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Stellar limits on keV pseudoscalars
Volume 218, number l
PHYSICS LETTERS B
9 February 1989
S T E L L A R L I M I T S O N keV P S E U D O S C A L A R S ~ Eric D. C A R L S O N 1 and Pierre SALATI z,2 Department of Physics and Lawrence Berkeley Laboratory. Universityof California, Berkeley, CA 94720, USA Received 14 November 1988
Based on the decay rate of orthopositronium and go- 2 for the electron, Samuel has suggestedthe existenceof a new pseudoscalar with coupling to electrons o~'--2× l0 s and mass less than 5 keV. We demonstrate that this contradicts stellar models of the sun and horizontal branch stars. For the coupling required to explain g~-2 and positronium, we demonstrate that a mass of at least 34 keV ( 175 keV) is required to avoid ruining models of the sun (horizontal branch stars).
1. Introduction Samuel has recently suggested [ 1 ] the existence of a new pseudoscalar particle R with coupling to electrons o ~ ' = g 2 / 4 n ~ - 2 × l O -8 and mass m R < 5 keV. This paper will demonstrate that such a particle would ruin stellar models applied to the sun and to horizontal branch stars. In particular, both the sun and horizontal branch stars would have more of the heat in their cores transported by the new pseudoscalar than by radiation or convection. For horizontal branch stars, this would spoil successful predictions of how these stars age. For the sun, this would spoil successful predictions of either the age or the initial hydrogen abundance. Therefore, no such pseudoscalar is allowed. The evidence for such a particle is primarily based on a m e a s u r e m e n t of the lifetime of orthopositroni u m by Westbrook et al. [2]. The measured value of the decay rate is 2 (exp.) = 7.0516 ( 13 ) MHz, which can be compared with the theoretical value 2(th. ) = 7.03830(7) MHz. It should be noted that the theo,~ This work was supported in part by the Director, Office of Energy Research, Office of High Energyand Nuclear Physics, Division of High EnergyPhysicsof the US Department of Energy under Contract DE-AC03-76SF00098and in part by the National ScienceFoundation under grant PHY85-15857. Miller Research Fellow at the University of California at Berkeley. 2 On leaveof absencefrom LAPP and from Universit6de Savoie; F-73000 Chambdry, France. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
retical calculation does not include the two-loop contribution in its value or its error. This ten standard deviation discrepancy was resolved by Samuel by introducing the aforementioned pseudoscalar. A light pseudoscalar allows an additional decay channel Ps-~TR with rate 2 R ( t h . ) = 7.34 × 105 o~'MHz. Samuel correctly points out that R also contributes to ae = (ge - 2 ) / 2 for the electron. The contribution for a very light pseudoscalar is given by a ~ ( t h . ) = - o ~ ' / 4 n . Samuel has argued [ 3 ] that there may be a discrepancy in the g o - 2 of the electron of size a e ( t h . ) a¢(exp.) = 6 0 4 ( 1 4 9 ) × 10 -12, and this discrepancy can also be resolved by the pseudoscalar R. Assuming that the correct values of these discrepancies lie within three standard deviations of the measured discrepancies, we find (in the limit of a very light pseudoscalar) c~' = 1.3× 10 - s .
(1.1)
Samuel then points out that, via a triangle diagram, the pseudoscalar decays to two photons with the lifetime
\ mR /
\sin-I(mR/2me)J (1.2)
Then the SLAC beam d u m p experiment [4] can be used to place a mass limit on the pseudoscalar of 79
Volume 218, number 1
mR ~<7.0 keV
1 " ~ _G! ' 0-8 "
PHYSICS LETTERS B
9 February 1989
.R /
(1.3)
\
/ /
e
/
-
/
e
,,e
t
e
,
Ze
2. Production rates in stars
~ 4-
.Ze . .." R / / /
The sun and other stars have been used [ 5 ] to constrain the properties of various particles, most notably axions and light neutrinos. However, those particles have either an extremely small mass or a weak coupling to fermions. In contrast, the mass m R and the electron coupling a ' of the pseudoscalar R are free parameters. Rather than limiting ourselves to the small range of values advocated by Samuel, we have decided to explore the entire domain of phase space which can be ruled out on the basis of stellar arguments. In particular, we analyze the problem of energy transfer inside stellar cores due to pseudoscalar particles more massive than the ambient temperature, where Boltzmann suppression plays a crucial role. The first three processes of fig. 1 contribute to the creation and destruction of R particles. These reactions and elastic scattering (fig. 1d) attempt to bring the R particles into thermal equilibrium with the rest of the star. For the conditions of stellar interiors ( T ~ 1-10 keV a n d p - 102-104g/cm3), the Compton reaction (fig. la) dominates over elastic scattering (fig. ld) (which is suppressed relative to fig. la by a factor a ' / a ) and over photon fusion (fig. 1c). In addition, for very light pseudoscalars, the Compton reaction (fig. la) and Bremsstrahlung (fig. lb) have been shown to occur at comparable rates [ 6 ]. As the mass of the pseudoscalar increases, the relative importance of the Compton reaction as compared to Bremsstrahlung also increases. For a mass significantly larger than the temperature of the medium in which the R particles propagate, the Compton reaction is always the dominant process responsible for the creation, destruction and thermal equilibrium of the pseudoscalars. Since we have only included in our calculations the effects of this reaction, our results should be considered as approximations, with likely errors of at most an order of magnitude. In the limit where the pseudoscalars have a mean free path much greater than the size of the star (i.e. c~' < 10- t ~for the sun and c~' < 10- ~5 for the core of a typical horizontal branch star), these particles are 80
/
e
4~
e
I
e
0
b R~.
+
,R x
T~ - _ - R T
e
T ~::>----R
R,
1 "x
i
r"
",
e
+
..R
x
I \
I x x
e
C
,
,
e
d Fig. 1. Processes contributing to the thermal and chemical equilibrium of pseudoscalars: Compton process (a), Bremsstrahlung (b), photon fusion (c), and elastic scattering (d).
thermally photo-produced inside the stellar core and freely escape, providing the star with an additional heat loss mechanism. If we treat the electrons as highly non-relativistic, then the rate at which energy escapes from the star is given, per unit mass, by T6 {mR'~ 2zc m4~m~,g~-) '
~R = (1 + X H ) p G O L '
(2.1)
where T and XH are the temperature and hydrogen mass fraction of the stellar matter and P is the suppression factor due to Fermi blocking of the electron final state. The thermodynamical function g accounts for the fact that the pseudoscalar mass mR and the temperature T are comparable: ~o
g ( a ) = .f (Y4+~x4) x d y eY_l ,
(2.2)
a
with x = x / y = - a 5. In this regime, the rate at which energy escapes is proportional to a ' because the bigger a ' is, the more particles are produced. On the other hand, in the limit where the pseudo-
Volume 218, number 1
PHYSICS LETTERSB
scalars have a mean free path much shorter than the size of the star, they are reabsorbed soon after they have been Compton produced. They transport heat on a distance typically given by their mean free path inside the stellar medium, therefore providing the star with an additional heat transport mechanism. In the presence of a temperature gradient, the m o m e n t u m distribution of the pseudoscalars departs from its isotropic equilibrium value and energy is carried by the pseudoscalars with a flux 1 SR =
mamp
6rC3OLC~' p ( 1 + X H ) P
rC('R J\T-,]
VT,
(2.3)
where p is the mass density a n d f i s defined by f ( a ) = i ( Y ~y ~_7~x4) (e,._ ) ~e ....1
x3 dy.
(2.4)
a
In this regime, the rate at which energy is transported is inversely proportional to the mean free path of the pseudoscalars and therefore to a ' because the smaller a ' is, the more transparent the core is to energy flow. In the following sections, we will apply these calculations to the sun and to horizontal branch stars. By asking that these stars should not be upset by the pseudoscalars R, we will derive bounds on the coupling of the particles ce' as a function of their mass
9 February 1989
Solar models require that the effective opacity, which is defined by (3.2)
Neffective
should be -~ 1 cm2/g. If the opacity drops below this value, the primordial hydrogen abundance has to be significantly increased in order for solar models to reproduce the real sun [7]. If Jqfrectiveis too low, the pseudoscalars will run into trouble because the primordial hydrogen abundance will soon exceed 90%, a value incompatible both with primordial nucleosynthesis and with the observations of the present hydrogen abundance in the universe. By demanding that the effective opacity should not drop by more than a factor of 2 below its standard value offer= 1 cm2/g (i.e. /(TR> 1 cm2/g) and taking Xn=0.35, T = 1.35 keV and P=0.93, we obtain the limit (3.3)
a ' > 106.4f(1.3~keX/) .
In the small a ' regime, the mean free path of the pseudoscalars is much larger than the solar radius and the particles will escape the sun without contributing Limits from the sun
mR.
-5
3. L i m i t s from the sun
The sun, being the most studied and best understood star, provides the most reliable astrophysical limit on pseudoscalars. Its mass ( M o - ~ 2 × 1033 g), its age ( ~ 4 . 5 × 10 9 yr) and its luminosity (Lo-~ 3.9 × 1033 erg/s) are well known, and any solar model has at least to reproduce these features in order to be successful. In the large o~' regime, the pseudoscalars supplement radiation transport inside the solar core with an additional process of heat diffusion. This translates into a decrease in the effective opacity of the stellar medium at the center of the sun. The equivalent opacity KR, which describes the diffusion of heat by the pseudoscalars, is defined by 4~ 2 T 3
SR . . . . VT. 45 pKR
(3.1)
-10 o o~ -15
-2o allowed
-55
....
I ....
10
I ....
I ....
20 30 mR (keY)
I ....
40
50
Fig. 2. Limits on the mass and coupling of the pseudoscalar R, coming from the sun and the SLAC beam dump experiment. The solid curve encloses a region where pseudoscalars would ruin standard solar models. The dotted curve is the boundary between the free streaming regime and the heat diffusion regime. The region above the dashed curve is excluded by the SLAC beam dump experiment. The dot-dashed curve is the value required by Samuel to explain g e - 2 and positronium.
81
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9 February 1989
to its radiative luminosity. The sun will therefore have to use its nuclear fuel to power both its visible and invisible radiations. It will then age much faster than in the standard case and will burn all its hydrogen on a time scale much shorter than its expected lifespan of 9 × 10 9 yr. If the energy carried off in pseudoscalars exceeds the energy which, in standard solar models [ 8 ], is produced at the center, i.e. ~,uc= 17.5 erg/g/s, the sun should have already evolved to the red giant stage and we would not be here discussing the properties of hypothetical pseudoscalars. Imposing the conservative condition CR< 17.5 erg/g/s, we obtain the bound
ficient in transporting heat by conduction, they will suppress core convection, a situation which is unacceptable. The convective part of the core is well modelled by an n = 3/2 polytrope [ 12 ]. The heat carried out of the center by the pseudoscalar obeys relation (2.3) with a temperature profile given, in the presence of convection, by
a,<9.24XlO-2OEg(1
VT= - z~ r Tcentral/3R2.
mR
.35keV]J
-I
"
(3.4)
These two limits are plotted in fig. 2.
6 \R] J"
(4.1)
z~ = 3.6537 is the first zero of the Lane-Emden function 03/2, and R is the radius of the helium core. This leads to the temperature gradient (4.2)
On the other hand, the heat flux carried out of the center by convection is simply given by 47~r3p6nuc
4. Limits from horizontal branch stars
S .... =
Horizontal branch stars, primarily due to their higher central temperatures, usually provide stronger limits on the interactions of weakly coupled particles. Unlike the sun, horizontal branch stars have convective cores, which exhibit an onion shell structure [ 9 ]. The helium core, which is burnt during the horizontal stage, extends up to 4 × 109 cm. At its very center, helium is cooked into carbon mainly through the triple ct reaction which is extremely temperature sensitive, since e3~ varies as T 4° [ 10]. The nuclear energy is therefore produced inside a fairly small region of the helium core. Indeed, 80% of the core luminosity originates from the inner ~ 8 × 108 cm. Convection and associated semi-convection develop up to 1.5 × 109 cm from the center. This convection is crucial because it constantly replenishes the central combustion region with flesh fuel from the more extended convective zone, which acts as a reservoir. If the core convection is suppressed, the star is starved since the fuel supply will be limited to the small amount in the center. The star will therefore pass through the horizontal branch much faster than indicated by standard stellar models. However, stellar counts in nearby globular clusters definitely point to the occurrence of convection ~. In the large a' regime, if pseudoscalars are too el'-
where enuc= 104 erg/g/s is the energy generated per unit mass at the center of the HB star. If, in the presence of convection, pseudoscalars are potentially able to withdraw from the center more energy than is actually produced by nuclear reactions, that is if SR (with convection) > & .... then there will be no need at all for convection to develop. The helium core will just be purely radiative, a large part of the energy being transported by the pseudoscalars. In that case, since these particles will lead to a contradiction with the observed existence of convection inside HB stars, they will run into trouble. By demanding that SR, in the presence of convection, be less than & .... we obtain a limit
~t Seeref. [11] for a review.
82
4rcr 2
=lrpe . . . .
°~' > 0"0251 f ( 1 0 ~ k e V )
-
(4.3)
(4.4)
We have used R = 5 X 1 0 4 km, p = 2 X 1 0 4 g/cm 3, XH=0, P=0.75, and T = 10.3 keV. Raffelt and Dearborn [ 13 ] have argued, in the case where the mean free path is long compared to the core size, that the energy carried offby exotic particles in the core cannot exceed the energy production per unit mass averaged over the volume of the core (e,uc) = 100 erg/g/s. Indeed, if on the one hand eR only depends gently on the temperature (i.e. eR0CT 6 ) , and on the other hand, E3~ is extremely temperature sensitive, then the HB star will just increase slightly
Volume 218, number 1 Limits from 0
.
.
.
PHYSICS LETTERS B
Horizontal .
.
.
.
.
Branch .
.
.
unknown experimental error. At present these represent the only plausible explanations. Finally, this fairly simple analysis has shown once more how powerful stellar arguments can be. The thermodynamics of the inner parts of stars have provided stringent constraints on the properties of the R pseudoscalars.
Stars
.
.
.
-5
o 2 -15
y
-~0
9 February 1989
Acknowledgement --
excluded
allowed
100
200
300
400
m R (keV)
Fig. 3. Limits on the mass and coupling of the pseudoscalar R, coming from horizontal branch stars and the SLAC beam dump experiment. The curves have the same interpretation as in fig. 2.
We would like to thank D. Dearborn for helpful comments. One of us (E.D.C.) would like to thank U. Sarid for bringing to our attention the paper by Samuel. We would like to thank the Miller Institute for Basic Research in Science at the University of California at Berkeley for its generous support.
References its central temperature in order to cope with any heat loss due to pseudoscalars. This will result in a shortening of the lifetime of the HB star. In order for this lifetime not to be decreased by more than a factor of 2, ~R should not exceed 100 erg/g/s, which translates into the limit on c( O(<4"5×10-24
g
10.3keVJJ
"
These two limits are graphed in fig. 3.
5. Conclusions To explain the positronium and g e - 2 discrepancies, a pseudoscalar must have coupling to electrons c ( = 1.3× 10 -8. To avoid contradictions with solar models, this would require mR> 34 keV. To avoid contradictions with horizontal branch star models, we must demand mR> 175 keV. However, the SLAC beam dump experiment forces mR< 7 keV. Therefore, no pseudoscalar can explain the positronium and g e - 2 discrepancies. What is the source of these seeming disagreements with the standard model? As Samuel himself points out, there is some dispute whether there is any g ~ - 2 discrepancy at all. The positronium may be explainable in terms of second order loop effects, or some
[ 1 ] M.A. Samuel, Mod. Phys. Lett. A 11 (1988) 1177. [ 2 ] CA. Westbrook, D.W. Gidley, R.S. Conti and A. Rich, Phys. Rev. Lett. 58 (1987) 1328, [3] M.A. Samuel, Phys. Rev. Len. 57 (1986) 3133. [4]J. Bjorken, Search for neutral penetrating metastable particles produced in the SLAC beam dump, Moriond Conf. (1985). [5] P. Sutherland et al., Phys. Rev. D 13 (1976) 2700; J. Ellis and K.A. Olive, Nucl. Phys. B 223 (1983) 252; D.A. Dicus et al., Phys. Rev. D 18 ( 1978 ) 1829; 22 (1980) 839; M. Fukugita, S. Watamura and M. Yoshimura, Phys. Rev. Lett. 48 (1982) 1522; Phys. Rev. D 26 (1982) 1840; A. Barroso and G.C. Branco, Phys. LeU. B 116 ( 1982 ) 247; A. Pantziris and K. Kang, Phys. Rev. D 33 (1986) 3509; H.M. Georgi, S.L. Glashow and S. Nussinov, Nucl. Phys. B 193 (1981) 297; A. Bouquet and C.E. Vayonakis, Phys. Len. B 116 (1982) 219. [6] G.G. Raffelt, Phys. Rev. D 33 (1986) 897. [7] D.S.P. Dearborn, private communication. [8] S. Turck-Chi6ze, S. Cahen, M. Cass6 and C. Doom, Revisiting the standard solar model, submitted to Astrophys. J. [ 9 ] A.V. Sweigart and P.G. Gross, Astrophys. J. 190 (1974) 10 I. [10]D.D. Clayton, Principles of stellar evolution and nucleosynthesis (McGraw-Hill, New York, 1968 ). [ 11 ] I. lben Jr. and A. Renzini, Phys. Rep. 105 (1984) 329. [12 ] S. Chandrasekhar, An introduction to the study of stellar structure (University of Chicago Press, Chicago, 1939). [ 13 ] G.G. Raffeh and D.S.P. Dearborn, preprint UCRL-96457 (July 1987).
83
PHYSICS LETTERS B
9 February 1989
S T E L L A R L I M I T S O N keV P S E U D O S C A L A R S ~ Eric D. C A R L S O N 1 and Pierre SALATI z,2 Department of Physics and Lawrence Berkeley Laboratory. Universityof California, Berkeley, CA 94720, USA Received 14 November 1988
Based on the decay rate of orthopositronium and go- 2 for the electron, Samuel has suggestedthe existenceof a new pseudoscalar with coupling to electrons o~'--2× l0 s and mass less than 5 keV. We demonstrate that this contradicts stellar models of the sun and horizontal branch stars. For the coupling required to explain g~-2 and positronium, we demonstrate that a mass of at least 34 keV ( 175 keV) is required to avoid ruining models of the sun (horizontal branch stars).
1. Introduction Samuel has recently suggested [ 1 ] the existence of a new pseudoscalar particle R with coupling to electrons o ~ ' = g 2 / 4 n ~ - 2 × l O -8 and mass m R < 5 keV. This paper will demonstrate that such a particle would ruin stellar models applied to the sun and to horizontal branch stars. In particular, both the sun and horizontal branch stars would have more of the heat in their cores transported by the new pseudoscalar than by radiation or convection. For horizontal branch stars, this would spoil successful predictions of how these stars age. For the sun, this would spoil successful predictions of either the age or the initial hydrogen abundance. Therefore, no such pseudoscalar is allowed. The evidence for such a particle is primarily based on a m e a s u r e m e n t of the lifetime of orthopositroni u m by Westbrook et al. [2]. The measured value of the decay rate is 2 (exp.) = 7.0516 ( 13 ) MHz, which can be compared with the theoretical value 2(th. ) = 7.03830(7) MHz. It should be noted that the theo,~ This work was supported in part by the Director, Office of Energy Research, Office of High Energyand Nuclear Physics, Division of High EnergyPhysicsof the US Department of Energy under Contract DE-AC03-76SF00098and in part by the National ScienceFoundation under grant PHY85-15857. Miller Research Fellow at the University of California at Berkeley. 2 On leaveof absencefrom LAPP and from Universit6de Savoie; F-73000 Chambdry, France. 0 3 7 0 - 2 6 9 3 / 8 9 / $ 03.50 © Elsevier Science Publishers B.V. ( N o r t h - H o l l a n d Physics Publishing Division )
retical calculation does not include the two-loop contribution in its value or its error. This ten standard deviation discrepancy was resolved by Samuel by introducing the aforementioned pseudoscalar. A light pseudoscalar allows an additional decay channel Ps-~TR with rate 2 R ( t h . ) = 7.34 × 105 o~'MHz. Samuel correctly points out that R also contributes to ae = (ge - 2 ) / 2 for the electron. The contribution for a very light pseudoscalar is given by a ~ ( t h . ) = - o ~ ' / 4 n . Samuel has argued [ 3 ] that there may be a discrepancy in the g o - 2 of the electron of size a e ( t h . ) a¢(exp.) = 6 0 4 ( 1 4 9 ) × 10 -12, and this discrepancy can also be resolved by the pseudoscalar R. Assuming that the correct values of these discrepancies lie within three standard deviations of the measured discrepancies, we find (in the limit of a very light pseudoscalar) c~' = 1.3× 10 - s .
(1.1)
Samuel then points out that, via a triangle diagram, the pseudoscalar decays to two photons with the lifetime
\ mR /
\sin-I(mR/2me)J (1.2)
Then the SLAC beam d u m p experiment [4] can be used to place a mass limit on the pseudoscalar of 79
Volume 218, number 1
mR ~<7.0 keV
1 " ~ _G! ' 0-8 "
PHYSICS LETTERS B
9 February 1989
.R /
(1.3)
\
/ /
e
/
-
/
e
,,e
t
e
,
Ze
2. Production rates in stars
~ 4-
.Ze . .." R / / /
The sun and other stars have been used [ 5 ] to constrain the properties of various particles, most notably axions and light neutrinos. However, those particles have either an extremely small mass or a weak coupling to fermions. In contrast, the mass m R and the electron coupling a ' of the pseudoscalar R are free parameters. Rather than limiting ourselves to the small range of values advocated by Samuel, we have decided to explore the entire domain of phase space which can be ruled out on the basis of stellar arguments. In particular, we analyze the problem of energy transfer inside stellar cores due to pseudoscalar particles more massive than the ambient temperature, where Boltzmann suppression plays a crucial role. The first three processes of fig. 1 contribute to the creation and destruction of R particles. These reactions and elastic scattering (fig. 1d) attempt to bring the R particles into thermal equilibrium with the rest of the star. For the conditions of stellar interiors ( T ~ 1-10 keV a n d p - 102-104g/cm3), the Compton reaction (fig. la) dominates over elastic scattering (fig. ld) (which is suppressed relative to fig. la by a factor a ' / a ) and over photon fusion (fig. 1c). In addition, for very light pseudoscalars, the Compton reaction (fig. la) and Bremsstrahlung (fig. lb) have been shown to occur at comparable rates [ 6 ]. As the mass of the pseudoscalar increases, the relative importance of the Compton reaction as compared to Bremsstrahlung also increases. For a mass significantly larger than the temperature of the medium in which the R particles propagate, the Compton reaction is always the dominant process responsible for the creation, destruction and thermal equilibrium of the pseudoscalars. Since we have only included in our calculations the effects of this reaction, our results should be considered as approximations, with likely errors of at most an order of magnitude. In the limit where the pseudoscalars have a mean free path much greater than the size of the star (i.e. c~' < 10- t ~for the sun and c~' < 10- ~5 for the core of a typical horizontal branch star), these particles are 80
/
e
4~
e
I
e
0
b R~.
+
,R x
T~ - _ - R T
e
T ~::>----R
R,
1 "x
i
r"
",
e
+
..R
x
I \
I x x
e
C
,
,
e
d Fig. 1. Processes contributing to the thermal and chemical equilibrium of pseudoscalars: Compton process (a), Bremsstrahlung (b), photon fusion (c), and elastic scattering (d).
thermally photo-produced inside the stellar core and freely escape, providing the star with an additional heat loss mechanism. If we treat the electrons as highly non-relativistic, then the rate at which energy escapes from the star is given, per unit mass, by T6 {mR'~ 2zc m4~m~,g~-) '
~R = (1 + X H ) p G O L '
(2.1)
where T and XH are the temperature and hydrogen mass fraction of the stellar matter and P is the suppression factor due to Fermi blocking of the electron final state. The thermodynamical function g accounts for the fact that the pseudoscalar mass mR and the temperature T are comparable: ~o
g ( a ) = .f (Y4+~x4) x d y eY_l ,
(2.2)
a
with x = x / y = - a 5. In this regime, the rate at which energy escapes is proportional to a ' because the bigger a ' is, the more particles are produced. On the other hand, in the limit where the pseudo-
Volume 218, number 1
PHYSICS LETTERSB
scalars have a mean free path much shorter than the size of the star, they are reabsorbed soon after they have been Compton produced. They transport heat on a distance typically given by their mean free path inside the stellar medium, therefore providing the star with an additional heat transport mechanism. In the presence of a temperature gradient, the m o m e n t u m distribution of the pseudoscalars departs from its isotropic equilibrium value and energy is carried by the pseudoscalars with a flux 1 SR =
mamp
6rC3OLC~' p ( 1 + X H ) P
rC('R J\T-,]
VT,
(2.3)
where p is the mass density a n d f i s defined by f ( a ) = i ( Y ~y ~_7~x4) (e,._ ) ~e ....1
x3 dy.
(2.4)
a
In this regime, the rate at which energy is transported is inversely proportional to the mean free path of the pseudoscalars and therefore to a ' because the smaller a ' is, the more transparent the core is to energy flow. In the following sections, we will apply these calculations to the sun and to horizontal branch stars. By asking that these stars should not be upset by the pseudoscalars R, we will derive bounds on the coupling of the particles ce' as a function of their mass
9 February 1989
Solar models require that the effective opacity, which is defined by (3.2)
Neffective
should be -~ 1 cm2/g. If the opacity drops below this value, the primordial hydrogen abundance has to be significantly increased in order for solar models to reproduce the real sun [7]. If Jqfrectiveis too low, the pseudoscalars will run into trouble because the primordial hydrogen abundance will soon exceed 90%, a value incompatible both with primordial nucleosynthesis and with the observations of the present hydrogen abundance in the universe. By demanding that the effective opacity should not drop by more than a factor of 2 below its standard value offer= 1 cm2/g (i.e. /(TR> 1 cm2/g) and taking Xn=0.35, T = 1.35 keV and P=0.93, we obtain the limit (3.3)
a ' > 106.4f(1.3~keX/) .
In the small a ' regime, the mean free path of the pseudoscalars is much larger than the solar radius and the particles will escape the sun without contributing Limits from the sun
mR.
-5
3. L i m i t s from the sun
The sun, being the most studied and best understood star, provides the most reliable astrophysical limit on pseudoscalars. Its mass ( M o - ~ 2 × 1033 g), its age ( ~ 4 . 5 × 10 9 yr) and its luminosity (Lo-~ 3.9 × 1033 erg/s) are well known, and any solar model has at least to reproduce these features in order to be successful. In the large o~' regime, the pseudoscalars supplement radiation transport inside the solar core with an additional process of heat diffusion. This translates into a decrease in the effective opacity of the stellar medium at the center of the sun. The equivalent opacity KR, which describes the diffusion of heat by the pseudoscalars, is defined by 4~ 2 T 3
SR . . . . VT. 45 pKR
(3.1)
-10 o o~ -15
-2o allowed
-55
....
I ....
10
I ....
I ....
20 30 mR (keY)
I ....
40
50
Fig. 2. Limits on the mass and coupling of the pseudoscalar R, coming from the sun and the SLAC beam dump experiment. The solid curve encloses a region where pseudoscalars would ruin standard solar models. The dotted curve is the boundary between the free streaming regime and the heat diffusion regime. The region above the dashed curve is excluded by the SLAC beam dump experiment. The dot-dashed curve is the value required by Samuel to explain g e - 2 and positronium.
81
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to its radiative luminosity. The sun will therefore have to use its nuclear fuel to power both its visible and invisible radiations. It will then age much faster than in the standard case and will burn all its hydrogen on a time scale much shorter than its expected lifespan of 9 × 10 9 yr. If the energy carried off in pseudoscalars exceeds the energy which, in standard solar models [ 8 ], is produced at the center, i.e. ~,uc= 17.5 erg/g/s, the sun should have already evolved to the red giant stage and we would not be here discussing the properties of hypothetical pseudoscalars. Imposing the conservative condition CR< 17.5 erg/g/s, we obtain the bound
ficient in transporting heat by conduction, they will suppress core convection, a situation which is unacceptable. The convective part of the core is well modelled by an n = 3/2 polytrope [ 12 ]. The heat carried out of the center by the pseudoscalar obeys relation (2.3) with a temperature profile given, in the presence of convection, by
a,<9.24XlO-2OEg(1
VT= - z~ r Tcentral/3R2.
mR
.35keV]J
-I
"
(3.4)
These two limits are plotted in fig. 2.
6 \R] J"
(4.1)
z~ = 3.6537 is the first zero of the Lane-Emden function 03/2, and R is the radius of the helium core. This leads to the temperature gradient (4.2)
On the other hand, the heat flux carried out of the center by convection is simply given by 47~r3p6nuc
4. Limits from horizontal branch stars
S .... =
Horizontal branch stars, primarily due to their higher central temperatures, usually provide stronger limits on the interactions of weakly coupled particles. Unlike the sun, horizontal branch stars have convective cores, which exhibit an onion shell structure [ 9 ]. The helium core, which is burnt during the horizontal stage, extends up to 4 × 109 cm. At its very center, helium is cooked into carbon mainly through the triple ct reaction which is extremely temperature sensitive, since e3~ varies as T 4° [ 10]. The nuclear energy is therefore produced inside a fairly small region of the helium core. Indeed, 80% of the core luminosity originates from the inner ~ 8 × 108 cm. Convection and associated semi-convection develop up to 1.5 × 109 cm from the center. This convection is crucial because it constantly replenishes the central combustion region with flesh fuel from the more extended convective zone, which acts as a reservoir. If the core convection is suppressed, the star is starved since the fuel supply will be limited to the small amount in the center. The star will therefore pass through the horizontal branch much faster than indicated by standard stellar models. However, stellar counts in nearby globular clusters definitely point to the occurrence of convection ~. In the large a' regime, if pseudoscalars are too el'-
where enuc= 104 erg/g/s is the energy generated per unit mass at the center of the HB star. If, in the presence of convection, pseudoscalars are potentially able to withdraw from the center more energy than is actually produced by nuclear reactions, that is if SR (with convection) > & .... then there will be no need at all for convection to develop. The helium core will just be purely radiative, a large part of the energy being transported by the pseudoscalars. In that case, since these particles will lead to a contradiction with the observed existence of convection inside HB stars, they will run into trouble. By demanding that SR, in the presence of convection, be less than & .... we obtain a limit
~t Seeref. [11] for a review.
82
4rcr 2
=lrpe . . . .
°~' > 0"0251 f ( 1 0 ~ k e V )
-
(4.3)
(4.4)
We have used R = 5 X 1 0 4 km, p = 2 X 1 0 4 g/cm 3, XH=0, P=0.75, and T = 10.3 keV. Raffelt and Dearborn [ 13 ] have argued, in the case where the mean free path is long compared to the core size, that the energy carried offby exotic particles in the core cannot exceed the energy production per unit mass averaged over the volume of the core (e,uc) = 100 erg/g/s. Indeed, if on the one hand eR only depends gently on the temperature (i.e. eR0CT 6 ) , and on the other hand, E3~ is extremely temperature sensitive, then the HB star will just increase slightly
Volume 218, number 1 Limits from 0
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PHYSICS LETTERS B
Horizontal .
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unknown experimental error. At present these represent the only plausible explanations. Finally, this fairly simple analysis has shown once more how powerful stellar arguments can be. The thermodynamics of the inner parts of stars have provided stringent constraints on the properties of the R pseudoscalars.
Stars
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-5
o 2 -15
y
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9 February 1989
Acknowledgement --
excluded
allowed
100
200
300
400
m R (keV)
Fig. 3. Limits on the mass and coupling of the pseudoscalar R, coming from horizontal branch stars and the SLAC beam dump experiment. The curves have the same interpretation as in fig. 2.
We would like to thank D. Dearborn for helpful comments. One of us (E.D.C.) would like to thank U. Sarid for bringing to our attention the paper by Samuel. We would like to thank the Miller Institute for Basic Research in Science at the University of California at Berkeley for its generous support.
References its central temperature in order to cope with any heat loss due to pseudoscalars. This will result in a shortening of the lifetime of the HB star. In order for this lifetime not to be decreased by more than a factor of 2, ~R should not exceed 100 erg/g/s, which translates into the limit on c( O(<4"5×10-24
g
10.3keVJJ
"
These two limits are graphed in fig. 3.
5. Conclusions To explain the positronium and g e - 2 discrepancies, a pseudoscalar must have coupling to electrons c ( = 1.3× 10 -8. To avoid contradictions with solar models, this would require mR> 34 keV. To avoid contradictions with horizontal branch star models, we must demand mR> 175 keV. However, the SLAC beam dump experiment forces mR< 7 keV. Therefore, no pseudoscalar can explain the positronium and g e - 2 discrepancies. What is the source of these seeming disagreements with the standard model? As Samuel himself points out, there is some dispute whether there is any g ~ - 2 discrepancy at all. The positronium may be explainable in terms of second order loop effects, or some
[ 1 ] M.A. Samuel, Mod. Phys. Lett. A 11 (1988) 1177. [ 2 ] CA. Westbrook, D.W. Gidley, R.S. Conti and A. Rich, Phys. Rev. Lett. 58 (1987) 1328, [3] M.A. Samuel, Phys. Rev. Len. 57 (1986) 3133. [4]J. Bjorken, Search for neutral penetrating metastable particles produced in the SLAC beam dump, Moriond Conf. (1985). [5] P. Sutherland et al., Phys. Rev. D 13 (1976) 2700; J. Ellis and K.A. Olive, Nucl. Phys. B 223 (1983) 252; D.A. Dicus et al., Phys. Rev. D 18 ( 1978 ) 1829; 22 (1980) 839; M. Fukugita, S. Watamura and M. Yoshimura, Phys. Rev. Lett. 48 (1982) 1522; Phys. Rev. D 26 (1982) 1840; A. Barroso and G.C. Branco, Phys. LeU. B 116 ( 1982 ) 247; A. Pantziris and K. Kang, Phys. Rev. D 33 (1986) 3509; H.M. Georgi, S.L. Glashow and S. Nussinov, Nucl. Phys. B 193 (1981) 297; A. Bouquet and C.E. Vayonakis, Phys. Len. B 116 (1982) 219. [6] G.G. Raffelt, Phys. Rev. D 33 (1986) 897. [7] D.S.P. Dearborn, private communication. [8] S. Turck-Chi6ze, S. Cahen, M. Cass6 and C. Doom, Revisiting the standard solar model, submitted to Astrophys. J. [ 9 ] A.V. Sweigart and P.G. Gross, Astrophys. J. 190 (1974) 10 I. [10]D.D. Clayton, Principles of stellar evolution and nucleosynthesis (McGraw-Hill, New York, 1968 ). [ 11 ] I. lben Jr. and A. Renzini, Phys. Rep. 105 (1984) 329. [12 ] S. Chandrasekhar, An introduction to the study of stellar structure (University of Chicago Press, Chicago, 1939). [ 13 ] G.G. Raffeh and D.S.P. Dearborn, preprint UCRL-96457 (July 1987).
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