Loop Integration in Spinor Language

[earth2]

Hi guys,

i'm looking at one-loop calculations in terms of helicity spinor (basically a paper by Brandhuber, Travglini and others) language but i have no idea how to integrate them :)

For instance

<br /> \int FeynParam\int d^D L \frac{\langle a|L|b]^2}{(L^2-\Delta^2)^3}<br />

How would I do the loop intergation here?

Cheers,
earth2

 

[fzero]

If these are standard loop integrals, then most QFT texts will discuss their treatment via dimensional reduction. For example, following Ramond's discussion, we can use beta functions to derive

\int \frac{d^N\ell}{(\ell^2 + a^2)^A }= \pi^{N/2} \frac{\Gamma(A-N/2)}{\Gamma(A)} \frac{1}{(a^2)^{A-N/2}}

By shifting \ell = \ell&#039; + p and setting b^2 = a^2 + p^2, we find

\int \frac{d^N\ell}{(\ell^2 +2p\cdot\ell + b^2)^A }= \pi^{N/2} \frac{\Gamma(A-N/2)}{\Gamma(A)} \frac{1}{(b^2-p^2)^{A-N/2}}.

Integrals with factors of \ell_\mu can now be obtained by differentiating with respect to p_\mu:


\int \frac{d^N\ell~\ell_{\mu_1}\cdots \ell_{\mu_n}}{(\ell^2 +2p\cdot\ell + b^2)^A }=\pi^{N/2} \frac{\Gamma(A-N/2)}{\Gamma(A)} \left( \frac{1}{2} \frac{\partial}{\partial p_{\mu_1}} \right) \cdots \left( \frac{1}{2} \frac{\partial}{\partial p_{\mu_n}} \right) \frac{1}{(b^2-p^2)^{A-N/2}}.

 

[earth2]

Hi! Thanks for your reply!

I understand how to treat these integrals if the numerator of the integrand is expressed in terms of four-vectors. But how do I proceed if the numerator is written in the spinor bra-ket language above? I don't really know how to handle these expressions if the loop momentum in the numerator is written via spinors...Any idea how to handle them? (So, my question really is: what do i do with the numerator?)

Cheers,
earth2

 

[fzero]

Can you give a more specific reference? I've been looking at papers like hep-th/0407214 and I haven't found an expression where the loop momentum wasn't given as a 4-vector.

 

[earth2]

Hi and thanks for your reply. Look for instance at hep-th/0612007. They never do the integrals (they reduce them to scalar integrals using PV) but i was wondering how to do them without reducing them.

Look for instance at eq (3.9). If one is given such a type of integral but has no idea about PV reduction and only knows numerators written via 4-momenta how does one integrate this thing directly? Or even eq 3.10. (which looks like what I've written above).

As I've said I know how to treat loop integrals in terms of the standard Peskin/Schroeder textbook way...I just don't really know how to deal with numerators if they are given in terms of these spinor brackets. :)

Cheers,
earth2

 

[fzero]

These are ordinary loop integrals, but you have to look at them the right way. You need to use the definition of the inner products for spinors of \pm helicity as well as the use of sigma matrices to write a 4-vector as a bispinor.

Take (3.9), which has a [ \eta |L_3| 3 \rangle in it. We can write this in terms of (L_3)_\mu by putting all of the indices in:

[ \eta |L_3| 3 \rangle = - \eta_{\dot{b}}\epsilon^{\dot{b}\dot{a}} (L_3)_\mu (\sigma^\mu)_{a\dot{a}} \epsilon^{ab} (3)_b.

I might have a minus sign wrong, you might want to check yourself using the conventions from the Witten paper or something. Up to signs then, we identify

- \eta_{\dot{b}}\epsilon^{\dot{b}\dot{a}} (\sigma^\mu)_{a\dot{a}} \epsilon^{ab} (3)_b = [ \eta |\mu| 3 \rangle.

as appears in (3.11).

 

[earth2]

Ah, thanks for the explanation. One more question about this:

So coming back to the numerator, i could rewrite is in terms of a four-vector product as:

\langle a |L|b]^2 = (2q\cdot L )^2

where q is a four-vector build from the spinors \langle a| and |b].

Under the integral sign i could write this as

\langle a |L|b]^2 = (2*q\cdot L )^2=4q^\alpha q^\beta L_\alpha L_\beta=\frac{4}{D} q^\alpha q^\beta g_{\alpha\beta}L^2 and after integration the result would be proportional to
q^2 times stuff coming from the integrak. But this would vanish since q can by expressed in terms of spinors and must so be massless, right? I.e. q^2=0? Or is this reasoning too naive?

Cheers

 

[fzero]

That looks about right. The integral in (4.19) is an example that leads to a vanishing contraction.