitude, where u denotes the number interaction vertices, which have minimal valency v .](https://thumb-ap.123doks.com/thumbv2/123dok/10412343.0/30.918.168.756.104.155/dimensionality-coupling-constant-plitude-denotes-interaction-vertices-minimal.webp)
Examples
8], but we can be more parsimonious by choosing (d+1)-line or less instead of a shift over all lines. 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 the gauge theory with the F3 operator given in Eq.
Outlook
The scaling shift in Eq. 3.4), thereby transforming the soft behavior as a degree σ of zero amplitude. In this appendix, we give a detailed proof of the fuzzy theorem mentioned in Section 3.
Recursion Relations for Effective Field Theories
Introduction
As we have seen from the previous chapter, recursion relations in the shell are usually inapplicable to EFTs. We derive a new class of recursion relations that completely construct the S-matrices of certain scalar effective field theories by exploiting an additional physical component: the vanishing of amplitudes in the soft limit.
Recursion and Factorization
This approach is logical because the soft behavior of the S matrix actually encodes the interactions and symmetries of the corresponding effective field theory [51] , providing a theory classification purely in terms of on-shell data. PI2A¯nI(zI), (3.3), which establishes a recursion relation in the form of the lower point amplitudes AnI and An¯I, wherenI+n¯I =n+2.
Recursion and Soft Limits
To calculate the amplitude, we use Cauchy's theorem for the contour that encloses all poles at finite z. Thus, the integrand of Eq. 3.9) has poles only from the factorization channels, so by analogy with BCFW the amplitude.
Criteria for On-Shell Constructibility
The derived power count parameter ρ in Eq. 3.17), together with the soft boundary degreeσ defined in Eq. 3.4) define a two-parameter classification of scalar effective field theories. For the extraordinary theories, Eq. 3.19) is more than satisfied, giving better largezfalloff than is even necessary to build them.
Example Calculations
The six-point scattering amplitude takes the form. where the ellipses indicate permutations, totaling the ten factorization channels of the six-point amplitude. 3.22), each factorization channel has two roots inz, so recursion yields. which can be shown as before to be equivalent to the Feynman diagram expression. Permutation symmetry implies that the amplitude again has the form of Eq. Finally, let's calculate the amplitudes in the general Galileon.
Outlook
Therefore, the behavior of the amplitude in the soft limit NGB p → 0 can be deduced from the properties of the residual function Rµ(p). Because p·R(p) is related to the amplitude via Eq. A.4), we can derive additional information on the smooth behavior of the amplitude at the top of Eq.
A Periodic Table of Effective Field Theories
Introduction
Here ρ characterizes the number of derivatives for interaction, with a corresponding Lagrangian of the schematic form. Soft non-trivial bounds require that the valence of the main interaction be bounded by the spatial dimension, so that v ≤d+1.
Classification Scheme
For such peculiar values of ρ, the leading valence v of the theory can be very high. By fixing the power counting parameter ρ, the soft degree σ, the valence of the leading interaction v, and the spacetime dimension d, we can now place tight constraints on the space of scalar EFTs.
From Symmetries to Soft Limits
An important component of the proof is an observation that the Noether currents of the shear symmetry and of the improved symmetry in Eq. The explicit form of Γµα1..αn(x), which depends on ∆α1..αn(x), is irrelevant for the proof of the soft theorem.
On-shell Reconstruction
So for n ≥ d+2 it is possible to implement a soft shift with all lines that probes all soft kinematic amplitude limits. We use smooth linear variation in all but one so that our results apply to general dimensions.
Bounding Effective Field Theory Space
This is chosen so that the 4pt amplitude carries a factor f1ρ+1(z) which will exceed the f1σ(z) factor in the denominator of the recursion. Therefore, the leading spurious pole in the residual occurs when all derivatives act on PI2(z) but not on the counters.
Classification of Scalar EFTs
The set of all possible values of ρ is ρ = m−2v−2 where m is the number of derivatives in the interaction with the constraint. For v > 6, which is only relevant for d > 5, there is a possibility that new theories may emerge, but they must stay at the same points (ρ, σ, v, d) degenerate with WZW and Galileo .
More Directions
The first question is what is the meaning of the word "non-trivial" from the point of view of the double soft limit. The other natural limit to consider is the collinear limit where two of the moments become proportional.
Outlook
Our claim is that the tree amplitude of the NLSM is equal to the tree amplitudes. The Euler-Lagrange equations of motion for Eq. where the divergence of the first equation gives 2∂µZµ = 0.
Symmetry for Flavor-Kinematics Duality from an Action . 87
Warmup
The above derivation actually corresponds to the diagrammatic representation of the Jacobi identity that typically appears in the study of scattering amplitudes. The resulting triplet of objects—each equal to Feynman's diagrammatic counter associated with a given cubic topology—satisfies the Jacobi identity.
Action
Scattering Amplitudes
We can generalize to the distribution of n particles by computing all kinematic counters for half-scale topologies, which form a complete basis for all tree amplitudes [32, 112]. We checked that these expressions reproduce the amplitudes of the NLSM trees up to the ten-particle distribution.
Equations of Motion
If Z is on the shell, then according to the prescription of Eq. 5.13) is longitudinally polarized, so Eq. 5.23) is still valid due to shell condition. In particular, due to the off-diagonal kinetic term, the Z source outside the shell propagates into X, which can then only act through the field strength combination in Eq. 5.23) simply says that the longitudinal polarizations of X are projected out.
Symmetries
In any case, the bottom line is that the draw. 5.23) is generally true whenever the equations of motion are satisfied. 5.23) is a restriction on classical fields, its implications for amplitudes are actually somewhat subtle. While θXµ is technically space-time dependent, it always enters with a derivative, so the symmetry is still global.
Kinematic Algebra
Note that the cubic interactions are also invariant under the local versions of the δX, δY, δZ transformations. At the level of the scattering amplitudes, these manipulations imply that all Feynman diagrams calculated from Eq. 5.10) will automatically satisfy Jacobi identities up to terms that vanish on the transverse condition in Eq.
Double Copy
Infrared Structure
In particular, consider the soft limit of a Nambu-Goldstone boson, here conceived as a longitudinal Z of the NLSM amplitude in Eq. Although the action lacks the required quartic interactions necessary for true Z-meter symmetry, the soft Z-limit is determined solely by cubic interactions.
Summary
It is known that the divergence of the amplitude of one loop is related to the intersection of two particles. The form of the matrix of anomalous dimension for all operators of dimension five and six is shown in Tab.
Non-renormalization Theorem without Supersymmetry
Introduction
In this chapter we derive a new class of non-renormalization theorems for non-supersymmetric theories. The resulting non-renormalization theorems for all dimension five and six operators are shown in Tab.
Weighing Tree Amplitudes
Most four-point tree amplitudes with w4 = 1 or 3 vanish since they have no Feynman diagrams, so. These "exceptional amplitudes" are the only four-point tree amplitudes with w4<4 that do not vanish identically.
Weighing One-Loop Amplitudes
Generalized unitarity [1, 2] determines integral coefficients by connecting kinematic singularities of single-loop amplitude with products of tree amplitudes. As mentioned earlier, the amplitudes on both sides of the cut must be four-point or higher for a non-trivial uniform cut, so Eq. 6.18) then implies that wi ≥ wj and wi≥wj, which is the non-normalization theorem of Eq.
Infrared Divergences
Our non-renormalization theorems imply that operators can only renormalize operators of equal or greater weight, which in Figure 6.1 prohibits transitions that move down or to the left. Here y2 and ¯y2 label items that are non-zero due to non-holomorphic Yukawa couplings, × labels items that vanish because there are no diagrams [127], and ×∗ labels items that vanish due to a combination of counterterm analysis and our non-renormalization theorems .
Application to the Standard Model
As a result, the interference term Sj→i∗ 0Si→i0 carries net small group weight relative to the parent particle initiating the collinear emission. In summary, since real emission is infrared finite for wi < wj, there are no corresponding ultraviolet divergences from massless bubbles.
Outlook
Proof of the Soft Theorem
As we will see in the next subsection, Eq. A.20) is the key formula for deriving fuzzy theorems for NGB. Note that it depends only on the c-number part of the general symmetric transformation of Eq. Therefore, theories invariant with respect to the transformation equation. A.11) with the same αj(x) form universality classes with the same fuzzy behavior.
Bounds on ρ from Bonus Relations
These strict bounds shorten the range of numerical checks for the false pole cancellation in Eq. Combining the two provides proof of ρ < 3 for all non-trivial theories, independent of the flavor structure.
Catalog of Scalar Effective Field Theories
The natural generalization of the single scalar DBI Lagrangian can be obtained as the lowest order action of the d−dimensional brane propagating in d+N dimensional flat space. However, the structure of the Lagrangian does not allow the introduction of flavored amplitudes.
