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

[TopQuark38]

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:

u + $\overline{d}$ $\rightarrow$ W$^{+}$ + $\gamma$​

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$^{2}$. I performed this calculation piece by piece, i.e. I calculated M1$\cdot$M1*, M2$\cdot$M2*, M3$\cdot$M3*, M1$\cdot$M2*, M1$\cdot$M3* and M2$\cdot$M3*. (All by hand, which was rather tedious…)

Page 9 shows my obtained Mtot$^{2}$. According to my professor all the terms seem reasonable and could, in principle, be correct. Except for one term! M3$\cdot$M3*, namely, has a different energy behavior. All the other terms (M1$\cdot$M1*, M2$\cdot$M2*, M1$\cdot$M2*, M1$\cdot$M3* and M2$\cdot$M3*) approach a constant as E → ∞, whereas M3$\cdot$M3* contains a term that goes with E$^{2}$ for large E. My professor has pointed out (and also proven) that this energy behavior cannot be correct and that M3$\cdot$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$\cdot$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$\cdot$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$\cdot$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$\cdot$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$\cdot$M3*, please mail me: tvdaal(at)science.ru.nl .

Thanks!

 

[Bill_K]

I'm confused. I assume this calculation is supposed to be in the context of the Salam-Weinberg electroweak theory. How does what you've done differ from just a gauge theory with a massive gauge particle? Gauge theory of course normally requires the gauge particles to be massless. If you add a mass to the gauge particle by hand you'll surely violate unitarity. Which is what you're seeing. So what makes this electroweak theory?

Note that the E² dependence in M₃ arises from an extra factor of E²/M_W². That came from the k^μk^ν/M_W² term in the propagator, and appears only in M₃ because M₃ is the only one of the three diagrams that has a W propagator. In the full electroweak theory this E² dependence should appear here, only to be canceled by a similar term elsewhere. But I can't see what term you're missing that is supposed to cancel it.

 

[Hepth]

Since you're not working in the Unitary gauge, do you have to add back in the goldstone boson from the symmetry breaking(the one that was absorbed to give the W mass in the Unitary gauge?, into the W's longitudinal polarization). I believe that diagram will generate a counter term that exactly cancels what you're getting in the longitudinal part of the W.

There are a number of other ways of looking at it, but i think it has something to do with this.

 

[TopQuark38]

Thanks Hepth and Bill_K for replying!
You guys clearly know more about this kind of calculations than I do as I do not understand all your problems. But perhaps this helps: I forgot to mention that the effect of longitudinally polarized W bosons cancels because of the choice for the gauge vector (r) I made, which is what my professor showed... Therefore the energy behavior of M3^2 must be the same as for the other terms!

 

[Hepth]

Using feyncalc in mathematica, without thinking about splitting the polarizations I get:

$$\Sigma_\epsilon |M_3|^2 = \frac{8 g_w^2 Q_w^2 (E^4-4 E^2 M_W^2 - M_W^4)}{(M_W(M_W^2-E^2))^2}$$

Somehow the theta dependence falls out. I took the amplitude you wrote right under the feynman rules on page 3.

You might not be able to run it but you can at least look at it in mathematica.

I'm not sure if its right, its just what I got when I ran through it really fast.

 

[TopQuark38]

Hepth, thank you very much for your efforts! I really appreciate that.
I looked at your code and it appears that you missed a factor of E^2 in your reply. Because of this additional factor, the energy behavior of M3^2 is the same as what I obtained...
I might overlook things, but I do not see where the photon polarization sums come in and where the explicit choice for the gauge vector is made. I haven't had much time yet to look at your code, but I installed feyncalc and am able to run it now.
No theta dependence seems a bit strange... It in turn implies that the diff. cross-section does not depend on theta. But should the diff. cross-section not always be larger for angles near 0 or pi? Or is this something one only expects for the sum of the three diagrams to be true? Perhaps one can generally not make such claims about a single Feynman diagram in case more diagrams contribute...

 

[Hepth]

I had an error in the definition of the amplitude, (a minus; I had p3+p4, should be p3-p4)
.
Yeah, I also missed the $E_n^2$ when typing it in the previous response.

The polarization sums are in the M3CC2= line. First is photon, second massive vector.

Ill attach it, it now has theta dependence.

But this still blows up at En-> inf. So does some of your other products? Doesn't M1M2 have the same behavior?

 

[Bill_K]

TopQuark38, This is a gauge issue. I suggest you read Weinberg Sects 21.1 and 21.2, or some other place where they discuss ξ-gauge. In general the electroweak theory does require a Higgs scalar φ and a ghost field ω. And in general the propagator of the gauge field has a numerator η_μν - (1 - ξ)k_μk_ν /(k² + M²ξ). Thus it is well behaved at large k, but the price you pay for this is that there is an additional gauge-dependent pole at M√ξ. This pole must be canceled by something, and what cancels it is a corresponding pole in the ghost propagator.

ξ can take on any value from 0 to ∞. If you let ξ → ∞ (unitary gauge) the extra poles move off to infinity, and in this limit the ghost field is not needed. However in this limit the propagator once again has the wrong behavior at large k.

 

[TopQuark38]

Hepth, fortunately there is now a theta dependence! However, still the energy behavior cannot be correct...

The other partial matrix elements are:

$\begin{align*} \langle \mathscr{M}_1 \mathscr{M}_1^* \rangle =& - \frac{4 \,\hbar^2 g_{\mathrm{w}}^2 Q_{\mathrm{d}}^2}{9} \left[ \left( \frac{m_{\mathrm{w}}^2}{E^2 - m_{\mathrm{w}}^2} \right) \csc^2 \left( \frac{\theta}{2} \right) + 1 \right] \\ \langle \mathscr{M}_2 \mathscr{M}_2^* \rangle =& - \frac{4 \,\hbar^2 g_{\mathrm{w}}^2 Q_{\mathrm{u}}^2}{9} \left[ \left( \frac{m_{\mathrm{w}}^2}{E^2 - m_{\mathrm{w}}^2} \right) \sec^2 \left( \frac{\theta}{2} \right) + 1 \right] \\ \langle \mathscr{M}_1 \mathscr{M}_2^* \rangle =& \frac{8 \,\hbar^2 g_{\mathrm{w}}^2 Q_{\mathrm{d}} Q_{\mathrm{u}}}{9} \left[ \left( \frac{E^2 \left( E^2 + m_{\mathrm{w}}^2 \right)}{\left( E^2 - m_{\mathrm{w}}^2 \right) ^2} \right) \csc^2\theta \right] \\ \langle \mathscr{M}_1 \mathscr{M}_3^* \rangle =& \frac{2 \,\hbar^2 g_{\mathrm{w}}^2 Q_{\mathrm{d}} Q_{\mathrm{w}}}{9} \left[ \frac{E^4 - m_{\mathrm{w}}^4}{\left( E^2 - m_{\mathrm{w}}^2 \right) ^2} + \left( \frac{E^2 \left( E^2 + m_{\mathrm{w}}^2 \right) }{\left( E^2 - m_{\mathrm{w}}^2 \right) ^2} \right) \csc^2 \left( \frac{\theta}{2} \right) - \cos\theta \right] \\ \langle \mathscr{M}_2 \mathscr{M}_3^* \rangle =& - \frac{2 \,\hbar^2 g_{\mathrm{w}}^2 Q_{\mathrm{u}} Q_{\mathrm{w}}}{9} \left[ \frac{2 \,m_{\mathrm{w}}^2}{E^2 - m_{\mathrm{w}}^2} + \left( \frac{E^2 \left( E^2 + m_{\mathrm{w}}^2 \right) }{\left( E^2 - m_{\mathrm{w}}^2 \right) ^2} \right) \sec^2 \left( \frac{\theta}{2} \right) + \cos\theta \right] \end{align*}$

As you can see, all these products approach a constant for E →∞. Hence the cross-section goes to zero in this limit as

$\begin{align*} \sigma =& \,\frac{E^2 - m_{\mathrm{w}}^2}{32 \,\pi E^4} \int_{-1}^1 \langle |\mathscr{M}|^2 \rangle \,\mathrm{d} \cos\theta \,. \end{align*}$

From the code I doubt whether you use my (axial gauge) convention for the photon polarization sum...

$\begin{align*} \sum_{s} \epsilon^\mu_{(s)} \,\epsilon^{\nu\,*}_{(s)} = -g^{\mu\nu} + \frac{p^\mu r^\nu + r^\mu p^\nu}{p \cdot r} \,, \end{align*}$

with

$\begin{align*} r = \begin{pmatrix} 0 \\ 1 \\ i \\ 0 \end{pmatrix} \,, \end{align*}$

or do you? In this gauge, the energy behavior of M3^2 must be the same as for the other products.
Frankly, I don't understand the assignment phi = theta in your code. These angles are completely independent, aren't they?

@ Bill_K: thanks for helping. I don't have Weinberg unfortunately and the things you come up with are a bit beyond my knowledge. I consider myself to be a beginner in QFT. Anyway, I do not think I fully understand your point...

Thanks guys!

 

[Bill_K]

My point is that in some gauges, M does grow with E, so you should not assume this result is incorrect. IMHO you're overly involved with the algebra, and would benefit by backing off and reconsidering the reasoning behind what you're doing.

 

[TopQuark38]

@ Bill_K: You're right. I would definitely benefit from learning more about the physical interpretations of what I am actually doing. I understand that in some gauges M3^2 does indeed grow with E. However, in this case, being almost finished with this cross-section calculation, I am only interested in one particular gauge for which I know, for sure, that M3^2 should not grow with E ... Calculating M3^2 in other gauges would disable me to compare it with the other products that I calculated previously and, as a consequence, it would not be possible to obtain an expression for the cross-section.

 

[Hepth]

There, With your gauge I get:

$$-\frac{4 \text{gw}^2 \text{Qw}^2 \left(7 \text{En}^4+\left(\text{En}^2-\text{Mw}^2\right)^2 \cos (2 \theta )+2 \text{En}^2 \text{Mw}^2-\text{Mw}^4\right)}{\left(\text{En}^2-\text{Mw}^2\right)^2}$$

Which goes to
$$-4 \text{gw}^2 \text{Qw}^2 (\cos (2 \theta )+7)$$

as En -> inf.

So maybe you just did your calculation wrong? All I did was replace the photon sum with yours and defined "r" and the scalar products.

**note : It may take a while to do the amplitude squaring if you don't have a decent computer. A few minutes probably on most normal machines.

 

[TopQuark38]

@ Hepth: I carefully analyzed your code: I understand the structure and I did not find any mistakes, so I guess that the answer is right! I made a plot of the total cross-section as a function of the total energy (W boson mass is the threshold energy). Sigma goes to zero for E to infinity, as it should :) Thanks a lot!