I am in trouble… I am a master student (first year) in theoretical particle physics and I have been working for several months on the calculation of the total cross-section at leading order for the process:(adsbygoogle = window.adsbygoogle || []).push({});

u+ [itex]\overline{d}[/itex] [itex]\rightarrow[/itex]W[itex]^{+}[/itex] + [itex]\gamma[/itex]

I performed the whole calculation by hand and a couple of weeks ago I obtained my final answer. However, my professor has actually proven that one term in my total matrix element squared cannot be correct. After spending weeks on checking this term, I really cannot find the mistake. My question is whether someone would like to check my calculation or do this calculation by computer software e.g. Mathematica or Form. (I do not have much experience with calculating such quantities by a computer, neither do I have time to figure out how that could be done. That’s why I chose to do everything by hand.)

I have attached a pdf-file containing all the relevant information. Below I will sometimes refer to a certain page of this file.

The problem is as follows:

There are three tree level Feynman diagrams for this process: M1, M2 and M3 (see page 1). In order to calculate the cross-section, one first needs to calculate the total matrix element squared: Mtot[itex]^{2}[/itex]. I performed this calculation piece by piece, i.e. I calculated M1[itex]\cdot[/itex]M1*, M2[itex]\cdot[/itex]M2*, M3[itex]\cdot[/itex]M3*, M1[itex]\cdot[/itex]M2*, M1[itex]\cdot[/itex]M3* and M2[itex]\cdot[/itex]M3*. (All by hand, which was rather tedious…)

Page 9 shows my obtained Mtot[itex]^{2}[/itex]. According to my professor all the terms seem reasonable and could, in principle, be correct. Except for one term! M3[itex]\cdot[/itex]M3*, namely, has a different energy behavior. All the other terms (M1[itex]\cdot[/itex]M1*, M2[itex]\cdot[/itex]M2*, M1[itex]\cdot[/itex]M2*, M1[itex]\cdot[/itex]M3* and M2[itex]\cdot[/itex]M3*) approach a constant as E → ∞, whereas M3[itex]\cdot[/itex]M3* contains a term that goes with E[itex]^{2}[/itex] for large E. My professor has pointed out (and also proven) that this energy behavior cannot be correct and that M3[itex]\cdot[/itex]M3* should also approach a constant for E → ∞. (This by the way implies that the cross-section must approach zero for E → ∞.)

A very important thing is that I have chosen to work in a specific gauge! When you sum over photon polarization vectors, some gauge vector, r, enters the calculation (see page 4). By making the following clever choice for the gauge vector: r = (0,1,i,0), a lot of terms drop out during the calculation. Of course, once you have made this explicit choice for the gauge vector you will have to stick it!

Would someone be willing to either check/calculate M3[itex]\cdot[/itex]M3*? Perhaps someone has already a computer code for calculating such quantities... In principle I am only interested in the answer and not necessarily in the calculation. Again, it is very important that the calculation of M3[itex]\cdot[/itex]M3* is performed in the gauge that I have chosen (see page 4), as all the other partial matrix elements were also calculated in this specific gauge. I only need a correct analytic expression for M3[itex]\cdot[/itex]M3*, so an expression for the total cross-section, for instance, is useless for me. I did not "TeX" my whole calculation: the vertical dots on page 8 represent the hand-written part, which I unfortunately cannot attach because of the size.

I realize that I ask a lot… In order to increase your motivation for this check/calculation, I award the person that provides me with a satisfying analytical expression for M3[itex]\cdot[/itex]M3* with EUR 20, besides eternal glory, of course! :) Not being able to finish this calculation is really frustrating and therefore your help would be greatly appreciated. If you have any questions, please feel free to ask them. Or if you need (or are interested) in the hand-written part of M3[itex]\cdot[/itex]M3*, please mail me:tvdaal(at)science.ru.nl.

Thanks!

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# Matrix element squared for up + antidown -> photon + W

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