Free Scalar Fields

Abstract

The concepts of Sect. 1 are generalized to the case of a free massive scalar field living in a d + 1 dimensional spacetime. This can be viewed as a system of infinitely many coupled harmonic oscillators. The resulting imaginary-time path integral for the partition function is expressed in Fourier representation. Matsubara sums are evaluated both in a low-temperature and a high-temperature expansion. The numerical convergence of these expansions, as well as some of their general properties, are discussed.

Appendices

Appendix A: Properties of the Euler $\Gamma$ and Riemann ζ functions

$\fbox{\mbox{$\displaystyle \varGamma(s)$}}$

The function $\Gamma (s)$ is to be viewed as a complex-valued function of a complex variable s. For $\mathop{\mathrm{Re}}(s) > 0$, it can be defined as

$$\displaystyle{ \Gamma (s) \equiv \int _{0}^{\infty }\!\mathrm{d}x\,x^{s-1}e^{-x}\;, }$$

(2.96)

whereas for $\mathop{\mathrm{Re}}(s) \leq 0$, the values can be obtained through the iterative use of the relation

$$\displaystyle{ \Gamma (s) = \frac{\Gamma (s + 1)} {s} \;. }$$

(2.97)

On the real axis, $\Gamma (s)$ is regular at s = 1; as a consequence of Eq. (2.97), it then has first-order poles at s = 0, −1, −2, … .

In practical applications, the argument s is typically close to an integer or a half-integer. In the former case, we can use Eq. (2.97) to relate the desired value to the behavior of $\Gamma (s)$ and its derivatives around s = 1, which can in turn be worked out from the convergent integral representation in Eq. (2.96). In particular,

$$\displaystyle{ \Gamma (1) = 1\;,\quad \Gamma '(1) = -\gamma _{\mbox{ E}}\;, }$$

(2.98)

where γ _E is the Euler constant, γ _E = 0. 577215664901… . In the latter case, we can similarly use Eq. (2.97) to relate the desired value to $\Gamma (s)$ and its derivatives around $s = \frac{1} {2}$, which can again be worked out from the integral representation in Eq. (2.96), producing

$$\displaystyle{ \Gamma {\bigl (\frac{1} {2}\bigr )} = \sqrt{\pi }\;,\quad \Gamma '{\bigl (\frac{1} {2}\bigr )} = \sqrt{\pi }(-\gamma _{\mbox{ E}} - 2\ln 2)\;. }$$

(2.99)

The values required for Eq. (2.91) thus become

$$\displaystyle\begin{array}{rcl} \Gamma {\bigl ( -\frac{1} {2}+\epsilon \bigr )}& =& -2\sqrt{\pi } + \mathcal{O}(\epsilon )\;,{}\end{array}$$

(2.100)

$$\displaystyle\begin{array}{rcl} \Gamma {\bigl (\frac{1} {2}+\epsilon \bigr )}& =& \sqrt{\pi }{\Bigl [1 -\epsilon (\gamma _{\mbox{ E}} + 2\ln 2) + \mathcal{O}(\epsilon ^{2})\Bigr ]}\;,{}\end{array}$$

(2.101)

$$\displaystyle\begin{array}{rcl} \Gamma {\bigl (\frac{3} {2}+\epsilon \bigr )}& =& \frac{\sqrt{\pi }} {2} + \mathcal{O}(\epsilon )\;.{}\end{array}$$

(2.102)

We have gone one order higher in the middle expansion, because this function is multiplied by 1∕ε in the result (cf. Eq. (2.112)).

$\fbox{\mbox{$\displaystyle \zeta(s)$}}$

The function ζ(s) is also to be viewed as a complex-valued function of a complex argument s. For $\mathop{\mathrm{Re}}(s) > 1$, it can be defined as

$$\displaystyle{ \zeta (s) =\sum _{ n=1}^{\infty }n^{-s} = \frac{1} {\Gamma (s)}\int _{0}^{\infty }\!\frac{\mathrm{d}x\,x^{s-1}} {e^{x} - 1} \;, }$$

(2.103)

where the equivalence of the two forms can be seen by writing $1/(e^{x} - 1) = e^{-x}/(1 - e^{-x}) =\sum _{ n=1}^{\infty }e^{-nx}$, and using the definition of the $\Gamma$-function in Eq. (2.96). Some remarkable properties of ζ(s) follow from the fact that by writing

$$\displaystyle{ \frac{1} {e^{x} - 1} = \frac{1} {(e^{x/2} - 1)(e^{x/2} + 1)} = \frac{1} {2}{\biggl [ \frac{1} {e^{x/2} - 1} - \frac{1} {e^{x/2} + 1}\biggr ]}\;, }$$

(2.104)

and then substituting integration variables through x → 2x, we can find an alternative integral representation,

$$\displaystyle{ \zeta (s) = \frac{1} {(1 - 2^{1-s})\Gamma (s)}\int _{0}^{\infty }\!\frac{\mathrm{d}x\,x^{s-1}} {e^{x} + 1} \;, }$$

(2.105)

defined for $\mathop{\mathrm{Re}}(s) > 0$, s ≠ 1. Even though the integral here clearly diverges at s → 0, the function $\Gamma (s)$ also diverges at the same point, making ζ(s) regular around origin:

$$\displaystyle\begin{array}{rcl} \zeta (0)& =& -\frac{1} {2}\;,{}\end{array}$$

(2.106)

$$\displaystyle\begin{array}{rcl} \zeta '(0)& =& -\frac{1} {2}\ln (2\pi )\;.{}\end{array}$$

(2.107)

Finally, for $\mathop{\mathrm{Re}}(s) \leq 0$, an analytic continuation is obtained through the relation

$$\displaystyle{ \zeta (s) = 2^{s}\pi ^{s-1}\sin {\Bigl ( \frac{\pi s} {2}\Bigr )}\Gamma (1 - s)\zeta (1 - s)\;. }$$

(2.108)

On the real axis, ζ(s) has a pole only at s = 1. Its values at even arguments are “easy”; in fact, at even negative integers, Eq. (2.108) implies that

$$\displaystyle{ \zeta (-2n) = 0\;,\quad n = 1,2,3,\ldots \;, }$$

(2.109)

whereas at positive even integers the values can be related to the Bernoulli numbers,

$$\displaystyle{ \zeta (2) = \frac{\pi ^{2}} {6}\;,\quad \zeta (4) = \frac{\pi ^{4}} {90}\;,\ldots \;. }$$

(2.110)

Negative odd integers can be related to positive even ones through Eq. (2.108), which also allows us to determine the behaviour of the function around the pole at s = 1. In contrast, odd positive integers larger than unity, i.e. s = 3, 5, …, yield new transcendental numbers.

The values required in Eq. (2.91) become

$$\displaystyle\begin{array}{rcl} \zeta (-1 + 2\epsilon )& =& -\frac{1} {2\pi ^{2}}\Gamma (2)\zeta (2) + \mathcal{O}(\epsilon ) = -\frac{1} {12} + \mathcal{O}(\epsilon )\;,{}\end{array}$$

(2.111)

$$\displaystyle\begin{array}{rcl} \zeta (1 + 2\epsilon )& =& 2^{1+2\epsilon }\pi ^{2\epsilon }{\Bigl [\sin {\Bigl ( \frac{\pi } {2}\Bigr )} +\pi \epsilon \cos {\Bigl ( \frac{\pi } {2}\Bigr )}\Bigr ]}{\biggl ( -\frac{1} {2\epsilon }\biggr )}\Gamma (1 - 2\epsilon )\zeta (-2\epsilon ) \\ & =& 2(1 + 2\epsilon \ln 2)(1 + 2\epsilon \ln \pi ){\biggl ( -\frac{1} {2\epsilon }\biggr )}(1 + 2\epsilon \gamma _{\mbox{ E}}){\bigl ( -\frac{1} {2}\bigr )}(1 - 2\epsilon \ln 2\pi ) \\ & & +\mathcal{O}(\epsilon ) \\ & =& \frac{1} {2\epsilon } +\gamma _{\mbox{ E}} + \mathcal{O}(\epsilon )\;, {}\end{array}$$

(2.112)

$$\displaystyle\begin{array}{rcl} \zeta (3 + 2\epsilon )& =& \zeta (3) + \mathcal{O}(\epsilon ) \approx 1.2020569031\ldots + \mathcal{O}(\epsilon )\;,{}\end{array}$$

(2.113)

where in the first two cases we made use of Eq. (2.108), and in the second also of Eqs. (2.106) and (2.107).

Appendix B: Numerical Convergence

We complete here the derivation of Eq. (2.91), and sketch the regimes where the low and high-temperature expansions are numerically accurate by inspecting J _T (m) from Eq. (2.82).

First of all, for the term l = 0 in Eq. (2.90), we make use of the results of Eqs. (2.100) and (2.111):

$$\displaystyle{ \left.I'(m,T)\right \vert _{l=0} = \frac{2T} {(4\pi )^{3/2}}(2\pi T)\frac{-2\sqrt{\pi }} {1} {\biggl ( - \frac{1} {12}\biggr )} + \mathcal{O}(\epsilon ) = \frac{T^{2}} {12} + \mathcal{O}(\epsilon )\;. }$$

(2.114)

For the term l = 1, we on the other hand insert the values of Eqs. (2.101) and (2.112):

$$\displaystyle\begin{array}{rcl} \left.I'(m,T)\right \vert _{l=1}& = & 2T \frac{(4\pi )^{\epsilon }} {(4\pi )^{3/2}}(2\pi T)^{1-2\epsilon }{\biggl [ \frac{-m^{2}} {(2\pi T)^{2}}\biggr ]} \\ & & \times \sqrt{\pi }{\Bigl [1 -\epsilon (\gamma _{\mbox{ E}} + 2\ln 2)\Bigr ]}\frac{1} {2\epsilon }(1 + 2\epsilon \gamma _{\mbox{ E}}) + \mathcal{O}(\epsilon ) \\ & \stackrel{1 =\mu ^{-2\epsilon }\mu ^{2\epsilon }}{=}& -\frac{m^{2}\mu ^{-2\epsilon }} {(4\pi )^{2}} {\biggl \{\frac{1} {\epsilon } +\ln \frac{\mu ^{2}} {T^{2}} +\ln (4\pi ) -\gamma _{\mbox{ E}} + 2[\gamma _{\mbox{ E}} -\ln (4\pi )]\biggr \}} \\ & & +\mathcal{O}(\epsilon ) \\ & \stackrel{\mbox{ (2.71)}}{=} & -\frac{m^{2}\mu ^{-2\epsilon }} {(4\pi )^{2}} {\biggl \{\frac{1} {\epsilon } +\ln \frac{\bar{\mu }^{2}} {T^{2}} + 2\ln {\Bigl (\frac{e^{\gamma _{\mbox{ E}}}} {4\pi } \Bigr )}\biggr \}} + \mathcal{O}(\epsilon )\;. {}\end{array}$$

(2.115)

Finally, for the term l = 2, we make use of Eqs. (2.102) and (2.113), giving:

$$\displaystyle{ \left.I'(m,T)\right \vert _{l=2} = \frac{2T} {(4\pi )^{3/2}}(2\pi T) \frac{m^{4}} {(2\pi T)^{4}} \frac{\frac{1} {2}\sqrt{\pi }} {2} \zeta (3) + \mathcal{O}(\epsilon ) = \frac{2m^{4}\zeta (3)} {(4\pi )^{4}T^{2}} + \mathcal{O}(\epsilon )\;. }$$

(2.116)

For the numerical evaluation of J _T (m), we again denote y ≡ m∕T and inspect the function

$$\displaystyle{ \mathcal{J} (y) \equiv \frac{J_{T}^{}(m)} {T^{4}} = \frac{1} {2\pi ^{2}}\int _{0}^{\infty }\!\mathrm{d}x\,x^{2}\,\ln {\Bigl (1 - e^{-\sqrt{x^{2 } +y^{2}} }\Bigr )}\;. }$$

(2.117)

We contrast this with the low-temperature result from Eq. (2.79),

$$\displaystyle{ \mathcal{J} (y)\stackrel{y \gg 1}{\approx }-{\biggl (\frac{y} {2\pi }\biggr )}^{\frac{3} {2} }e^{-y}\;, }$$

(2.118)

as well as with the high-temperature expansion from Eq. (2.82),

$$\displaystyle{ \mathcal{J} (y)\stackrel{y \ll 1}{\approx } = - \frac{\pi ^{2}} {90} + \frac{y^{2}} {24} -\frac{y^{3}} {12\pi } - \frac{y^{4}} {2(4\pi )^{2}}{\biggl [\ln {\biggl (\frac{ye^{\gamma _{\mbox{ E}}}} {4\pi } \biggr )} -\frac{3} {4}\biggr ]} + \frac{y^{6}\zeta (3)} {3(4\pi )^{4}}\;. }$$

(2.119)

The result of the comparison is shown in Fig. 2.1. We observe that if we keep terms up to y ⁶ in the high-temperature expansion, its numerical convergence is good for $y\mathop{\mbox{ $ <$ $\sim $}}3$. On the other hand, the low-temperature expansion converges reasonably well as soon as $y\mathop{\mbox{ $ >$ $\sim $}}6$. In between, a numerical evaluation of the integral is clearly necessary. It should be stressed that these statements are to be understood in a pragmatic sense, rather than as mathematically defined convergence radii.