A Wilson Line Propagator: Understanding Eqtn 5.7 & 5.8

  • A
  • Thread starter Thread starter Elmo
  • Start date Start date
  • Tags Tags
    Line Propagator
Elmo
Messages
37
Reaction score
6
TL;DR Summary
An undefined term in the wilson line propagator
I asked this from a number of people but no one knew what to do about this exponential with iota infinity in the power,in Eqtn 5.7
The textbook seems to imply that it is zero but cos and sine are undefined at infinity.
Also,all the exponentials seem to vanish from the final result of Eqtn 5.8 whereas the integration of exponential function should still leave behind the function.
Where do all the exponentials go ?
Screenshot (1).png
 
Physics news on Phys.org
I think the author intoduces a small variable ##\epsilon > 0## so to write the integral in 5.7 as:

$$\lim_{\epsilon \rightarrow 0^+} \int_{-\infty}^{\lambda_2} d\lambda e^{-i(nk + i \epsilon)\lambda}$$

In this way you will have your original exponential multiplied by ##e^{\epsilon \lambda}##. This will assure the convergence of the function as ##\lambda## goes to ##-\infty## because it will exponentially go to zero.
The result of the integral is to be understood with a ##\lim_{\epsilon \rightarrow 0^+}## in front.
 
dRic2 said:
I think the author intoduces a small variable ##\epsilon > 0## so to write the integral in 5.7 as:

$$\lim_{\epsilon \rightarrow 0^+} \int_{-\infty}^{\lambda_2} d\lambda e^{-i(nk + i \epsilon)\lambda}$$

In this way you will have your original exponential multiplied by ##e^{\epsilon \lambda}##. This will assure the convergence of the function as ##\lambda## goes to ##-\infty## because it will exponentially go to zero.
The result of the integral is to be understood with a ##\lim_{\epsilon \rightarrow 0^+}## in front.
Seems good though it introduces another undefined term as the lower limit, in the form of
$$e^{-n\epsilon\infty} $$
And as ##\epsilon \rightarrow 0## is being multiplied with infinity ,should that not again be undefined ?
Also,my other question remains that what happens to the exponentials in the upper limit ?
Why don't they show up in the final expression or the list of Feynman rules.
 
Elmo said:
should that not again be undefined
No. You take the ##\lim_{\epsilon->0^+}## at the end of all the calculations. So, first, you do the definite integral (that is first you take the limit as ##\lambda \rightarrow - \infty## which annihilates the exponential).

As for your second question, I don't see it immediately. If I have time I'll check more carefully. Sorry
 
Elmo said:
Also,my other question remains that what happens to the exponentials in the upper limit ?
You are performing the integrals in (5.6) one after the other. The explicit example is for the innermost one (over ##\lambda_1##). Have you verified the claim about the ##\lambda_2## integral? The upper limit in the last integral (the leftmost integral symbol) is 0, giving you ##e^0 = 1##.
 
Toponium is a hadron which is the bound state of a valance top quark and a valance antitop quark. Oversimplified presentations often state that top quarks don't form hadrons, because they decay to bottom quarks extremely rapidly after they are created, leaving no time to form a hadron. And, the vast majority of the time, this is true. But, the lifetime of a top quark is only an average lifetime. Sometimes it decays faster and sometimes it decays slower. In the highly improbable case that...
I'm following this paper by Kitaev on SL(2,R) representations and I'm having a problem in the normalization of the continuous eigenfunctions (eqs. (67)-(70)), which satisfy \langle f_s | f_{s'} \rangle = \int_{0}^{1} \frac{2}{(1-u)^2} f_s(u)^* f_{s'}(u) \, du. \tag{67} The singular contribution of the integral arises at the endpoint u=1 of the integral, and in the limit u \to 1, the function f_s(u) takes on the form f_s(u) \approx a_s (1-u)^{1/2 + i s} + a_s^* (1-u)^{1/2 - i s}. \tag{70}...
Back
Top