Solving Integral with Logarithm Identity

[parton]

I read the following expression in a book:

\int_{-\infty}^{\infty} \dfrac{1}{t(1-t)} \log \left| \dfrac{t^{2} q^{2}}{(p-tq)^{2}} \right| ~ dt = - \pi^{2}

p and q are both timelike four-vectors, so p², q² > 0

This integral was solved by using the identity

\lim_{s \to \infty} \int_{-s}^{s} \dfrac{\log \vert t-a \vert}{t-b} ~ dt = \dfrac{\pi^{2}}{2} sign(b - a)

But I don't know how I can apply this identity to the integral above. I don't find any substitution or anything else to 'convert' the integral
in such a shape where I can use these formula.

Could anyone help me please?

 

[tiny-tim]

Hi parton!

Try splitting the 1/t(1-t) in particle fractions, and the log into the difference of two logs, and then apply the identity several times.

 

[parton]

Hi tiny-tim ! Thanks for your answer

I tried to split the integral as you suggested:

\int_{-\infty}^{\infty} \dfrac{1}{t(1-t)} \log \left| \dfrac{t^{2} q^{2}}{(p-tq)^{2}} \right| ~ dt = \int_{-\infty}^{\infty} \left[ \dfrac{1}{t} - \dfrac{1}{t-1} \right] \left[ \log \vert t^{2} q^{2} \vert - \log \vert (p-tq)^{2} \vert \right] ~ dt

For the first part, I've got:

\int_{-\infty}^{\infty} \left[ \dfrac{1}{t} - \dfrac{1}{t-1} \right] \log \vert t^{2} q^{2} \vert \right] ~ dt= - \pi^{2}

Here, I decomposed the logarithm again, i.e.

\log \vert t^{2} q^{2} \vert^{2} = 2 \log \vert t \vert + \log \vert q^{2} \vert

I just used the identity for computing
- 2 \int_{-\infty}^{\infty} \dfrac{1}{t-1} \log \vert t \vert = - \pi^{2}

The other integrals coming from this part will vanish, for example:

2 \int_{-\infty}^{\infty} \dfrac{1}{t} \log \vert t \vert ~ dt = \left[ \log \left( \vert t \vert \right) ^{2} \right]_{-\infty}^{\infty} = 0

I argued that: \left[ \log \left( \vert t \vert \right) ^{2} \right]_{-s}^{s} = 0, so it has also to be valid for s--> infinity. But I'm not sure wheter this argument is correct.

Nevertheless, I don't know how to compute the second part of the originial integral:

- \int_{-\infty}^{\infty} \left[ \dfrac{1}{t} - \dfrac{1}{t-1} \right] \log \vert (p-tq)^{2} \vert ~ dt

There is this nasty square appearing in the logarithm, and now I don't know how to continue

To you know some way to solve this?

 

[Hurkyl]

There is this nasty square appearing in the logarithm, and now I don't know how to continue

Don't you know something about log x^e?

 

[parton]

Don't you know something about log xe?

ok, maybe I could simply write:

-\int_{-\infty}^{\infty} \left[ \dfrac{1}{t} - \dfrac{1}{t-1} \right] \log \vert (p-tq)^{2} \vert ~ dt = -2 \int_{-\infty}^{\infty} \left[ \dfrac{1}{t} - \dfrac{1}{t-1} \right] \log \vert p -t q \vert ~ dt

But how do I solve that? p and q are four-vectors, so I can't simply apply a substitution !?

 

[tiny-tim]

There is this nasty square appearing in the logarithm, and now I don't know how to continue

To you know some way to solve this?

Hi parton!

The difficulty is that p and q are 4-vectors.

But (p - tq)² is a quadratic in t (with coefficients which are combinations of the coordinates of p and q), so it must be of the form (t - a)(t - b), where (p - aq)² = (p - bq)² = 0.

Does that help?

 

[parton]

Yes, it does help, thanks a lot

So, I wrote:

(p-tq)^{2} = (t-t_{1}) (t-t_{2})

where t_{1,2} = \dfrac{1}{q^{2}} \left[ pq \pm \sqrt{ (pq)^{2} - p^{2} q^{2} } \right]

And I've got:

- \int_{-\infty}^{\infty} \dfrac{1}{t} \left[ \log \vert t - t_{1} \vert + \log \vert t - t_{2} \vert \right] = - \dfrac{\pi^{2}}{2} \left[ sign(-t_{1}) + sign(-t_{2}) \right] = 0

This is of course equal to zero, because of the opposite signs of t_{1} and t_{2}.

But the next one isn't so easy:

+ \int_{-\infty}^{\infty} \dfrac{1}{t-1} \left[ \log \vert t - t_{1} \vert + \log \vert t - t_{2} \vert \right] = \dfrac{\pi^{2}}{2} \left[ sign(1-t_{1}) + sign(1-t_{2}) \right]

Ok, I know that if t_{2} < 0 \Rightarrow 1-t_{2} >0 \Rightarrow sign(1-t_{2}) = 1

But what is sign(1-t_{1}) ?

 

[parton]

sorry, I made a mistake.

The equation above is not equal to zero.

If I choose a frame where p = (p_{0}, \vec{0}) than I have:

t_{1,2} = \dfrac{p_{0}}{q^{2}} \left[ q^{0} \pm \vert \vec{q} \vert \right] < 0 because of q^{2} = q_{0}^{2} - \vec{q} \, ^{2} > 0 and q_{0} < 0.

So I obtain:

- \int_{-\infty}^{\infty} \dfrac{1}{t} \left[ \log \vert t - t_{1} \vert + \log \vert t - t_{2} \vert \right] ~ dt = - \dfrac{\pi^{2}}{2} \left[ sign(-t_{1}) + sign(-t_{2}) \right] = - \pi^{2}

and

+ \int_{-\infty}^{\infty} \dfrac{1}{t-1} \left[ \log \vert t - t_{1} \vert + \log \vert t - t_{2} \vert \right] ~ dt = \dfrac{\pi^{2}}{2} \left[ sign(1-t_{1}) + sign(1-t_{2}) \right] = \pi^{2}

and finally:

-\int_{-\infty}^{\infty} \left[ \dfrac{1}{t} - \dfrac{1}{t-1} \right] \log \vert (p-tq)^{2} \vert ~ dt = 0

Is anything wrong in my argumentation or is it correct?

 

[parton]

By the way,

does anyone know where I can find such identities like

\lim_{s \to \infty} \int_{-s}^{s} \dfrac{\log \vert t-a \vert}{t-b} ~ dt = \dfrac{\pi^{2}}{2} sign(b - a)

Is there any book where such formulas are "collected"? I looked in different tables of integrals, but couldn't find them.