# Problem 14.5c5

Counting states in heterotic SO(32) string theory | A First Course in String Theory

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c) Are there tachyonic states in heterotic string theory?

Write out the massless states of the theory (bosons and fermions) …

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We have the following well-known bosonic string mass formulae:

open string:

$\displaystyle{\alpha' M^2 = N - a}$

closed string:

$\displaystyle{\frac{1}{4} \alpha' M^2 = N_L - a}$
$\displaystyle{\frac{1}{4} \alpha' M^2 = N_R - a}$

p.55

— Solutions to K. Becker, M. Becker, J. Schwarz String Theory And M-theory

— Mikhail Goykhman

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How come there is an extra $\displaystyle{\frac{1}{4}}$ at the beginning of the closed string formula?

p.322

$\displaystyle{\frac{1}{2} \alpha' M^2 = \alpha' M_L^2 + \alpha' M_R^2}$

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\displaystyle{\begin{aligned} \alpha' M_L^2 &= \alpha' M_R^2 \\ \frac{1}{2} \alpha' M^2 &= \alpha' M_L^2 + \alpha' M_R^2 \\ \alpha' M^2 &= 2 \left( \alpha' M_L^2 + \alpha' M_R^2 \right) = 4 \alpha' M_L^2 \\ \end{aligned}}

— Me@2018-12-15 08:59:18 PM

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2018.12.15 Saturday (c) All rights reserved by ACHK