Interacting Scalar Fields

Abstract

The key concepts of a perturbative or weak-coupling expansion are introduced in the context of evaluating the imaginary-time path integral representation for the partition function of an interacting scalar field. The issues of ultraviolet and infrared divergences are brought up. These problems are cured through renormalization and resummation, respectively.

Appendix: An Alternative Method for Resummation

In this appendix we show that the previous resummation can also be implemented through the following steps:

• (i) Following the computation of $m_{\mbox{ phys}}^{2}$ in Eq. (3.71) but working now at finite temperature, we determine a specific T-dependent pole mass in the m _B → 0 limit. The result can be called an effective thermal mass, $m_{\mbox{ eff}}^{2}$.

• (ii) We argue that in the weak-coupling limit ($\lambda _{\mbox{ R}}^{} \ll 1$), the thermal mass is important only for the Matsubara zero mode [3].

• (iii) Writing the Lagrangian (for m _B ² = 0) in the form

  $$\displaystyle\begin{array}{rcl} L_{E}^{}& =& \mathop{\underbrace{\frac{1} {2}\partial _{\mu }\phi \,\partial _{\mu }\phi + \frac{1} {2}m_{\mbox{ eff}}^{2}\,\phi _{ n=0}^{2}}}\limits +\mathop{\underbrace{ \frac{1} {4}\lambda _{\mbox{ B}}^{}\phi ^{4} -\frac{1} {2}m_{\mbox{ eff}}^{2}\,\phi _{ n=0}^{2}}}\limits \;, {} \\ & & L_{0}^{} L_{\mbox{ I}}^{} \\ \end{array}$$

  (3.94)

  we treat L ₀ as the free theory and $L_{\mbox{ I}}^{}$ as an interaction of order $\lambda _{\mbox{ R}}^{}$. With this reorganization of the theory, we write down the contributions f ₍₀₎ and f ₍₁₎ to the free energy density, and check that we obtain a well-behaved perturbative expansion that produces a result agreeing with what we got in Eq. (3.92).

Starting with the effective mass parameter, the computation proceeds precisely like the one leading to Eq. (3.71), with just the replacement $\mbox{ $\int _{P}$} \rightarrow \Sigma \!\!\!\!\mbox{ $\int $}_{\mbox{ $P$}}$. Consequently,

$$\displaystyle{ m_{\mbox{ eff}}^{2} =\lim _{ m_{\mbox{ B}}^{2}\rightarrow 0}{\Bigl [m_{\mbox{ B}}^{2} + 3\lambda _{\mbox{ B}}^{}I(m_{\mbox{ B}}^{},T)\Bigr ]} = 3\lambda _{\mbox{ B}}^{}I(0,T) = \frac{\lambda _{\mbox{ R}}^{}T^{2}} {4} + \mathcal{O}(\lambda _{\mbox{ R}}^{2})\;. }$$

(3.95)

We note that for the non-zero Matsubara modes, with ω _n ≠ 0, we have $m_{\mbox{ eff}}^{2} \ll \omega _{n}^{2}$ in the weak-coupling limit $\lambda _{\mbox{ R}}^{} \ll (4\pi )^{2}$, so that the thermal mass plays a subdominant role in the propagator. In contrast, for the Matsubara zero mode, $m_{\mbox{ eff}}^{2}$ modifies the propagator significantly for $p^{2} \ll m_{\mbox{ eff}}^{2}$, removing any infrared divergences. This observation justifies the fact that the thermal mass was only introduced for the n = 0 mode in Eq. (3.94).

With our new reorganization, the free propagators become different for the Matsubara zero $(\tilde{\phi }_{n=0})$ and non-zero $(\tilde{\phi }')$ modes:

(3.96)

(3.97)

Consequently, Eq. (3.17) gets replaced with

$$\displaystyle\begin{array}{rcl} f_{(0)}(T)& =& \mbox{ $\sum $}\!\!\!\!\!\!\!\int \,_{\!\!\!\!\mbox{ $P$}}'\frac{1} {2}\ln (P^{2}) + T\int _{\mathbf{ p}}\frac{1} {2}\ln (p^{2} + m_{\mbox{ eff}}^{2}) -\mbox{ const.} \\ & =& J'(0,T) + J^{(n=0)}(m_{\mbox{ eff}},T) \\ & =& -\frac{\pi ^{2}T^{4}} {90} -\frac{m_{\mbox{ eff}}^{3}T} {12\pi } \;. {}\end{array}$$

(3.98)

In the massless first term, the omission of the zero mode made no difference. Similarly, with f ₍₁₎ now coming from $L_{\mbox{ I}}^{}$ in Eq. (3.94), Eq. (3.19) is modified into

$$\displaystyle\begin{array}{rcl} f_{(1)}(T)& =& \frac{3} {4}\lambda _{\mbox{ B}}^{}\langle \phi (0)\phi (0)\rangle _{0}^{}\langle \phi (0)\phi (0)\rangle _{0}^{} -\frac{1} {2}m_{\mbox{ eff}}^{2}\langle \phi _{ n=0}(0)\phi _{n=0}(0)\rangle _{0}^{} \\ & =& \frac{3} {4}\lambda _{\mbox{ B}}^{}{\Bigl [I'(0,T) + I^{(n=0)}(m_{\mbox{ eff}},T)\Bigr ]}^{2} -\frac{1} {2}m_{\mbox{ eff}}^{2}\,I^{(n=0)}(m_{\mbox{ eff}},T) \\ & =& \frac{3} {4}\lambda _{\mbox{ B}}^{}{\biggl [ \frac{T^{4}} {144} -\frac{m_{\mbox{ eff}}T^{3}} {24\pi } + \frac{m_{\mbox{ eff}}^{2}T^{2}} {16\pi ^{2}} \biggr ]} + \frac{1} {2}m_{\mbox{ eff}}^{2}\frac{m_{\mbox{ eff}}T} {4\pi } \;. {}\end{array}$$

(3.99)

Inserting Eq. (3.95) into the last term of Eq. (3.99), we see that this contribution precisely cancels against the linear term within the square brackets. As we recall from Eq. (3.84), the linear term was part of the problematic series that needed to be resummed. Combining Eqs. (3.98) and (3.99), we instead get

$$\displaystyle{ f(T) = -\frac{\pi ^{2}T^{4}} {90} + \frac{3} {4}\lambda _{\mbox{ R}}^{} \frac{T^{4}} {144} -\frac{m_{\mbox{ eff}}^{3}T} {12\pi } + \mathcal{O}(\lambda _{\mbox{ R}}^{2})\;, }$$

(3.100)

which agrees with Eq. (3.93).

The cancellation that took place in Eq. (3.99) can also be verified at higher orders. In particular, proceeding to $\mathcal{O}(\lambda _{\mbox{ R}}^{2})$, it can be seen that the structure in Eq. (3.85) gets cancelled as well. Indeed, the resummation of infrared divergences that we carried out explicitly in Eq. (3.91) can be fully captured by the reorganization in Eq. (3.94).