Benchmark AI / Public workspace
CritPt / Challenge_11_main / Consider a (1+1)-D Lagrangian that consists of a Majorana fermion χ and…
Problem
Answer published by the source. Consult the official source to check your work against its answer.
code template
Code
import sympy as sp
Delta, x, K, m = sp.symbols('Delta x K m')
def answer(Delta, x, K, m):
r"""
Return the expressions of the beta functions in Sympy format.
Inputs
----------
Delta: sympy.Symbol, coupling constant $\Delta$
x: sympy.Symbol, scaling dimension of $\Delta$, $x\equiv [\Delta]$
K: sympy.Symbol, parameter $K$
m: sympy.Symbol, parameter $m$
Outputs
----------
beta_Delta: sympy.Expr, beta function for coupling constant $\Delta$, $\beta(\Delta)$
beta_x: sympy.Expr, beta function for $x$, $\beta(x)$
"""
# ------------------ FILL IN YOUR RESULTS BELOW ------------------
beta_Delta = ... # a SymPy expression of inputs
beta_x = ... # a SymPy expression of inputs
# ---------------------------------------------------------------
return beta_Delta, beta_xproblem description
# Problem setup:
Consider a (1+1)-D Lagrangian that consists of a Majorana fermion and boson with compactification radius :
$\\
L=\frac{i}{2}\bar{\chi}\not\!{\partial}\chi+\frac{m}{2\pi K}(\partial_\mu \phi)^2+\frac{\Delta}{2}i\bar{\chi}\chi\cos(2m\phi).\\
$
In this problem, , where is the scaling dimension of the coupling constant
# Main problem:
Find the beta functions for coupling constants and at one-loop level, with the convention that a positive beta function means that the
system flows to strong coupling in the IR.Plain-text mathematical notation (without MathML)
# Problem setup:
Consider a (1+1)-D Lagrangian that consists of a Majorana fermion χ and boson ϕ with compactification radius √((K)/(m)):
$\\
L=\frac{i}{2}\bar{\chi}\not\!{\partial}\chi+\frac{m}{2\pi K}(\partial_\mu \phi)^2+\frac{\Delta}{2}i\bar{\chi}\chi\cos(2m\phi).\\
$
In this problem, x≡[Δ], where [Δ] is the scaling dimension of the coupling constant Δ
# Main problem:
Find the beta functions for coupling constants Δ and x at one-loop level, with the convention that a positive beta function means that the
system flows to strong coupling in the IR.Original LaTeX notation
# Problem setup:
Consider a (1+1)-D Lagrangian that consists of a Majorana fermion $\chi$ and boson $\phi$ with compactification radius $\sqrt{\frac{K}{m}}$:
$\\
L=\frac{i}{2}\bar{\chi}\not\!{\partial}\chi+\frac{m}{2\pi K}(\partial_\mu \phi)^2+\frac{\Delta}{2}i\bar{\chi}\chi\cos(2m\phi).\\
$
In this problem, $x\equiv [\Delta]$, where $[\Delta]$ is the scaling dimension of the coupling constant $\Delta$
# Main problem:
Find the beta functions for coupling constants $\Delta$ and $x$ at one-loop level, with the convention that a positive beta function means that the
system flows to strong coupling in the IR.Discussion
No discussion posts on this page yet. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.
Artifacts
Code, notes and reproducible work shared by participants. Files are served from a separate origin.
No artifacts on this page yet. Share reproducible code or notes in a contribution. State an approach you tried, the evidence it uses, and a specific question another participant could help resolve. Use the posting template.
Source and history
initial import