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After reading part of this paper by Nima Arkani-Hamed, http://arxiv.org/abs/1212.5605, I cannot understand what is the precise meaning of the $d\log(\alpha)$ integrations.

Any on-shell diagram is associated with a "differential form" (see 6.1 of the above paper):

$$f = \int \prod_{i=1}^{dim} \frac{d\alpha_i}{\alpha_i } \delta^{k \times 4}(C . \tilde{\eta}) \delta^{k \times 2}(C . \tilde{\lambda}) \delta^{n - k \times 2}(\lambda . C^\dagger).\tag{6.1}$$

Now, these $d\alpha$ integrations, how are to be understood? As integral over a complex variable (i.e. as an integral over $\mathbb{R^2}$), as a path integral of a complex variable over a not specified contour in $\mathbb{C}$? As an integral over $\mathbb{R}$?

It seems to me that all of the above intepretations are used in the paper: the first time these on-shell diagrams are given an integral value the integrations is over (complex?) matrices, so these should integral over several complex planes. Moreover if one thinks of these variable arising as a BCFW shift they should be complex.

Later on (pag. 87) it is explicitly stated that the integral is a contour integral so that in the case there is only one free integration (just a single variable left after the delta's costraints have been resolved) it yields zero. I imagine that it is meant therefore that the contour encircle all the residues an there is no pole at infinity.

Finally, at a certain point (in the middle of chapter 5) these variables are explicitly stated to be real, moreover positive. (So that the matrix C appearing in the delta functions lives in the positive grassmannian).

BONUS QUESTION: why these integrated expressions are called "differential forms"? It is meant that the integrand is the differential form and one still calls the result of its integration "differential form"?

Qmechanic
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giulio bullsaver
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1 Answers1

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First, the meaning of ${\rm d}\log x$ is nothing else than ${\rm d}x /x$, just like in the elementary calculus: just the meaning of $x$ may be complicated.

Second, none of the integrals in these papers is ever an integral over a two-dimensional complex plane. Such 2-dimensional integrals would be written as $\int d^2z\, f(z,\bar z)$ or $\int dz\,d\bar z\,f(z,\bar z)$ which doesn't occur anywhere in the papers.

Third, the actual integration of the $p$-forms (completely antisymmetric tensors with $p$ indices) may occur along any contour you like or find relevant.

Fourth, the integration needed to get the physical amplitude is evaluated by integrating over any contour which restricts the loop momenta to be real, i.e. belonging to the Minkowski space ${\mathbb R}^{3,1}$. It means that various variables are real and/or obey other reality conditions, perhaps quadratic ones.

Fifth, both the integrands and the partial integrals are differential forms in the space of the variables describing both the internal and external momenta of their diagrams. One may integrate over the variables encoding the internal momenta but even after that, the object remains a differential form in the variables encoding the external momenta (which may be finally integrated along contours described in the previous paragraph).

Sixth, all these integrands are nontrivial differential forms ($p$-forms with $p$ which is neither zero nor the dimension of the space but something in between) because they are products of complicated delta-functions in a multi-dimensional space and a delta-function is naturally a differential form. Consider ordinary real 3-dimensional space given by $x^i,i=1,2,3$. The function $\delta(\sum_i k_i x^i)$ is a 1-form because it's the derivative of the step function $\theta(\sum_i k_i x^i)$ with respect to the argument. But the argument is a combination of all $x^i$ so the delta-function is a combination of the gradients, partial derivatives with respect to $x^i$, weighted by $k_i$. If integrated over a 1-line, the integral of this delta-function gives one, a typical behavior of the integrals of general differential forms.

Luboš Motl
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