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Another question from Srednicki's QFT book.

by MathematicalPhysicist
Tags: book, srednicki
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Jul16-14, 09:39 AM
P: 3,220
Sorry for my questions, (it does seem like QFT triggers quite a lot of questions :-D).

Anyway, on page 103 (it has a preview in google books), I am not sure how did he get equation (14.40), obviously it should follow from (14.39), but I don't understand where did -ln(m^2) disappear ?

Shouldn't we have an expression like (14.40) but the term (linear in k^2 and m^2) be instead:
(linear in k^2,m^2 and ln(m^2)).

Cause as far as I can tell from (14.39) [tex]\Pi(k^2)[/tex] depends also on ln(m^2), and thus the term "linear in..." should be replaced with "linear also in ln(m^2)", cause as far as I can tell ln(m^2) isn't linear function of m^2, right?

Hope someone can enlighten me.
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Jul18-14, 04:19 AM
P: 754
No that's not possibly the case, because even if you have the [itex]- \ln (m^{2}) [/itex] it's multiplied with a [itex]D[/itex].. After integration of [itex]D dx [/itex] you will get [itex](k^{2}+m^{2})ln(m^{2})/6[/itex]
so I think the [itex]ln(m^{2})[/itex] is absorbed within the [itex]κ_{A,B}[/itex]

But I hope someone can be more helpful
Jul18-14, 01:42 PM
P: 3,220
Actually the integral on Ddx yields 1/6 k^2 +m^2, it's written previously.

Jul18-14, 01:45 PM
P: 754
Another question from Srednicki's QFT book.

and that's what I've written? I just didn't take in account the minus from the ln....Ah yes, you are write, just put a 6 in front of m^2 then
Jul18-14, 02:59 PM
P: 3,220
I think the task of solving the problem of mass gap from clay institute which relates to QFT looks a lot more intimidating as I keep reading. :-D
Jul20-14, 04:43 AM
Sci Advisor
P: 303
The mass gap problem is essentially that you have to prove Yang-Mills exists and after you have done that, you must prove it describes massive particles (rather than massless ones as you would naively think from the Lagrangian). So, yes it is very difficult!
Jul21-14, 01:36 AM
P: 3,220
Ok, I plan to ask all of my questions from QFT books here (if the moderators want to include my other posts into this thread it's ok by me, but make it chronological orderd (or as we say use the time-ordering operator).
Jul21-14, 01:49 AM
P: 3,220
In this case we don't have a book preview of pages 166-167 .

So I'll write the equations:

[tex](27.23) ln |\mathcal{T}|^2_{obs} = C_1+2ln \alpha +3\alpha (ln \mu +C_2)+O(\alpha^2)[/tex]

Now he says that "Differentiating wrt ln \mu then gives":

[tex](27.24)0=\frac{d}{dln \mu} ln |\mathcal{T}|^2_{obs} = \frac{2}{\alpha} \frac{d\alpha}{dln \mu} +3\alpha +O(\alpha^2) [/tex]

Now as far as I can tell when you differentiate: [tex]\frac{d}{dln \mu} (3\alpha(ln \mu +C_2))=3\alpha + 3 \frac{d\alpha}{dln \mu} (ln \mu +C_2)[/tex]
so where did [tex]3 \frac{d\alpha}{dln \mu} ln \mu[/tex] disapper from eq. (27.24)?

Don't see why he didn't include this term in eq. (27.24).


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