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Loop Integration in Spinor Language

  1. May 2, 2012 #1
    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

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

    How would I do the loop intergation here?

    Cheers,
    earth2
     
  2. jcsd
  3. May 2, 2012 #2

    fzero

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    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

    [tex] \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}}[/tex]

    By shifting [itex]\ell = \ell' + p[/itex] and setting [itex]b^2 = a^2 + p^2[/itex], we find

    [tex] \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}}.[/tex]

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


    [tex] \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}}.[/tex]
     
  4. May 2, 2012 #3
    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
     
  5. May 2, 2012 #4

    fzero

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    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.
     
  6. May 3, 2012 #5
    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
     
  7. May 3, 2012 #6

    fzero

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    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 [itex]\pm[/itex] helicity as well as the use of sigma matrices to write a 4-vector as a bispinor.

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

    [tex] [ \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.[/tex]

    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

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

    as appears in (3.11).
     
  8. May 3, 2012 #7
    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:

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

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

    Under the integral sign i could write this as

    [tex]\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[/tex] and after integration the result would be proportional to
    [tex] q^2 [/tex] 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
     
  9. May 3, 2012 #8

    fzero

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