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

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

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 $\ell = \ell' + p$ 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.