- #1
Sonny Liston
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This isn't a homework problem, but something from a set of notes that I'd like to better understand. My confusion starts on page 23 here: http://isites.harvard.edu/fs/docs/icb.topic1146665.files/III-9-RenormalizationGroup.pdf. I'm having trouble reproducing his calculation for the marginal/irrelevant couplings from the notes, and would like to solve problem 5 at the end. I'm hoping that understanding what he does in the notes will help my current confusion about that problem, which I think promises a really nice understanding of why relevant operators are problematic.
Schwartz starts with the general renormalization group equations \Lambda \frac{d}{d\Lambda} \lambda_{4} = a\lambda_{4} +b\lambda_{6} and \Lambda \frac{d}{d\Lambda} \lambda_{6} -2\lambda_{6} = c\lambda_{4} + (d+2)\lambda_{6} where \lambda_{4},\lambda_{6} are dimensionless couplings and a,b,c,d are small constants. At this point, the problem just becomes solving this set of coupled ODEs, which Schwartz does by diagonalizing the matrix \bigl( \begin{smallmatrix} <br /> a & b\\<br /> c & d+2 <br /> \end{smallmatrix} \bigr)
This eventually leads to exact solutions in the basis we began with, which are given by eqns. (118) and (119) in his notes.
The part of the reasoning that I'm confused about is how he goes from eqn. (120) to eqn. (121), and later from (122) to (123).
Eqn (120) is \lambda_{6}(\Lambda)=\lambda_{4}(\Lambda) \frac{2c[(\Lambda/\Lambda_{H})^{\Delta}-1]}{(2+d−a+\Delta)-(2+d−a-\Delta)(\Lambda/\Lambda_{H})^{\Delta}}
and Schwartz claims that Setting \Lambda=\Lambda_{L}\ll\Lambda_{H} and assuming a, b, c, d\ll 2 so that \Delta\approx 2 we find Eqn(121), which is \lambda_{6}(\Lambda_{L})=\lambda_{4}(\Lambda_{L})[\frac{c}{2}((\Lambda_{L}/\Lambda_{H})^2 -1]
I'll list my confusion for this case below, but similar confusion extends to moving from (122) to (123).
I can come up with a justification for getting eqn (121) from eqn (120), but it seems likely wrong: since both d,a are small and \Delta\approx 2 and (\Lambda_{L}/\Lambda_{H})^2 is very small since \Lambda=\Lambda_{L}\ll\Lambda_{H} we can neglect the d-a and (d-a)(\Lambda_{L}/\Lambda_{H}) terms which allow us to approximate the denominator in (121) by 4. This then gives us eqn (121).
I mentioned I'm suspicious of this reasoning, but does this seem sound? It also seems justify the move from (122) to (123). And if so, does anyone have guidance on how to tackle problem 5 in those notes?
Homework Statement
Schwartz starts with the general renormalization group equations \Lambda \frac{d}{d\Lambda} \lambda_{4} = a\lambda_{4} +b\lambda_{6} and \Lambda \frac{d}{d\Lambda} \lambda_{6} -2\lambda_{6} = c\lambda_{4} + (d+2)\lambda_{6} where \lambda_{4},\lambda_{6} are dimensionless couplings and a,b,c,d are small constants. At this point, the problem just becomes solving this set of coupled ODEs, which Schwartz does by diagonalizing the matrix \bigl( \begin{smallmatrix} <br /> a & b\\<br /> c & d+2 <br /> \end{smallmatrix} \bigr)
This eventually leads to exact solutions in the basis we began with, which are given by eqns. (118) and (119) in his notes.
Homework Equations
The part of the reasoning that I'm confused about is how he goes from eqn. (120) to eqn. (121), and later from (122) to (123).
Eqn (120) is \lambda_{6}(\Lambda)=\lambda_{4}(\Lambda) \frac{2c[(\Lambda/\Lambda_{H})^{\Delta}-1]}{(2+d−a+\Delta)-(2+d−a-\Delta)(\Lambda/\Lambda_{H})^{\Delta}}
and Schwartz claims that Setting \Lambda=\Lambda_{L}\ll\Lambda_{H} and assuming a, b, c, d\ll 2 so that \Delta\approx 2 we find Eqn(121), which is \lambda_{6}(\Lambda_{L})=\lambda_{4}(\Lambda_{L})[\frac{c}{2}((\Lambda_{L}/\Lambda_{H})^2 -1]
I'll list my confusion for this case below, but similar confusion extends to moving from (122) to (123).
The Attempt at a Solution
I can come up with a justification for getting eqn (121) from eqn (120), but it seems likely wrong: since both d,a are small and \Delta\approx 2 and (\Lambda_{L}/\Lambda_{H})^2 is very small since \Lambda=\Lambda_{L}\ll\Lambda_{H} we can neglect the d-a and (d-a)(\Lambda_{L}/\Lambda_{H}) terms which allow us to approximate the denominator in (121) by 4. This then gives us eqn (121).
I mentioned I'm suspicious of this reasoning, but does this seem sound? It also seems justify the move from (122) to (123). And if so, does anyone have guidance on how to tackle problem 5 in those notes?
