Another question from Srednicki's QFT book

[MathematicalPhysicist]

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) $$\Pi(k^2)$$ 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.

 

[ChrisVer]

No that's not possibly the case, because even if you have the $- \ln (m^{2})$ it's multiplied with a $D$.. After integration of $D dx$ you will get $(k^{2}+m^{2})ln(m^{2})/6$
so I think the $ln(m^{2})$ is absorbed within the $κ_{A,B}$

But I hope someone can be more helpful

 

[MathematicalPhysicist]

Actually the integral on Ddx yields 1/6 k^2 +m^2, it's written previously.

 

[ChrisVer]

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

 

[MathematicalPhysicist]

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

 

[DarMM]

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!

 

[MathematicalPhysicist]

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).

 

[MathematicalPhysicist]

eqs. (27.23)-(27.24) on page 166 of Srednicki's QFT.

In this case we don't have a book preview of pages 166-167 .

So I'll write the equations:

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

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

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

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

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

Anyone?

 

[MathematicalPhysicist]

Another question from Srednicki.

Hi, so I hope there are still some folks who look at this thread of mine.

So now I am looking at Srednicki's solution to question 48.4b, here:
http://www.scribd.com/doc/87916496/Srednicki-Ms-Qft-Solutions-Rev

I don't understand how in eq (48.53) he pulled from the trace the term (1+s1s2), shouldn't it be (1-s1s2)?

I mean if you simplify the term inside the trace in the first line you should get:
Tr((1-s1s2)(...)) = (1-s1s2)Tr(...)

So how did he get the plus sign there?

Perhaps x^2=1 and not -1 as he wrote there?

Puzzled.

 

[ChrisVer]

I think there is a mistake in the signs...
$\bar{x} ( -\bar{p}_{1} -m) \bar{x}$

$-\bar{x} \bar{p}_{1} \bar{x} - m \bar{x} \bar{x}$

$+\bar{x}\bar{x} \bar{p}_{1}- 2 xp_{1} + m$

$-\bar{p}_{1}+m$

So I think the minus in front of s1s2 in the first line of 48.53 is wrong? Obviously by bared quantities I mean slash, it's just \slash{} didn't work...

 

[MathematicalPhysicist]

Another question from Srednicki's text.

On page 316, I am not sure how did he get equation (51.10) from eq. (51.9)?

Here's what I have done :
eqaution (51.10) is the same as the next equation:

(I am writing it down without a slash, since I am not sure how to use it here)

$$\frac{(-p^{\mu}\gamma_{\mu}+m)}{(p^{\mu}\gamma_{\mu}+m-i\epsilon)(-p^{\mu}\gamma_{\mu}+m)}+\int \frac{...}{p^2+s-\epsilon^2-2i\epsilon\sqrt{s}}$$

So as you can see it's a bit different than (51.9), the denominator in the integral, the epsilon doesn't have as a factor, the coefficient, i .
And the first term after rearranging it we should have:

$$\frac{ -p^{\mu}\gamma_{\mu} + m}{p^2+m^2-i\epsilon \cdot (-p^{\mu}\gamma_{\mu} +m)}$$

It's not the same term as in eq (51.9), perhaps he stated previously that it's meaningless factor multiplies the epsilon, I might have forgotten it.

What do you think?

P.S
There's a preview in google books:
http://books.google.co.il/books?id=5OepxIG42B4C&printsec=frontcover&hl=iw#v=onepage&q&f=false

 

[ChrisVer]

the page is unfortunately not previewed.

 

[MathematicalPhysicist]

I can see the preview in google books of pages 316-315 does exist, I think you have a problem in your computer.

Cheers!

 

[MathematicalPhysicist]

Now I am not sure as for the solution of question 51.2) in Srednicki.

It's in here:
http://www.scribd.com/doc/87916496/Srednicki-Ms-Qft-Solutions-Rev#logout

If I am not mistaken he uses the next first order approximation:

$$V_Y(p,p') = V_Y(0,0) + V_Y'(0,0) p\cdot p'$$

But I don't understand what did he plug instead of $$V_Y'(0,0)$$

Anyone, or if you have a more lengthy detailed calculation as to how did he get eq. (51.58)

 

[nrqed]

On page 316, I am not sure how did he get equation (51.10) from eq. (51.9)?

Here's what I have done :
eqaution (51.10) is the same as the next equation:

(I am writing it down without a slash, since I am not sure how to use it here)

$$\frac{(-p^{\mu}\gamma_{\mu}+m)}{(p^{\mu}\gamma_{\mu}+m-i\epsilon)(-p^{\mu}\gamma_{\mu}+m)}+\int \frac{...}{p^2+s-\epsilon^2-2i\epsilon\sqrt{s}}$$

So as you can see it's a bit different than (51.9), the denominator in the integral, the epsilon doesn't have as a factor, the coefficient, i .
And the first term after rearranging it we should have:

$$\frac{ -p^{\mu}\gamma_{\mu} + m}{p^2+m^2-i\epsilon \cdot (-p^{\mu}\gamma_{\mu} +m)}$$

It's not the same term as in eq (51.9), perhaps he stated previously that it's meaningless factor multiplies the epsilon, I might have forgotten it.

What do you think?

The key point is that if you multiply any positive quantity by epsilon, you may still write the result as epsilon since it is taken much smaller than any other quantity. $\sqrt{s}$ is positive so we may write $\sqrt{s} \epsilon \simeq \sqrt{s}$. Likewise for the other term, multiplying or dividing epsilon by $p^2 + m^2$ does not matter.

Hope this helps.

 

[MathematicalPhysicist]

Well, if you can write $$\sqrt{s} \epsilon \approx \sqrt{s}$$ then presumably you can divide by $\sqrt{s}$ both sides (besides when s=0 which we have equality), and thus you'll get: $$\epsilon \approx 1$$ but we've taken it to be approaching zero from above, so it should be $$\approx 0$$.

Best regards.

 

[nrqed]

Well, if you can write $$\sqrt{s} \epsilon \approx \sqrt{s}$$ then presumably you can divide by $\sqrt{s}$ both sides (besides when s=0 which we have equality), and thus you'll get: $$\epsilon \approx 1$$ but we've taken it to be approaching zero from above, so it should be $$\approx 0$$.

Best regards.

Sorry, I had a typo in the equation. As I wrote in words, I meant that $$\sqrt{s} \epsilon \approx \epsilon$$

 

[MathematicalPhysicist]

ok, now it makes sense.
Thanks!

 

[MathematicalPhysicist]

Another question from Srednicki's.

 

[MathematicalPhysicist]

I have got another question from Srdnicki's book.
from chapter 54, pages 338-337.

He arrives at the next form of the lagrangian:

[tex][/tex]

 

[nrqed]

What is your question? :)

 

[MathematicalPhysicist]

I scanned my question, hopefully you can watch the question, it's a pdf you just need to flip it 180 degrees, becasue I scanned it in the opposite direction.

 

[nrqed]

I scanned my question, hopefully you can watch the question, it's a pdf you just need to flip it 180 degrees, becasue I scanned it in the opposite direction.

I did not have any problem showing that Eq 54.24 = Eq 54.23. Did you try this?
Note that you can always rename dummy indices so you are free to exchange $\mu$ and $\nu$. Note also that you can always switch a pair of indices that is upstair and downstair when they are summed over. For example,
$A_\mu \partial^\mu \partial^\nu A_\nu = A^\nu \partial^\mu \partial_\nu A_\mu$
where I have switched order of the two partial derivatives

 

[MathematicalPhysicist]

Indeed, I get the equality as well, not sure why I didn't succeed in doing it before. Thanks!

 

[MathematicalPhysicist]

On page 347 of Srednicki.

First I'll highlight the equations that are at hand.

$$(56.25)\tilde{\Delta}^{\mu\nu} (k) = -\frac{\hat{t}^{\mu} \hat{t}^{\nu}}{k^2+(\hat{t}\cdot k)^2}+\frac{g^{\mu\nu} +\hat{t}^{\mu}\hat{t}^{\nu}-\hat{z}^{\mu}\hat{z}^{\nu}}}{k^2-i\epsilon$$

Now he aregues that from the next sub:
$$(56.26)\hat{z}^{\mu} \rightarrow \frac{(\hat{t}\cdot k ) \hat{t}^{\mu}}{[k^2+(\hat{t}\cdot k)^2]^{0.5}}$$

Then equation (56.25) becomes:
$$\tilde{\Delta}^{\mu\nu} (k) =\frac{1}{k^2-i\epsilon}[g^{\mu\nu}+(-\frac{k^2}{k^2+(\hat{t}k)^2} +1-\frac{(\hat{t}\cdot k )^2}{k^2+(\hat{t}\cdot k)^2})\hat{t}^{\mu}\hat{t}^{\nu}]$$

But I don't get this result.
BTW, I must say a few things about the expressions (though I believe that most of you have the books (electronic copy or otherwise).

$$(56.19)\hat{t}^{\mu} = (1,\vec{0})$$
where $z$ is a unit vector in the direction of $$\vec{k}$$.

Well, I get after some algebraic manipulations:
that $$\Delta=\frac{i\epsilon\hat{t}^{\mu}\hat{t}^{\nu}+(k^2+(\hat{t}\cdot k)^2)g^{\mu \nu} }{(k^2+(\hat{t}\cdot k )^2)(k^2-i\epsilon)$$

In a PDF is attached what I have done.

 

[ChrisVer]

Just make the substitution... $\hat{z}^\mu \hat{z}^\nu = \hat{t}^{\mu} \hat{t}^{\nu}\frac{(\hat{t}\cdot k )^2}{k^2+(\hat{t}\cdot k)^2}$
So:
$\frac{g^{\mu\nu} +\hat{t}^{\mu}\hat{t}^{\nu}-\hat{z}^{\mu}\hat{z}^{\nu}}{k^2-i\epsilon} =\frac{g^{\mu\nu} +\hat{t}^{\mu}\hat{t}^{\nu} (1-\frac{(\hat{t}\cdot k )^2}{k^2+(\hat{t}\cdot k)^2})}{k^2-i\epsilon}$

I'll drop the $g^{ab}$ term since it's already in the form that you want it. So there should be a small work to be done with:

$\frac{1}{k^2-i\epsilon} \Big(1-\frac{(\hat{t}\cdot k )^2}{k^2+(\hat{t}\cdot k)^2} \Big) \hat{t}^{\mu}\hat{t}^{\nu} -\frac{\hat{t}^{\mu} \hat{t}^{\nu}}{k^2+(\hat{t}\cdot k)^2}$

Which gives [trivially] the one you want to have... if for one moment you forget the $i\epsilon$ in the nominator

 

[MathematicalPhysicist]

I got after some algebra that the propagator should be:

$$\frac{i\epsilon t^{\mu} t^{\nu}+(k^2+(k\cdot t)^2)g^{\mu\nu}}{(k^2-i\epsilon)(k^2+(k\cdot t)^2)}$$

Can anyone confirm or disconfirm my result?

 

[ChrisVer]

Since the prop is
$$\tilde{\Delta}^{\mu\nu} (k) =\frac{1}{k^2-i\epsilon}[g^{\mu\nu}+(-\frac{k^2}{k^2+(\hat{t}k)^2} +1-\frac{(\hat{t}\cdot k )^2}{k^2+(\hat{t}\cdot k)^2})\hat{t}^{\mu}\hat{t}^{\nu}]$$

Do you want to start manipulating it?
For example [just by looking at your input] I say that $k \cdot \hat{t}$ is giving $k_0$ alone... So

$$\tilde{\Delta}^{\mu\nu} (k) =\frac{1}{k^2-i\epsilon}[g^{\mu\nu}+(-\frac{k^2}{k^2+k_0^2} +1-\frac{k_0^2}{k^2+k_0^2})\hat{t}^{\mu}\hat{t}^{\nu}]$$

or

$$\tilde{\Delta}^{\mu\nu} (k) =\frac{1}{k^2-i\epsilon}[g^{\mu\nu}+(1-\frac{k^2+k_0^2}{k^2+k_0^2})\hat{t}^{\mu}\hat{t}^{\nu}]=\frac{g^{\mu\nu}}{k^2-i\epsilon}$$

Which is practically the same as yours, except for I didn't carry the $i \epsilon$ in the nominator.

 

[MathematicalPhysicist]

Can one of the experts let me know when can we take epsilon as zero and when put it as it is. I understand that it's negligble but I don't understand when is it legal to neglect it, anyone?

 

[ChrisVer]

Recall: Why did you put the epsilon in the propagator in the first place?
The straight answer, I guess, would be that there are no problems for neglecting it when it appears in the nominator... in contrast to neglecting it in the denominator.