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I'm experiencing some troubles with one of the exercises in Kardars book on Statical physics of fields (problem 5 Ch3) or see https://ocw.mit.edu/courses/physics/8-334-statistical-mechanics-ii-statistical-physics-of-fields-spring-2014/assignments/MIT8_334S14_pset2.pdf

Consider an n-component vector field $m(\textbf{x})$ coupled to a scalar field $A(\textbf{x})$, through the effective Hamiltonian \begin{equation} \beta \mathcal{H} = \int d^d \textbf{x} \left[ \frac{K}{2} (\nabla m)^2 +\frac{t}{2}m^2 +u m^4 +e^2 m^2 A^2 +\frac{L}{2}(\nabla A)^2\right] \end{equation} with $K$, $L$ and $u$ positive. First, you should simply do some saddle point calculations, quite straightforward. Next (question c), you include some fluctuations and expand $\beta \mathcal{H}$ to quadratic order in the fluctuations $\phi$ and $a$. From this you can also find the correlation lengths, for the different fluctuations. But I get stuck for the correlation length of the fluctuation $a$ of the scalar field. If I do the calculations I obtain \begin{equation} \frac{L}{\xi_a^2}= \frac{\partial^2 \Psi (\bar{m})}{\partial a^2}=4e^2 \bar{m}^2 \end{equation} Now from the saddle point calculations you find that $\bar{m}=0$ for $t>0$. This would mean that the correlation length for the scalar field fluctuation $a$ becomes infinite for $t>0$?

What am I doing wrong in my calculations of the correlation length? If it would be correct, what is the physical meaning of this divergence? More general, what is the expected behavior of this system? How will this coupling of a scalar field lead to the Higgs mechanism; where will this be observed, in the fluctuations?

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What you found is completely sensible. In the context of particle physics, the inverse correlation length corresponds to the mass of the particle. You found the correlation length is infinite for $t > 0$, which corrresponds to massless particles. The Higgs mechanism occurs and gives mass to the particles when $t < 0$.

Also, you probably know this, but a divergent correlation length doesn't mean the correlator is diverging; nothing about this situation is pathological. It just means that it falls off as a power law rather than an exponential.

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