Timeline for What would change if we add a linear term in the Klein-Gordon Lagrangian?
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| when toggle format | what | by | license | comment | |
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| Oct 5, 2019 at 9:21 | vote | accept | jak | ||
| Aug 26, 2017 at 9:23 | comment | added | AccidentalFourierTransform | (2/2) If you take the vev of your equation (3), you get $m^2\langle\Phi\rangle=C$, and therefore $\langle\Phi\rangle=0$ if and only if $C=0$. Finally, recall that if $\langle\Phi\rangle=v\neq 0$, you can always redefine $\Phi'=\Phi-v$, which has $\langle\Phi'\rangle=0$ by construction. This shift is, by the previous argument, equivalent to $\Phi\to\Phi-C/m^2$. That's why I said that the two hints are actually the same. Again, $v\neq 0$ is not bad per-se. If you insist that it should vanish, then take $C=0$. Otherwise, you may keep $C$ arbitrary and use it to tune $v$ to any value you want. | |
| Aug 26, 2017 at 9:19 | comment | added | AccidentalFourierTransform | @JakobH recall that for the validity of the LSZ formula we require $\langle\Phi\rangle=0$. In general we use $\Phi$ to calculate $S$-matrix elements and therefore we need its vev to vanish. If you don't want to use $\Phi$ to calculate $S$-matrix elements, then it is perfectly valid for its vev to be different from zero (but bear in mind that that would require one more renormalisation condition). If the vev of $\Phi$ is not forced to vanish, then a linear term in the Lagrangian is valid. But in general we do require $\langle\Phi\rangle$ to vanish, and this explains why we use $C\equiv 0$.(1/2) | |
| Aug 26, 2017 at 8:49 | comment | added | jak | thanks for your answer! I would love to know your thoughts on what the correct answers to these questions are. Regarding "Hint 1": A linear term arises sometimes in GUTs, and thus, as you note one gets $\langle\Phi\rangle\neq 0$, which is called an induced VEV. Usually these additional contributions to the symmetry breaking are tiny, because they are surpressed by some large scale. "Hint 2:" I'm not sure about the meaning of such a shift here... | |
| Aug 3, 2017 at 11:52 | history | answered | AccidentalFourierTransform | CC BY-SA 3.0 |