Cosmological upper limit to neutrino magnetic moments

Volume 102B, number 4

PHYSICS LETTERS

18 June 1981

COSMOLOGICAL UPPER LIMIT TO NEUTRINO MAGNETIC MOMENTS John A. MORGAN Astronomy Centre, Physics Building, University o f Sussex, Falmer, Brighton, Sussex BN1 9QH, United Kingdom Received 26 March 1981

An upper limit to a possible neutrino magnetic moment of (1-2) × 10-11 Bohr magnetons is obtained by requiring that synthesis of 4He in the Big Bang not be disrupted 'by the excitation of additional neutrino helicity states.

If, as suggested by recent experiments [1 ], at least one species of neutrino possesses nonvanishing mass, then on quite general grounds one expects that neutrinos will also possess nonzero magnetic moments. In addition, various investigators have invoked nonvanishing neutrino magnetic moments to resolve the solar neutrino problem [2]. It is therefore of interest to examine possible constraints on the size of a hypothetical neutrino magnetic moment, independently of any theoretical expectations. We argue in this letter that the excitation through a magnetic moment of additional (right-handed) neutrino helicity states in the early universe would have potentially grave consequences for nucleosynthesis and that an upper limit /Av ~ (1--2) X 10 -11 Bohr magnetons follows in their absence. It is clear that a substantial neutrino magnetic moment can alter the development of the early universe. An earlier paPer [3] showed that the best available laboratory upper limit/AVe < 1.4 X 10 - 9 #B [4] does not rule out dramatic alterations to the thermal history of the universe: A moment of this size suffices to provide thermal contact between neutrinos and all other matter to late times, t > 100 s, forcing photons and neutrinos to share the heat liberated by pair annihilation and thus altering the conditions for nucleosynthesis. However, a magnetic moment also breaks chirality invariance of the neutrino [5] ; the vertex coupling neutrinos to the electromagnetic current (fig. 1) connects states of opposite helicity only, so that electron-neutrino scattering always changes the

/ / /

~(~) /

/

/ / q,

\ \ \

\ "~/t (r)

\

N

N

Fig. 1. Vertex coupling neutrinos to the electromagnetic current. helicity, e±+v r~e ±+vl,

(1)

and electron pair annihilation produces both helicities, e+ + e - ~ Ul +gr

(2)

t,,r+ ~ 1 . Here and in what follows neutrinos are assumed to be Dirac particles, and the term 'right (left)-handed neutrino' applies to the left (right)-handed helicity state of its antiparticle as well. Processes such as (1) and (2) (and analogues involving muons or even free quarks) would have occurred copiously in the early universe, given a large enough moment. The neutrino

0 031 - 9 1 6 3 / 8 1 / 0 0 0 0 - 0 0 0 0 / $ 02.50 © North -Holland Publishing Company

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component of the early universe would thus contribute to the equation of ~tate with its full (spin) statisdcai Weight ~ot~two,'rather than iJnity, as assumed in standard models of the Big Bang [6]. The resultant doubling of the neutrino energy density would then enhance the expansion of the universe as described by the first Friedman equation, (R/R) 2 = ~- lr G p / c 2 .

(3)

The level of 4He production in the Big Bang depends sensitively on a balance between the expansion rate dictated by (1) and the rate of weak interactions, which determine the n/p ratio at the time deuterium becomes stable. Helium synthesis proves more sensitive to the effect of additional neutrino helicities than to neutrino equilibration alone. Including the effects of a light tau neutrino, the change in 4He production for a present-day baryonic density of 2.26 X 10 -31 g cm -3 caused by complete thermalization is +0.007, whereas the change caused by doubling the statistical weight is +0.036, or roughly 15% of helium synthesized at that density * 1. This suggests that the main influence upon primordial nucleosynthesis of a netatrino magnetic moment comes from the latter effect. Yang et al. [7] have surveyed the sensitivity of 4He production to changes in the assumed number of light neutrinos (two in the standard case) and find

where the present-day baryonic density is (5)

(6)

Yang et al. argue that if X(4He) < 0.25 and ~2N h 2 > 0.01, then A N v ~ 1, allowing the tau neutrino, if 4He is not to be overproduced. Evidently, excitation of right-handed helicities in equilibrium with left-handed states in the early universe would compromise the standard picture of primordial nucleosynthesis.We assume that the magnetic moment provides the strongest coupling between right- and left-handed helicities and derive an upper ,1 Unpublished calculations by the author. 248

(7)

in terms of the temperature of all other matter after the annihilation of particles more massive than an electron. The upper limit on/a v comes from the latest possible choice for the decoupling epoch which will guarantee (7); decoupling must occur for T > 1012 K, t < 10 -4 s in order for muons and hadrons to be present in appreciable numbers, and for ordinary weak interactions to couple left-handed neutrinos to other matter. Right-handed helicity states will remain coupled to other matter as long as the rate of exciting interactions is greater than the local expansion rate, (8)

We assume in what follows that there are three light (m ,~ 1 MeV) neutrinos re, v u, and Vr, and that allthree have a magnetic moment lav = ~1~B = ( e h / 2 m e C ) ~ .

and Hubble's constant is H 0 = 100h 0 km/s/mpc.

r r / T ~ (1/3)1/4 = 0.76,

t

(4)

Po/C 2 = 2 X 10-29~Nh2 g cm - 3 ,

limit by requiring that right-handed neutrinos cease being excited by processes such as (1) or (2) at an early period in the history of the universe. The epoch of decoupling must be sufficiently early that the subsequent annihilation of massive pairs as the universe expands and cools will heat up all other matter, including left-handed neutrinos, so as to progressively dilute the relative contribution of right-handed states to the total energy density. If we generously assume that the gravitational equivalent of one neutrino in excess of ur ( A N v = 2 in eq. (4)) can be tolerated without dire effects on nucleosynthesis, the argument requires that the right-handed neutrino temperature be

~. n i avi c > I@R .

AX(4He) = (0.011 --2.3 X l O - 4 1 o g ( ~ N h 2 ) ) A N v ,

18 June 1981

,(9)

If neutrinos oscillate with sufficiently high frequency co ~ / ~ / R the latter assumption may be relaxed without altering the argument significantly. In that ca~e the physical neutrino will be described by a density matrix of the weak interaction eigenstates and if (say) only one kind of neutrino has a large magnetic moment, then the RaMS/1u will be taken from (9). The cross section for electron-neutrino scattering through a magnetic moment was derived by Bethe [8], it is approximately o = rt(e/hc) 2 la2 log ( q 2 a x / q 2 i n ) , 2 a x / q m2 i n ) , = 7r(e2 /meC2)2rl2 log ( q m

(10) (11)

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where qmax and qmin are the maximum and minimum momentum transfers, respectively. The same formula applies to/Iv scattering and to quark-v scattering if e 2 in (10) is replaced by the square of the quark charge. A conservative underestimate of the cross section for relativistic energies is a ~> 2r/2 b .

(12)

o tends to be somewhat larger for neutrino scattering off of muons or free quarks than off of electrons. The cross section for pair annihilation to neutrinos is of a similar magnitude, but its neglect will cause only a small error in what follows. The exact choice for the decoupling epoch depends on the equation of state applicable in the early universe (unfortunately highly uncertain), but the arguments leading to an upper limit on r/for different prescriptions differ only in detail. While k T > 100 MeV the universe contains a significant proportion of hadrons in its makeup. Near the end of the hadronic era this component can be regarded as a dilute mixture of ideal quantum fluids. However, an ideal gas approximation becomes suspect at a period when the number of hadrons per cubic pion Compton wavelength exceeds unity,which happens for T > 1.25 X 1012 K. Some investigators suggest that quark confinement may fail at roughly this point and that hadronic matter undergoes a phase transition to free quarks and gluons at some temperature Tc in consequence [ 9 11 ]. Estimates of Tc vary widely; Dicus et al. [ 10] quote a range 2 X 1011 K < T o < 2.7 × 1013 K. Here we consider two idealized cases: A cocktail of ideal fluids of particles with m < 1 GeV, and a mixture of relativistic free quarks and gluons, following ref. [11 ]. Curiously, the latest possible decoupling epoch proves virtually the same for these two very different pictures of hadronic matter, T = 5 X 1012 K , of kT = 430 MeV. (i) Ideal gas cocktail: The equation of state of an ideal quantum fluid may be expressed as a sum over modified Bessel functions of the second kind [12] :

Pi=2rt---2 Pi = - -

2rr2 ~

c

v=l 4o~[3K3(~)+gl(°Cb?~3)

mi¢2

v=l

,(p~)2 '

(14)

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rl i = _ _

sv+l

27r2

v=l

v~b

(15)

where gi = 2Ji + 1, s = +1 ( - 1 ) for bosons (fermions), and • = mic2/kT. For T > 1012 and m < 800 MeV, the sums may be approximated to a fraction of a percent by truncating after four or five terms. At 5 X 1012 K, significant contributions to the equation of state come from n -+, 7r0, K ± , K0 , I( 0, ~7,p, and co. Both muons and pions are highly relativistic and may be treated as extremely so with little error. (ii) Free quark gas: In this picture, following ref. [11 ], hadrons give way to free u, d, and s quarks and (massless) gluons for T > 5 X 1012 K. The equation of state is thus that of a mixture of relativistic fluids. Quark pairs contribute a statistical weight of 36 = 2×(2×½ +l)X3X3 and gluons a weight of 16 = 2X8, so that Phad

=

1

(a~ X36 + ~ X 16)aT 4

Phad = 31 Phad •

(16) (17)

In either case, the entropy in a comoving volume of all matter exclusive of right-handed neutrinos, which are assumed to have just decoupled at T = 5 X 1012, is

S = R3(p * p ) / T , =R3/T(Phad + Phad + 4 (.~) aT4) "

(lS) (19)

As the expansion of a friedmannian universe is isentropic, this will equal the entropy/comoving volume after the annihilation of states with m > 100 MeV, when the universe consists only of 7's, e -+'s, and neutrinos,

s = ~ a(nr)3.

(20)

Writing S ( > 1012 K) as

S = F(Tr)a(RTr) 3 ,

(21)

where the temperature of the distribution function for right-handed neutrinos Tr equals the temperature T of all other matter at the decoupling epoch, Tr and T will ultimately be related by

TilT -- [& F(Tr) 1-1/3

(22)

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Volume 102B, number 4

For the case of free quarks and gluons, one obtains immediately

Tr/T= (43/247) 1/3 = 0 . 5 6 .

(23)

Numerical evaluation of F ( T r ) for a mixture of ideal gases at the same decoupling temperature yields

Tr/T = 0 . 7 7 .

(24)

In calculating the expansion rate from (3), the effect of right-handed neutrinos must be included in p:

P = ~ aT4 + Phad + ~ aT4 •

(25)

The upper limit for 77 follows from (8):

[R / ~ niOvie~l/2

'7 < t 7 1, ,72

)

(26)

'

and is 7 / < 1.5 X 10 -11 ,

(27)

for the free quark case, where elastic neutrino scattering off electrons, muons, and free quarks is included in (8), or 7 / < 2.2 X 10 -11 ,

(28)

for the ideal gas case, considering scattering off electrons and muons only. Because (12) underestimates cross sections, particularly for/av and qv, and because only leptonic scattering is considered in the ideal gas case, b o t h upper limits are overestimates, by perhaps as much as a factor of two. It is of interest to compare these limits with those available from laboratory data. As noted earlier, reactor experiments require ~e < 1.4 X 10 - 9 . Kin~ [13] finds r/u < 10 - 8 - 1 0 - 9 , and no data concerning any magnetic moment o f the vr exist at all. The upper limits derived in this letter thus represent a considerable refinement over previously available estimates. They also serve to rule out most solutions of the solar neutrino problem which rely upon neutrino magnetic moments; with the exception of the mechanism proposed by Cisneros [2], which relies on an artificially

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large interior solar magnetic field, these require r / > 10 - 9 in the early universe [3]. It therefore appears that a nonvanishing neutrino magnetic moment will be of little aid in resolving the solar neutrino problem. I wish to thank Professor R.J. Tayler, Dr. N. Dombey, and Mr. D. Lindley for helpful discussions and criticisms, and Dr: D. Sciama for bringing to my attention a recent paper b y Shapiro et al, [14] on a related topic. A preliminary version of this work was presented at the RAL Topical Conference on Massive Neutrinos (February 1981). This research was supported by the Science Research Council.

References [1] F. Reines, H.W. Sobel and E. Pasierb, Phys. Rev. Lett. 45 (1980) 1307; V. Barger, K. Whisnant, D. Cline and R.J.N. Phillips, Phys. Lett. 93B (1980) 194, V.A. Lubimov, E.G. Novikov, V.Z. Nozik, E.F. Tretyakov and V.S. Kosik, Phys. Lett. 94B (1980) 266. [2] A. Cisneros, Astrophys. Space Sci. 10 (1971) 87, R.B. Clark and D. Pedigo, Phys. Rev. D8 (1973) 2261, D. Leiter and N. Glass, Phys. Rev. D16 (1977) 3380. [3] J.A. Morgan, Mon. Not. R. Astr. Soc. 194 (1981), to be published. [4] C.L. Cowan and F. Reines, Phys. Rev. 107 (1957) 528, J. Bernstein, M. Ruderman, and G. Feinberg, Phys. Rev. 132 (1963) 1227. [5] A. Salam, Nuovo Cimento 5 (1957) 299. [6] S. Weinberg, Gravitation and cosmology (Wiley, New York, 1973). [7] J. Yang, D.N. Schramm, G. Steigman and R.T. Rood, Astrophys. J. 227 (1979) 697. [8] H. Bethe, Proc. Camb. Phil. Soc. 31 (1935) 108. [9] C.G. Callen, R.F. Dashen and D.J. Gross, Phys. Rev. D 19 (1979) 1826. [ 10] D.A. Dicus, J.C. Pati and V.L. Teplitz, Phys. Rev. D 21 (1980) 922. [11] R.V. Wagoner and G. Steigman, Phys. Rev. D 20 (1979) 825. [12] H.Y. Chiu, Stellar physics (BlaisdeU, London). [13] J.E. Kim, Phys. Rev. D 14 (1976) 3000. [14] S.L. Shapiro, S.A. Teukolsky and I. Wasserman, Phys. Rev. Lett. 45 (1980) 669.