- #1
saadhusayn
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- TL;DR Summary
- Hi,
I am trying to calculate the ghost field from a paper by Becker and Becker. However, I seem to get an incorrect coefficient for a term and am not quite sure how certain quantities look like after a Wick rotation.
This is the paper that I refer to. I'm trying to figure out the ghost action (Equation 2.16) in the background field gauge. I am attempting to use Srednicki's (chapter 78) expression for the ghost field in the background gauge. However, I am missing out on a √g coefficient in front of the term
$$ \epsilon^{abc}(\partial_{\tau} \overline{c}^{a}) c^{b} A^{c}$$. Furthermore, how does the quantity b (Equation 2.7) transform under a Wick rotation? Does it become ib? Here is my work so far. Thank you in advance.
$$G^{a}t^{a} = \partial^{\mu} A_{\mu}^{a} t^{a} + [\ B^{\mu r} t^{r}, A_{\mu}^{s} t^{s} ]$$
The gauge condition is therefore given by:
$$ G^{a} = \partial^{\nu} A_{\nu}^{a}+ B^{\nu r} A_{\nu}^{s} \epsilon^{rsa}t^{a}$$
From Srednicki's book, we have the expression for the ghost term in the Lagrangian is given by:
$$ \mathcal{L}_{GH} = \overline{c}^{a} \frac{\partial G^{a}}{\partial A_{\mu}^{b}} D_{\mu}^{bc}c^{c} $$
Where the gauge covariant derivative is:
$$ D_{\mu}^{bc} c^{c} = (\delta^{bc} \partial_{\mu} + \epsilon^{bsc} A_{\mu}^{s})c^{c} = \partial_{\mu} c^{b} + \epsilon^{bsc} A_{\mu}^{s} c^{c}$$
And
$$ \frac{\partial G^{a}}{\partial A_{\mu}^{b}} = \delta^{ab} \partial^{\mu} + B^{\mu r} \epsilon^{rba}$$
Putting it all together,
$$ \mathcal{L}_{GH} = \overline{c}^{a}(\delta^{ab} \partial^{\mu} + B^{\mu r} \epsilon^{rba})(\partial_{\mu} c^{b} + \epsilon^{bsc} A_{\mu}^{s} c^{c})$$
$$\mathcal{L}_{GH} = \overline{c}^{a} \Box c^{a} + \epsilon^{asc}\overline{c}^{a} \partial^{\mu} (A_{\mu}^{s} c^{c}) + B^{\mu r} \epsilon^{rca} \overline{c}^{a} \partial_{\mu} c^{c} + B^{\mu r} \epsilon^{bsc} \epsilon^{rba} A_{\mu}^{s} \overline{c}^{a} c^{c}$$
$\because$ this is a dimensionally reduced Yang-Mills theory, the space derivatives all disappear. Also, $$B^{0} = 0$$.
$$ \mathcal{L}_{GH} = \overline{c}^{a} \partial^{t} \partial_{t} c^{a} + \epsilon^{asc}\overline{c}^{a} \partial^{t} (A^{s} c^{c}) + \cancelto{0}{B^{0 r} \epsilon^{rca} \overline{c}^{a} \partial_{t} c^{c}} + \cancelto{0}{B^{0 r} \epsilon^{bsc} \epsilon^{rba} A^{s}} + B^{i r} \epsilon^{bsc} \epsilon^{rba} A_{i}^{s} \overline{c}^{a} c^{c}$$
We make a Wick rotation $$t \rightarrow -i\tau \implies \partial_{t} \rightarrow i \partial_{\tau}$$. Also, $$A^{c} \rightarrow -i A^{c}$$.
Then, upto a total derivative, in Euclidean space,
$$ \mathcal{L}_{GH} = -\overline{c}^{a} \partial_{\tau}^{2} c^{a} + \epsilon^{abc}(\partial_{\tau} \overline{c}^{a}) c^{b} A^{c} + B^{i r} \epsilon^{cbx} \epsilon^{arx} A_{i}^{c} \overline{c}^{a} c^{b}$$
We can expand the last term above about the background field:
$$ \epsilon^{arx} \epsilon^{cbx}B^{ir}(B_{i}^{c} + \sqrt{g}Y_{i}^{c}) \overline{c}^{a} c^{b}$$
$$ = (\delta^{ac} \delta^{rb} -\delta^{ab}\delta^{rc})B^{ir} (B_{i}^{c} + \sqrt{g} Y_{i}^{c}) \overline{c}^{a} c^{b}$$
$$ \epsilon^{abc}(\partial_{\tau} \overline{c}^{a}) c^{b} A^{c}$$. Furthermore, how does the quantity b (Equation 2.7) transform under a Wick rotation? Does it become ib? Here is my work so far. Thank you in advance.
$$G^{a}t^{a} = \partial^{\mu} A_{\mu}^{a} t^{a} + [\ B^{\mu r} t^{r}, A_{\mu}^{s} t^{s} ]$$
The gauge condition is therefore given by:
$$ G^{a} = \partial^{\nu} A_{\nu}^{a}+ B^{\nu r} A_{\nu}^{s} \epsilon^{rsa}t^{a}$$
From Srednicki's book, we have the expression for the ghost term in the Lagrangian is given by:
$$ \mathcal{L}_{GH} = \overline{c}^{a} \frac{\partial G^{a}}{\partial A_{\mu}^{b}} D_{\mu}^{bc}c^{c} $$
Where the gauge covariant derivative is:
$$ D_{\mu}^{bc} c^{c} = (\delta^{bc} \partial_{\mu} + \epsilon^{bsc} A_{\mu}^{s})c^{c} = \partial_{\mu} c^{b} + \epsilon^{bsc} A_{\mu}^{s} c^{c}$$
And
$$ \frac{\partial G^{a}}{\partial A_{\mu}^{b}} = \delta^{ab} \partial^{\mu} + B^{\mu r} \epsilon^{rba}$$
Putting it all together,
$$ \mathcal{L}_{GH} = \overline{c}^{a}(\delta^{ab} \partial^{\mu} + B^{\mu r} \epsilon^{rba})(\partial_{\mu} c^{b} + \epsilon^{bsc} A_{\mu}^{s} c^{c})$$
$$\mathcal{L}_{GH} = \overline{c}^{a} \Box c^{a} + \epsilon^{asc}\overline{c}^{a} \partial^{\mu} (A_{\mu}^{s} c^{c}) + B^{\mu r} \epsilon^{rca} \overline{c}^{a} \partial_{\mu} c^{c} + B^{\mu r} \epsilon^{bsc} \epsilon^{rba} A_{\mu}^{s} \overline{c}^{a} c^{c}$$
$\because$ this is a dimensionally reduced Yang-Mills theory, the space derivatives all disappear. Also, $$B^{0} = 0$$.
$$ \mathcal{L}_{GH} = \overline{c}^{a} \partial^{t} \partial_{t} c^{a} + \epsilon^{asc}\overline{c}^{a} \partial^{t} (A^{s} c^{c}) + \cancelto{0}{B^{0 r} \epsilon^{rca} \overline{c}^{a} \partial_{t} c^{c}} + \cancelto{0}{B^{0 r} \epsilon^{bsc} \epsilon^{rba} A^{s}} + B^{i r} \epsilon^{bsc} \epsilon^{rba} A_{i}^{s} \overline{c}^{a} c^{c}$$
We make a Wick rotation $$t \rightarrow -i\tau \implies \partial_{t} \rightarrow i \partial_{\tau}$$. Also, $$A^{c} \rightarrow -i A^{c}$$.
Then, upto a total derivative, in Euclidean space,
$$ \mathcal{L}_{GH} = -\overline{c}^{a} \partial_{\tau}^{2} c^{a} + \epsilon^{abc}(\partial_{\tau} \overline{c}^{a}) c^{b} A^{c} + B^{i r} \epsilon^{cbx} \epsilon^{arx} A_{i}^{c} \overline{c}^{a} c^{b}$$
We can expand the last term above about the background field:
$$ \epsilon^{arx} \epsilon^{cbx}B^{ir}(B_{i}^{c} + \sqrt{g}Y_{i}^{c}) \overline{c}^{a} c^{b}$$
$$ = (\delta^{ac} \delta^{rb} -\delta^{ab}\delta^{rc})B^{ir} (B_{i}^{c} + \sqrt{g} Y_{i}^{c}) \overline{c}^{a} c^{b}$$