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IL NUOVO CIMENTO VOL. 112 B, N. 10 Ottobre 1997 NOTE BREVI

Zeeman splitting in the field of a cosmic string

C. WOLF

Department of Physics, North Adams State College - North Adams, MA 01247, USA (ricevuto il 14 Maggio 1997; approvato il 24 Luglio 1997)

Summary. — By taking into account the corrections to the magnetic moment of the

electron in the field of a topological defect (string) we demonstrate that the Zeeman splitting for electrons in the field of a string can serve to identify cosmic strings in a cosmological setting.

PACS 96.80 – Cosmology.

1. – Introduction

One of the most trusted and accurate probes to the structure of elementary particles is found in the difference between the experimental and theoretical value of the magnetic moment, the electric-dipole moment and the electric-quadrupole moment of elementary particles [1], [2]. Supersymmetry [3], grand unification [4] and higher field-theoretic corrections can account for the measured g factor of the electron to within a margin of a NgMeas2 gTheorN E 1029 leaving open the question of whether composite particles effects can account for the difference [5, 6]. Also it might be that in high magnetic fields environmental effects might produce corrections to the magnetic moment in spin phenomena that cannot be described by a Hamiltonian [7-9]. Even the Lamb shift with its precise experimental verification can receive temperature-dependent corrections that have no Hamiltonian origin to them [10]. In addition to these possibilities, there is another correction to the magnetic moment of the electron that has its origin rooted in topological notions, namely the corrections to the magnetic moment induced by a disclination produced by a cosmic string [11, 12]. The basic reason for this phenomena is the “Casimir effect”, basically the disclination modifies the vacuum state of the quantum fields in the neighborhood of the string due to the non-trivial topology which in turn modifies the magnetic moment of the electron. In what follows we point out that this effect can be used to identify cosmic strings in a cosmological setting by observing Zeeman lines due to spin flips which depend on the distance from the string. If a characteristic variation of the Zeeman frequency with distance from the string is found that varies inversely with the distance squared from the string, then the curvature of the plot can be used to calculate the disclination angle produced by the string and in turn the string mass density.

C.WOLF

1430

2. – Zeeman splitting in the field of cosmic string

We begin by considering the result of ref. [11]; for the cylindrically symmetric metric in the field of a string we have

( ds)2

4 C2dt2

2 dr22 ( dz)22 r2a2_{( df)}2_; (2.1)

if we set a2_df2

4 df2where f goes from 0 to 2 p , f goes from 0 to a2 p . Thus

2 pa 42p1l, a 411 l

2 p (2.2)

(l 4dihedral angle). If 0 GaG1 we have a positive-curvature disclination, if aD1 we have a negative-curvature disclination. In ref. [12] the correction to the magnetic moment of an electron in the field of a string was written as

dm 42m0

y

2 1 ) e2ˇ

48 p2_m2_r2_C3

z

here m04 eˇO2 mC , r 4 distance from center of string.

If 1 Da we have a negative correction to m, if aD1 we have a positive correction to m .

For an electron in a magnetic field directed along the string (B 4Bk) we have for the spin energy of an electron

E_{64 6} e mc

g

h

(6 refer to spin up and spin down). For a transition of spin up to spin down

DE 4 2 e mC

g

h

g

h

g

h

If the Zeeman wavelength for a series of lines exhibits a decrease with 1 Or2_{or an} increase with 1 Or2_{it would be evidence for the presence of an electron spin flip in the} presence of a disclination. If a E1 it would be evidence for a positive-curvature string. Even without a precise numerical value of B, a plot of l vs. 1 Or2could determine the sign of the disclination (a D1 or aE1). If B can be found from other phenomena such as an astrophysical Aharanov-Bohm effect or Zeeman splittings in nearby HI regions, then a plot of l vs. 1 Or2 _{would determine a and we could estimate the string density}

ZEEMAN SPLITTING IN THE FIELD OF A COSMIC STRING 1431

(m 4mass density per unit length) from a4124GmOC2 _{[13]. We note that even for}

a 4121026_{, m 410}22

gOcm (GUT string) [14]. In closing, small variations of the Zeeman wavelength coming from electron spin flips that vary as 1 Or2_{can be used as a} signature to identify a cosmic string in a cosmological setting.

* * *

I would like to thank the Physics Departments at Williams College and Harvard University for the use of their facilities.

R E F E R E N C E S

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[13] GAMBOAJ. and SEGUI-SANTONJAA. J., Class. Quantum Grav., 9 (1992) 1. [14] GUIMARAESM. E. X., Phys. Lett. B, 398 (1997) 281.