# Conformal deformations of N=4 Sym

Open Access

- Graduate Program:
- Physics
- Degree:
- Doctor of Philosophy
- Document Type:
- Dissertation
- Date of Defense:
- June 02, 2014
- Committee Members:
- Radu Roiban, Dissertation Advisor
- Radu Roiban, Committee Chair
- Murat Gunaydin, Committee Member
- Martin Bojowald, Committee Member
- Nigel David Higson, Committee Member

- Keywords:
- Scattering amplitude
- BCFW recursion
- Conformal field theory

- Abstract:
- This thesis focuses on the calculation and application of scattering amplitude. We present recursion relations which can be used to compute tree level scattering amplitude which the conventional BCFW recurion fails to apply. We also present the complete one loop and some higher loop scattering amplitude of $\beta$-deformed SYM theory, a superconformal deformation of the $\mathcal{N}=4$ super Yang-Mills theory. Color/kinematic duality is the conjecture that in a gauge theory, kinematics numerators of diagrams obey relations similar to Jacobi identities of color factors. We carry out a comprehensive search in theories with fields solely in the adjoint representation of the gauge group. It turns out in this family any theory with color/kinematics duality is a dimensional reduction of $\mathcal{N}=1$ SYM in 4D, 6D, 10D or pure Yang-Mills in any dimensions. We also test a weaker version of color/kinematics duality in which some Jacobi identities of color factors are violated(for example by deformations), but kinematics factors still obey all Jacobi identities. This new duality was confirmed in $\beta$-deformed SYM and massless QCD for tree amplitude with number of external legs less than seven. If a quantum field theory exhibits a symmetry which is absent in the original Lagrangian at its conformal fixed point, this symmetry will emerge as the result of renormalization group flow. $\gamma_i$-deformed SYM is a 3 parameter deformation of $\mathcal{N}=4$ SYM, and for general values of $\gamma_i$'s it has no supersymmetry. However, if $\gamma_i$ parameters take the same value, the theory has $\mathcal{N}=1$ supersymmetry, and is known as $\beta$-deformed SYM. We show that $\beta$-deformed SYM is an infrared fixed point of $\gamma_i$-deformed SYM, and supersymmetry emerges at low energy. We also developed a method to extract beta functions directly from on shell scattering amplitude when working on this project. We describe several new conformal deformations of $\mathcal{N}=4$ SYM. The one loop renormalization flow of coupling constants of deformation theories is characterized by a scalar potential, and conformal field theories lie in the minima of this scalar potential. When energy decrease, Yukawa couplings always flow to a fixed point, while scalar couplings may flow without bound. Around the $\mathcal{N}=1$ superconformal fixed point (Leigh-Strassler theory), for some special configurations of coupling constants, scalar couplings also flow to the fixed point, and $\mathcal{N}=1$ supersymmetry emerges as a result of renormalization flow.