benhaotang

/

phi4-qwq-sky-t1

like 2

OK, I see if I run with pytorch version with the system instruction by sky-T1

SYSTEM """Your role as an assistant involves thoroughly exploring questions through a systematic long thinking process before providing the final precise and accurate solutions. This requires engaging in a comprehensive cycle of analysis, summarizing, exploration, reassessment, reflection, backtracing, and iteration to develop well-considered thinking process.

Please structure your response into two main sections: Thought and Solution.

In the Thought section, detail your reasoning process using the specified format:

“‘
<|begin_of_thought|>
{thought with steps separated with "\n\n"}
<|end_of_thought|>
”’

Each step should include detailed considerations such as analisying questions, summarizing relevant findings, brainstorming new ideas, verifying the accuracy of the current steps, refining any errors, and revisiting previous steps.

In the Solution section, based on various attempts, explorations, and reflections from the Thought section, systematically present the final solution that you deem correct. The solution should remain a logical, accurate, concise expression style and detail necessary step needed to reach the conclusion, formatted as follows:
“‘
<|begin_of_solution|>
{final formatted, precise, and clear solution}
<|end_of_solution|>
”’
Now, try to solve the following question through the above guidelines:
"""

now looks like the completely wrong percentage will be lower(tried 5 times all at lease have one correct). But it cannot follow the instructed template very well, so no idea what will happen(e.g. hallucinations) reaching the end of the content length.

Haha, I guess this model is no more important, tried DeepSeek-R1-Distill-Qwen-14B@Q4_K_M, much stronger and robust reasoning chain. I mean, it's such a good time that we have so many opensource options to choose from.

>>> For a scalar field theory with interaction Lagrangian $\mathcal{L}_{int} = g\phi^3 + \lambda\phi^4$:
... 1. Enumerate all possible 1-loop Feynman diagrams contributing to the scalar propagator
... 2. For each diagram, write down its loop contribution
... 3. Provide Mathematica code to calculate these loop amplitudes with dimensional regularization at $d=4-\epsilon$
... Please explain your reasoning step by step.
<think>
Alright, let's tackle this problem about a scalar field theory with interaction Lagrangian 𝒫 int = gφ³ + λφ⁴. The task is to enumerate all possible one-loop Feynman diagrams contributing to the scalar 
propagator, write down their loop contributions, and provide Mathematica code for calculating these amplitudes using dimensional regularization at d=4−ε.

### 1. Enumerate All Possible 1-Loop Feynman Diagrams Contributing to the Scalar Propagator

First, I need to identify all possible one-loop diagrams that contribute to the two-point function (the propagator) of the scalar field φ. In this theory, we have two types of interactions: cubic (φ³) and 
quartic (φ⁴). Each interaction term contributes differently to Feynman diagrams.

- **Cubic Interaction (gφ³):** Each vertex in a cubic diagram has three legs. For a propagator correction at one loop, I need diagrams where the external lines attach to these vertices, and internal lines 
form a loop. Since each φ³ vertex has three legs, two of them can be connected externally, leaving one leg free for connecting to another vertex or forming a loop.

- **Quartic Interaction (λφ⁴):** Each vertex in a quartic diagram has four legs. For the propagator correction at one loop, the external lines attach to this vertex, and the remaining two legs form a loop. 
This is known as a "bubble" diagram.

After considering these interactions, there are **two distinct one-loop diagrams** contributing to the scalar propagator:

1. **φ³ Loop Diagram:** Two cubic vertices connected by internal propagators forming a loop.
2. **φ⁴ Bubble Diagram:** A single quartic vertex with two legs forming a loop while connecting external lines.

### 2. Loop Contributions for Each Diagram

Now, let's derive the loop contributions for each of these diagrams.

#### 1. φ³ Loop Diagram Contribution

- **Number of Vertices (V):** Two cubic vertices.
- **Number of Loops (L):** One loop formed by connecting two internal propagators between the two vertices.
- **Symmetry Factor (S):** The symmetry factor accounts for identical contributions from different topological arrangements. For this diagram, there is only one way to connect the two external legs and form 
a loop.

The contribution can be written as:

\[
\Sigma_{g} = (-i g)^2 \cdot \frac{i}{(2\pi)^d} \int d^d k \, \frac{1}{k^2 - m^2 + i\epsilon}
\]

Here:
- \( g \) is the coupling constant for the cubic interaction.
- The symmetry factor and combinatorial factors are absorbed into the loop integral.

#### 2. φ⁴ Bubble Diagram Contribution

- **Number of Vertices (V):** One quartic vertex.
- **Number of Loops (L):** One loop formed by two internal propagators connected to the same vertex.

The contribution can be written as:

\[
\Sigma_{\lambda} = (-i \lambda) \cdot \frac{i}{(2\pi)^d} \int d^d k \, \frac{1}{k^2 - m^2 + i\epsilon}
\]

Here:
- \( \lambda \) is the coupling constant for the quartic interaction.

### 3. Mathematica Code for Dimensional Regularization at d=4−ε

To calculate these loop amplitudes using dimensional regularization, we can use Mathematica with appropriate packages like FeynCalc or LoopIntegration. Below is a sample code outline:

(* Load necessary packages *)
<< FeynCalc`

(* Define the dimension *)
$DC = 4 - ε;

(* Define the scalar propagator in d dimensions *)
propagator[p_] := 1 / (p^2 - m^2 + I ε);

(* φ³ Loop Contribution *)
Sigma_g = (-i g)^2 * Integrate[ propagator[k] * propagator[k - p], {k, 0} ] // DimReg;

(* φ⁴ Bubble Contribution *)
Sigma_lambda = (-i λ) * Integrate[ propagator[k]^2, {k, 0} ] // DimReg;

(* Combine contributions *)
TotalSigma = Sigma_g + Sigma_lambda;

This code snippet outlines the steps to compute each contribution separately and then combine them. The actual implementation would require specific functions from FeynCalc or another package to handle loop 
integrals and dimensional regularization.

### Final Answer

For a scalar field theory with interaction Lagrangian 𝒫 int = gφ³ + λφ⁴, there are two one-loop diagrams contributing to the propagator:

1. **φ³ Loop Diagram:** Two cubic vertices connected by internal propagators forming a loop.
   - Contribution: \[
     \Sigma_{g} = (-i g)^2 \cdot \frac{i}{(2\pi)^d} \int d^d k \, \frac{1}{k^2 - m^2 + i\epsilon}
     \]

2. **φ⁴ Bubble Diagram:** A single quartic vertex with a loop formed by two internal propagators.
   - Contribution: \[
     \Sigma_{\lambda} = (-i \lambda) \cdot \frac{i}{(2\pi)^d} \int d^d k \, \frac{1}{k^2 - m^2 + i\epsilon}
     \]

Mathematica code using dimensional regularization can be structured to compute these contributions individually and then sum them for the total self-energy correction.
</think>

### Summary of Results

For a scalar field theory with interaction Lagrangian $\mathcal{L}_{int} = g\phi^3 + \lambda\phi^4$, we have identified two one-loop Feynman diagrams contributing to the scalar propagator:

1. **$\phi^3$ Loop Diagram:**
   - This diagram involves two cubic vertices connected by internal propagators forming a loop.
   - The contribution is given by:
     \[
     \Sigma_{g} = (-i g)^2 \cdot \frac{i}{(2\pi)^d} \int d^d k \, \frac{1}{k^2 - m^2 + i\epsilon}
     \]

2. **$\phi^4$ Bubble Diagram:**
   - This diagram consists of a single quartic vertex with two internal propagators forming a loop.
   - The contribution is given by:
     \[
     \Sigma_{\lambda} = (-i \lambda) \cdot \frac{i}{(2\pi)^d} \int d^d k \, \frac{1}{k^2 - m^2 + i\epsilon}
     \]

### Mathematica Code for Dimensional Regularization

Here is a Mathematica code snippet that calculates these loop contributions using dimensional regularization at $d = 4 - \epsilon$:

(* Load necessary packages *)
<< FeynCalc`

(* Set the dimension *)
$d = 4 - ε;

(* Define the scalar propagator in d dimensions *)
propagator[p_] := 1 / (p^2 - m^2 + I ε);

(* Compute φ³ Loop Contribution *)
Sigma_g = (-i g)^2 * Integrate[ propagator[k] * propagator[k - p], {k, 0} ] // DimReg;

(* Compute φ⁴ Bubble Contribution *)
Sigma_lambda = (-i λ) * Integrate[ propagator[k]^2, {k, 0} ] // DimReg;

(* Total self-energy contribution *)
TotalSigma = Sigma_g + Sigma_lambda;

This code outlines the steps to compute each contribution separately and then combines them to get the total self-energy correction at one loop.

Sure there is also minor flaws with the answer here, and the code is still concept-driven, and is still not runable, but this is so much more better than what I see with QwQ and even claude's CoT, this is how professor expect a student to think.