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I am reading A. Zee "Quantum Field Theory in a Nutshell" and I have solved the problem about inverse square law in $D$ dimensions. Unfortunately, I have been confused with some results. Let me desribe my derivations briefly and focuse on results. The energy of interaction has the following form: $$E(r)=-\int\frac{d^Dk}{(2\pi)^D}\frac{e^{i{\bf k}\cdot{\bf r}}}{k^2+m^2}\equiv -W(r),$$ where $W(r)$ can be calculated using Schwinger parametrization (see Wiki) with the parameter $A=k^2+m^2$. Then, I have obtained the result: $$W(r)=\frac{1}{(2\pi)^{D/2}}\left(\frac{m}{r}\right)^{D/2-1}K_{D/2-1}(mr),$$ where $K_{\nu}$ is the Bessel functions 2nd kind. The result demonstrates correct answer for massive force carrier ($m\neq 0$) in 3D but I don't understand how to obtain $W(r)$ in 2D case with massive carrier because $K_0(mr)\neq \ln(mr)$. Moreover, my calculation falls down in case of massless carrier ($m=0$), it is easy to see it. Can anybody explain how to evaluate correct answers for massless carrier from my calculation?

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    $\begingroup$ $(m/r)^{D/2-1}=1+\frac\epsilon2\log(m/r)$, with $d=2+\epsilon$. $\endgroup$ Mar 6, 2018 at 21:47
  • $\begingroup$ like the dimensional regularization? $\endgroup$ Mar 6, 2018 at 22:32
  • $\begingroup$ OK, let me try this. I expand $K_{\nu}$ and $(m/r)$ into series. Thus, I obtain: $$K_{0}(mr)-K_{0}(mr)\log\left(\frac{m}{r}\right)\epsilon+... $$ And...? It is not easy for me to see $\log(r)$ law for massless carrier. $\endgroup$ Mar 7, 2018 at 7:14
  • $\begingroup$ You should see it in Classical field theory by Mark Burgess. He has given these solutions. $\endgroup$ Mar 8, 2018 at 16:31
  • $\begingroup$ Dear Zohaib, can You be more specifif? Classical Covariant Fields by M. Burgess? $\endgroup$ Mar 8, 2018 at 17:26

2 Answers 2

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Thank You, AccidentalForierTransform & Sean E. Lake!

(1) To obtain the correct answer for massless carrier one can use Schwinger parametrization and obtain the following expression: $$E(r)=-\frac{2^{D/2-1}}{r^{D-2}}\Gamma\left(\frac{D}{2}-1\right)\frac{1}{2(2\pi)^{D/2}}.$$ (2) Unfortunately, both cases (massive and massless carriers) have "bad behavior" for $D=2$. The gamma function has the pole at $z=0$. To deal with it, one can use dimensional regularization: replace $D\rightarrow D+2\epsilon$. Thus, the integration measure is to change: $$\frac{d^{D}k}{(2\pi)^D}\rightarrow \frac{d^{D+2\epsilon}k}{(2\pi)^{D+2\epsilon}},$$ but with this replacement, one should correct the dimensionality and regularization parameter $\mu$. Finally, the measure has the following form: $$\frac{d^{D+2\epsilon}k}{(2\pi)^{D+2\epsilon}}\mu^{-2\epsilon}.$$ This regularization provides the physically correct answer. The gamma function should be expanded into series: $$\Gamma(\epsilon)\approx\frac{1}{\epsilon}-\gamma.$$ And the fraction $(1/(\mu r))^{\epsilon}$ should be expanded too: $$\left({\mu r}\right)^{-\epsilon}\approx 1 - \ln (\mu r)\epsilon.$$

Considering all the above, the answer is $$E(r)=\frac{1}{2\pi}\ln(\mu r),$$ which has the correct dimensionality (in contrast to $-\ln r/(2\pi)$ which is "unphysical" due to the logarithm of length).

Comments:

  1. dimensional regularization does not change the singularity character of the gamma function for $D=2$ because the expansion contains the pole at 0.
  2. the Schwinger parametrization is very convinient way to calculate propagator-type because it allows to avoid the charade with hyperpsherical coordinates
  3. of course, these tricks are easy for good physicists but I have not found any explanations and solutions for this problem
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  • $\begingroup$ See also: limiting forms of Bessel functions in standard texts. e.g. dlmf.nist.gov/10.30 $\endgroup$ Mar 11, 2018 at 19:54
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I'm also reading Zee's book. When attempting this question, I took a shortcut and considered the massless case $(m=0)$ from the get go. Then, I noticed that $E(r)$ is reduced to the Green's function for the $D$-dimensional Laplace equation. It is well known, or by Gauss' Law, one can find that $\nabla E(r) = \frac{1}{S_{D-1}} \propto 1/r^{D-1}$, where $S_{D-1}$ is the surface area of a $D$-dimensional sphere. Thus, $E(r)\propto 1/r^{D-2}$. In the case of $D=2$, $\nabla E(r) \propto 1/r \implies E(r) \propto \ln(r)$.

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  • $\begingroup$ If you would like to see calculations for D-dimensional case, notice me $\endgroup$ Aug 20, 2019 at 20:06
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    $\begingroup$ Additionly, your answer isn't correct completely. Indeed, you calculate log function of argument with length dimensionality. The argument of log functions should be dimensionless $\endgroup$ Aug 20, 2019 at 20:07
  • $\begingroup$ Ah you're right! thanks for pointing that out! $\endgroup$
    – bygolly
    Aug 21, 2019 at 22:30

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