- #1
JD_PM
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- TL;DR Summary
- I want to understand how to get Eq. 8.71a in Mandl & Shaw
I was studying how to compute an unpolarized cross-section (QFT Mandl & Shaw, second edition,https://ia800108.us.archive.org/32/items/FranzMandlGrahamShawQuantumFieldTheoryWiley2010/Franz%20Mandl%2C%20Graham%20Shaw-Quantum%20Field%20Theory-Wiley%20%282010%29.pdf) and came across the following expression
$$Z:=16 \{ 2(f_1p)(f_1p') -f_1^2(pp')+m^2[-4(pf_1)+4f_1^2]+m^2 [(pp')-4(f_1 p')] +4m^4 \} \tag 1$$
Where:
$$f_1:=p+k \tag 2$$
We're also given the on-shell and off-shell conditions:
$$p^2 = p'^2=m^2, \ k^2=k'^2=0, \ pk=p'k', \ pk'=p'k \tag 3$$
Plugging ##(2)## into ##(1)## I get (I include all steps so that we can find the mistake)
$$Z=16\{2(p+k)p(p+k)p'-(p+k)^2(pp')+m^2[-4p(p+k)+4(p+k)^2]+m^2[(pp')-4(p+k)p']+4m^4\} \tag 4$$
Expanding out ##(4)## I get
$$Z=16\{ 2p^3p'+2kpkp'+2p^2kp'+2kp^2p'-p^3p'-2pkpp'-k^2pp'-4m^2p^2-4m^2pk+4m^2p^2+8m^2pk+4m^2k^2+m^2pp'-4m^2pp'-4m^2kp'+4m^4\} \tag 5$$
Now it is about simplifying ##(5)##. I'll go step by step.
I first used ##k^2=k'^2=0##, ##p^2 = p'^2=m^2## conditions to get
$$Z=16\{ 2p^3p'+2kpkp'+2p^2kp'+2kp^2p'-p^3p'-2pkpp'-4m^2pk+8m^2pk+m^2pp'-4m^2pp'-4m^2kp'+4m^4\} \tag 6$$
Then I simplified ##Xm^2kp'##, ##Ym^2pp'## terms and got
$$Z=16\{ +4m^2(pk) -2m^2pp'+4m^4+2kpkp'-2pkpp'\} \tag 6$$
Mmm but this is not the provided solution
$$Z=32\{m^4+m^2(pk)+(pk)(pk')\}$$
At least I got ##m^4## and ##m^2(pk)## right but I got the extra term ##-2m^2pp'## and did not get ##(pk)(pk')##. Actually, if the term ##-p^3p'## in ##(5)## were to be positive, ##Xm^2kp'## would cancel out but it is not the case.
Where did I get wrong then?
Any help is appreciated.
Thank you.
PS: I asked it https://math.stackexchange.com/questions/3702158/simplifying-an-expression-given-certain-on-shell-off-shell-conditions but got no answer.
$$Z:=16 \{ 2(f_1p)(f_1p') -f_1^2(pp')+m^2[-4(pf_1)+4f_1^2]+m^2 [(pp')-4(f_1 p')] +4m^4 \} \tag 1$$
Where:
$$f_1:=p+k \tag 2$$
We're also given the on-shell and off-shell conditions:
$$p^2 = p'^2=m^2, \ k^2=k'^2=0, \ pk=p'k', \ pk'=p'k \tag 3$$
Plugging ##(2)## into ##(1)## I get (I include all steps so that we can find the mistake)
$$Z=16\{2(p+k)p(p+k)p'-(p+k)^2(pp')+m^2[-4p(p+k)+4(p+k)^2]+m^2[(pp')-4(p+k)p']+4m^4\} \tag 4$$
Expanding out ##(4)## I get
$$Z=16\{ 2p^3p'+2kpkp'+2p^2kp'+2kp^2p'-p^3p'-2pkpp'-k^2pp'-4m^2p^2-4m^2pk+4m^2p^2+8m^2pk+4m^2k^2+m^2pp'-4m^2pp'-4m^2kp'+4m^4\} \tag 5$$
Now it is about simplifying ##(5)##. I'll go step by step.
I first used ##k^2=k'^2=0##, ##p^2 = p'^2=m^2## conditions to get
$$Z=16\{ 2p^3p'+2kpkp'+2p^2kp'+2kp^2p'-p^3p'-2pkpp'-4m^2pk+8m^2pk+m^2pp'-4m^2pp'-4m^2kp'+4m^4\} \tag 6$$
Then I simplified ##Xm^2kp'##, ##Ym^2pp'## terms and got
$$Z=16\{ +4m^2(pk) -2m^2pp'+4m^4+2kpkp'-2pkpp'\} \tag 6$$
Mmm but this is not the provided solution
$$Z=32\{m^4+m^2(pk)+(pk)(pk')\}$$
At least I got ##m^4## and ##m^2(pk)## right but I got the extra term ##-2m^2pp'## and did not get ##(pk)(pk')##. Actually, if the term ##-p^3p'## in ##(5)## were to be positive, ##Xm^2kp'## would cancel out but it is not the case.
Where did I get wrong then?
Any help is appreciated.
Thank you.
PS: I asked it https://math.stackexchange.com/questions/3702158/simplifying-an-expression-given-certain-on-shell-off-shell-conditions but got no answer.