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PhD Review Jonathan Maloney


Academics ●

Qualification exams passed.

Research hours (Phys 799).

Phys 684 left to take.

Dissertation and Defense.


Preliminary Work ●

Loop integration techniques in QFT (i.e., regularization and renormalization). Select homework problems from Peskin & Schroeder. Reproducing already known results from select papers (e.g., F,J & J, Sher, Martin et. al., etc.). Develop habits necessary to handle large volumes of algebra, which are necessary for these types of calculations.


Loop Integration Integral Table

Dirac Equation

Gamma Identities

Symmetry


Summary of Methods in Loop Calculations 1. Draw the diagram(s) and write down the amplitude. 2. Introduce Feynman parameters to combine the denominators of the propagators. 3.Complete the square in the new denominator by shifting to a new loop momentum variable . 4. Write the numerator in terms of . Drop odd powers of and rewrite even powers using identities. 5.Perform the momentum integral by means of a Wick rotation and four-dimensional spherical coordinates (i.e., just look it up in a table!).

,


Reproduce Known Results SM Effective Potential at 1-Loop (i.e.,

}

):


Reproduce Known Results Cont.

Identities:

Linear Equations:

{


Reproduce Known Results Cont. Differential Equations:

Where,

& Where,


Reproduce Known Results Cont. Where,

(From previous result)

*


Reproduce Known Results Cont.

Where,

{


Mathematica ●

Program Basics. How to write functions and other important related objects. Create and save functions related to research (i.e., I(x,y,z) and A(x) integrals).


First Project Objective: Compute general 2-loop vacuum integrals with

complex squared propagator masses, by reducing them to a basis and then computing the basis integrals.

Ex) Through elementary manipulation, each individual Feynman diagram can be reduced to a sum of integrals of the form:

[1]


First Project Cont. Effective Potential: Plays a role in determining the nature of the vacuum in renormalizable field theories, which allows us to refine the masses for particles of interest. Further, it allows one to find the qualitative (e.g., whether a symmetry is spontaneously broken) and quantitative features (e.g., what the VEV is numerically) of the vacuum state.

,

[1] [1] Ford, Jack, Jones. The Standard Model Effective Potential at Two Loops. Nucl.Phys.B387:373-390,1992.


Moving Forward Project 1:

Compute general 2-loop vacuum integrals with complex squared propagator masses, by reducing them to a basis and then computing the basis integrals.

Project 2: Application of evaluated 2loop basis integrals with complex squared masses to the re-summation of 2-loop effective potentials in general gauges.

Project 3:

TBD. Possibly on search strategies for new particles at proton-proton colliders with energies higher than the LHC.


Questions?

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PhD Review  

I recently had to undergo what's known as a PhD review in order to demonstrate that I'm making adequate progress towards the degree. This is...

PhD Review  

I recently had to undergo what's known as a PhD review in order to demonstrate that I'm making adequate progress towards the degree. This is...

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