Primordial nucleosynthesis corners the Z′

Volume 241, number

3

PRIMORDIAL

PHYSICS

NUCLEOSYNTHESIS

LETTERS

B

17 May 1990

CORNERS THE Z’

Jorge L. LOPEZ ’ and D.V. NANOPOULOS

2,3

Centerfor Theoretical Physics, Department ofPhysics, Texas A&M University, College Station, TX 77843-4242. USA Received

28 November

1989; revised manuscript

received

18 December

1989

We investigate the implications on the Z’ gauge boson parameters that come from enforcing the primordial nucleosynthesis constraints on the effective number of neutrino-like degrees of freedom in the presence of a Z’ that couples to three types of light right-handed neutrinos. We obtain stringent bounds on the Z-Z’ mixing angle, -0.004Sog0, and on the mass of the heavy eigenstate, M& 1 TeV. These bounds are stronger than the current experimental limits derived from precise measurements of the electroweak gauge boson masses. In this case we also find that the effect of Z-Z’ mixing on the Z width is completely negligible, and hence that it could not account for any small (positive or negative) shift on the Z width with respect to the standard model value.

A repeated theme in many extensions of the standard model (SM ) is the possible existence of extra neutral (Z’ ) gauge bosons, not necessarily much heavier than the SM Z. In particular, in many superstring models [ 1 ] an extra Z’ is present naturally. In the last few years, a lot of effort has been devoted to constrain the mass of the Z’ and its couplings to the SM particles. Limits on the Z’ parameters come from different places, including precise measurements of A4,, Mw, and sin2& [ l-7 ] from particle physics, and restrictions on its couplings to new type of (e.g., righthanded) neutrinos from primordial nucleosynthesis [ 8,9]. In this paper we investigate the implications on the Z’ gauge boson parameters that come from enforcing the primordial nucleosynthesis constraints on the effective number of neutrino-like degrees of freedom, AN !” G 1, in the presence of a Z’ that couples to three types of light right-handed neutrinos. We obtain stringent bounds on the Z’ parameters which are stronger than the current experimental limits [ 6,7] derived from precise measurements of the electroweak gauge boson masses [ lo- 12 1. Furthermore, we find that in this case the effect of Z-Z’ mixing on the Z width is completely negligible. This precludes at-

tempts to ascribe a possible shift on the Z width, with respect to the SM value, to Z-Z’ mixing. Our results then cast serious doubts on the existence of such a Z’ with couplings to light right-handed neutrinos. The model we use for our analysis is the E6 superstring model of ref. [ 3 1. (Also known as model A in ref. [ 41 or the t-l model in refs. [ 5-71.) However, we believe our results are qualitatively model independent. The generic features of this type of models we exploit are: the downward shift in the mass of the light eigenstate Z, (the one the model predicts is being observed in e+e- collisions at &EM,: the heavy eigenstate is denoted by Z,) due to Z-Z’ mixing, and the existence of three types of light right-handed neutrinos which couple only to the Z’. The angle @ rotates from the neutral current eigenstates (Z, Z’ ) to the mass eigenstates (Z,, Z,). The allowed sign of @ in this model is obtained by examining the mass formulas for M, and M2 (see appendix B of ref. [ 41) =l+i

M2

’

sin0,(4-5r)

tan@,

=l-fsin&(4-55)cot@,

MZ (->

’ Supported by an ICSC-World Laboratory Scholarship. * Supported in part by DOE grant DE-AC02-76ER008 3 HARC fellow.

392

(la) (lb)

where <= V2/ ( v2+ fi2), with 0, v the vacuum expectation values of the two Higgs doublets giving masses

I.

0370-2693/90/$

03.50 0 Elsevier Science Publishers

B.V. (North-Holland

)

V o l u m e 241, n u m b e r 3

1 7 M a y 1990

PHYSICS LETTERS B

Table 1 The couplings o f the Z (ge, rt) a n d Z ' (g[.R) to the s t a n d a r d m o d e l fermions. Fermion

gL

gR

gL

g~

v~, v., v. e, g, z u, c, t d, s, b

1/2 - 1/2+sin20w 1 / 2 - ( 2 / 3 ) sin20, - 1 / 2 + ( 1 / 3 ) sin20~

0 sinZ0w - ( 2 / 3 ) sin20~ ( 1 / 3 ) sin20w

- 1/6 - 1/6 1/3 1/3

-5/6 - 1/3 - 1/3 1/6

to the bottom and top quarks respectively. The dynamical calculations of ref. [3] show that 0.2~< IUvl ~<0.6 in order to obtain an acceptable pattern of low-energy phenomenology. This implies 0.04~< ~<0.27 and hence M~ ~
(37.281 GeV) 2

v/2Gvm~v( 1 - Ar) ~ M2w(1 - Ar) '

Mw

- COS 0w '

where Ar contains the electroweak radiative corrections and depends on m, and mH. From the experimentally observed value of M~ one can then obtain ME and 0 (as functions o f f / v ) through ( 1 ). Taking the central values of Mt = 91.10 GeV [ 10,12 ] and Mw=80.0 GeV [ l l ] , and Ar=0.0606 [7] (which corresponds to mr= 90 GeV and m , = 100 GeV) one gets #1

~kN v _=

FA(Z 1--.all) - FSM (Z--+all) FSM ( Z ~} VL~' L )

= 4 Z [--(g2L+g2) sin20 + (g~ + g ~ ) sin2¢~ sin20w

+ (gLg'L+gRg'R)sin 20 sin 0w] Q C D ,

where the sum runs over all the SM fermions (excluding the top quark), QCD = 1 for leptons and 1 + as/Jr= 1.0382 for quarks, and the couplings gL,R, gLR are given in table 1 [2 ]. The result as a function of¢~ is shown in fig. 1. Negative values o f ~ (the only allowed ones) give ANy>0 for 101 not too large. In fact, for ~ ~ - 0.1, ANy ~ 0.1 and hence the Z - Z ' mixing could account for a possibly observed effect. However, this situation seems rather unlikely if the gauge boson masses settle around their current central values and the bound on ¢~ in (2) applies, or in any case in light of the primordial nucleosynthesis constraints obtained below (8). In this case the contribution of the mixing to the Zl width is negligible Lo

--0.007<0~<0,

M2>0.7TeV.

(3)

....

I ....

I ....

I ....

I '

(2)

The next thing we want to determine is the effect of Z - Z ' mixing on the width of the lighter mass eigenstate ZI. Recent studies of the Z width by SLC [ 10 ] and LEP [ 12 ] indicate that Nv is definitely less than four, with the most accurate measurement yet being Nv = 3.01 +_0.15 _+0.05 [ 13 ] (the first error is experimental while the second is theoretical). We obtain the following expression for A N ~ - N ~ - 3 (see also ref. [ 6 ] ): ~J Calculations allowing for the uncertainties in Mw and M~ have been p e r f o r m e d in refs. [6,7].

0,5

&N,,

o.o

.

.

.

.

.

.

.

.

-

-0.5

-t.o -0.3

....

I .... -0.2

I .... -O.I

I .... 0

I , 0.1

(rad) Fig. I. The additional number of neutrino-like species above N v = 3 as a function of the mixing angle 0-

393

Volume 241, number 3

PHYSICS LETTERS B

[ i.e., equivalent to ANy ~ 0.013, 0.007 when eqs. ( 2 ) or ( 8 ) apply respectively ]. A possible value ANv> 0 would then have to be explained as in ref. [ 14 ] by resorting to the lightest s u p e r s y m m e t r i c particle. F u r t h e r m o r e , a small n e g a t i v e value o f AN~ could not be explained in this model, since this would require q~>0 or 0 < - 0 . 2 . We now turn to the implications on the Z' p a r a m eters due to p r i m o r d i a l nucleosynthesis [ 15 ]. These calculations d e p e n d in an essential way on the effective n u m b e r of neutrino-like degrees o f freedom ~2 AN PN. Similar analyses for the cases AN PN = 1.0, 1.6 and AN PN ~ = 2.5 have been p e r f o r m e d in ref. [ 9 ] and ref. [8] respectively ~3. P r i m o r d i a l nucleosynthesis calculations o f light element a b u n d a n c e s in the big bang model [ 16 ] have been traditionally successful when the n u m b e r o f effective neutrino-like degrees o f freedom does not exceed N ~ N = 3 significantly [9 ], i.e., 0~< AN PN ~< 1. Extra feebly interacting light particles can, however, still be a c c o m m o d a t e d as long as they are effectively suppressed by decoupling at sufficiently early times before nucleosynthesis starts ( t ~ l s ) [17]. The c o n t r i b u t i o n o f the three species o f left- and right-handed neutrinos to the total n u m b e r o f effective degrees o f freedom is given by [ 17 ]

17 May 1990 4

4

gv(T) = 3-7(-T~) +3.~(~) 4

4

= 7(~)[3+3(~,~)

] .

(4)

It thus follows that

AN ~

-

\ TvL /

"

W h e n the weak interactions decouple at t e m p e r a t u r e Td ~ 3.5 MeV [ 18 ], the neutron to proton ratio freezes out and nucleosynthesis starts [ 15 ]. It also follows that at decoupling, TvL = Td. If the interactions o f the VR are weak enough, they will not be able to keep up with the expansion rate o f the U n i v e r s e a n d the VR will decouple at a t e m p e r a t u r e T~ >> Td, high enough for the VL tO have undergone m a n y reheatings relative to the VR. In this way it is possible to get small values o f A N ~ N . The first two columns of table 2 show this relationship ( 5 ) for various values o f AN PN. To relate these cosmological considerations to a specific particle physics model we need to know the decoupling t e m p e r a t u r e T~ for a given value o f AN PvN • We get Tdt from the exact numerical calculations o f ref. [ 17 ] which give (TvR/Td) as a function o f T~. The results are shown in the third column o f table 2. The uncertainty in T~ for some values o f ANPvTM is due to uncertainties in the q u a r k - g l u o n to h a d r o n phase transition, reflected on the choice o f constituent versus current algebra quark masses [ 17 ]. It is clear that there is a m i n i m u m value o f ~kN PN that can be o b t a i n e d by this mechanism, since there

~2 ANPN should not be confused with the above defined ANy, which is just a measure of a shift in the Z~ width with respect to the SM value. ~3 However, errors and omissions in those calculations led to much weaker constraints than the ones found here.

Table 2 Values of the cosmological and Z' parameters for a given value of AN ~N. (GeV)

A N ~ TM

TvR/T,._

T'o

1.0 0.75 0.50 0.35 0.30 0.25 0.22 0.20 0.19

0.76 0.71 0.64 0.58 0.56 0.54 0.52 0.51 0.50

0.15 0.17 0.19 0.21 0.4-0.5 0.5-0.7 1.36 2.63 7.20

394

g~

a'/e

29 34 44 53 63-68 68-74 78 82 86

2.09)< 10-5 1.55)< 10-5 1.21 )< 10-5 1.04)< 10-5 ( 16.2-8.63 ) )< 10-7 (8.63-3.28))< 10-7 4.58 × 10-8 6.53)< 10 -9

3.25)< I0 -~°

I~1max (rad)

M~'i" (TeV)

3.59)< 10-3

1.04 1.12 1.20 1.24 1.98-2.31 2.31-2.95 4.82 7.85 16.6

3.09)< 10 -3 2.73)< 10 -3

2.53)< 10-3 (9.9-7.3))< 10-4 ( 7 . 3 - 4 . 5 ) )< 10-4 1.67 )< 10-4 6.30)< 10-5 1.41 )< 10-5

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PHYSICS LETTERS B

are a finite number of particle thresholds that can be crossed to obtain a smaller T~/T~ L ratio. It is difficult to imagine how one would obtain values of ANPvN much smaller than zXN~ PN ~0.16, where T d = mt ( > M w ) ~4. The particle physics connection is achieved through the definition of the decoupling temperature for a given particle species. At this temperature the expansion rate of the universe H = []nGNp(T)] 1/2 equals the annihilation rate for this type o f particles F= n(av). Writing this relation for both VR and VL we get

Td '

(g'a)l/6( _

_

Td - - \ g d /

~,/3 a

\or'/

(6)

'

where ga = ~ [ 17 ] and g~ are the number of effective degrees of freedom at Td and T~ respectively• The values ofg~ are given in the fourth column of table 2 and have been calculated numerically as a function of T~ in ref. [ 17 ]. Also, we have parametrized 0"~ (O'U)e+e--~VLqL

=otT 2 ,

0" ~ (0"V)e+e-~.,a,~a + (O'V) VLVL~Vava= a ' T 2 .

where

ELa=sinOwg[,R(e) • gL,R(e)

, . , g[(v) I v = s i n ttw gL~VS'

(7d)

r r f = c ° s 2 0 + l~-r-r' f = s i n Z 0 + 1 - r '

(7e)

and r=M2/M 2. Note that when ¢ ~ 0 (r--,0), D E N gives the SM result and N U M ~ 0 ~s. Also, eqs. (7) are valid in the low-energy limit E << Mz. Using ( 1 ), a'/a can be expressed as a function o f ~ and O/v.This result is given in fig. 2 (see also column 5 in table 2). Putting things together, from a given value of ~tN~ N we obtain TvR/T~L (5) and then the VR decoupiing temperature T~ and g~ = g ( T ~ ) . Eq. (6) then gives the required value o f ot'/a=a'/a. From the particle physics end we also obtain a'/a for specific values of the Z' parameters ¢ and Uv (or ~ and M , / M2). The values of M2 obtained this way are plotted in fig. 3 for representative values o f AN ~N. The curves are parametrized by 0.2 ~ 1 T e V .

(8)

We see that the primordial nucleosynthesis conMsThe Z~ exchange diagram vanishes as 0--,0, and so does the Zz exchange diagram since M2-,~.

- - ~ V R 9 R ) "~ 0 " ( V L 9 L ---~VR 9 R )

a ( e + e - ~VLgL )

10-4

(gh(v)~ z NUM sinZ0~kgL(V),/ D E N '

[

. . . .

[

. . . .

[

. . . .

10-5 - - ~ 0 . 6 o'/o

N U M = g ~ ( e ) (sin 2 0 + 2lEg) z + [g,2 (e) +g,2 ( v ) ] (sin 2¢ + 2 f E L ) : , D E N = g ~ (e) [ 2 f + (ER + N) sin 20 +

+ g~(e)[2f+(EL+N)

. . . .

(7a)

with

g4

'

~l]V vPN~< 1 ::=> -- 0 . 0 0 4 ~ ¢ ~ 0 ,

For T>T'd ( > T d ) , a/a'=a/a', and the particle physics connection is made. In the model we are using, a'/a can be straightforwardly calculated and is given by (see also ref. [8] ) a(e+e -

17 May 1990

(7b)

10-8

2lEaN] 2

sin 2 ¢ + 2 f E L N ] 2 ,

(7c)

Thresholds for supersymmetric particles will become relevant at such high values of T~. However, as shown below, the resulting particle physics scenario is so ugly that we disregard this possibility.

10-7

] . . . . [ . -0.004. . . . -0,003 -0.00~

.

. . -0.001

0

¢ (raa)

Fig. 2. The ratio a'la=--((av)e+e_~v.~+(ov)vL~Vr~R)/ (av),+e-~vL,. as a function of the mixing angle 0 and for the dynamically favored range 0.2 ~
Volume 241, number 3 . . . .

PHYSICS LETTERS B I

. . . .

I

. . . .

[

'

,

,

Note added in proof A f t e r s u b m i t t i n g this m a n u script we r e c e i v e d a p a p e r by G o n z a l e z - G a r c i a a n d Valle [ 20 ], w h e r e s i m i l a r c a l c u l a t i o n s are p e r f o r m e d a r r i v i n g at s i m i l a r results. H o w e v e r , those a u t h o r s m a k e invalid c l a i m s a b o u t the correctness o f o u r calCulations. F u r t h e r m o r e , in t h e i r p a p e r they o m i t the a(VLgL--VRgR ) c o n t r i b u t i o n to t 7 ' / a in eq. ( 7 ) . T h i s o m i s s i o n m a k e s t h e i r c o m p u t a t i o n s inaccurate.

,

0.25k

\ 0.30

~o 35 "~o.v5 o.so 1

17 May 1990

/~I~=1.0

References o

. . . .

-0.004

[

-0.003

. . . .

I

. . . .

-0.002

I

-0.001

,

o

# (rad)

Fig. 3. The mass of the heavy eigenstate Z2 as a function of the mixing angle ¢ for representative values of AN~TM. For AN~TM = 0.30 (0.25 ) the allowed region is between the solid line and the upper (lower) dotted line. The curves are parametrized by 0.2~
straints on the Z ' p a r a m e t e r s are s t r o n g e r t h a n the c u r r e n t e x p e r i m e n t a l limits in ( 2 ) . A recent reassessm e n t [19] o f the s t a n d a r d p r i m o r d i a l n u c l e o s y n thesis c a l c u l a t i o n s c o n c l u d e s that A N ~N m u s t be v e r y small for the w h o l e p r o g r a m to work. T h i s result (see table 2 ) w o u l d only s t r e n g t h e n the b o u n d s in ( 8 ) . We c o n c l u d e that the c o u p l i n g o f the Z ' to t h r e e types o f light r i g h t - h a n d e d n e u t r i n o s severely constrains its p a r a m e t e r s , a result in a g r e e m e n t with current e x p e r i m e n t a l limits. T h e range o f values o f 0 is such that there is a negligible c o n t r i b u t i o n to the Z~ w i d t h f r o m the m i x i n g , a n d h e n c e a possible v a l u e A N > 0 w o u l d h a v e to be e x p l a i n e d as in ref. [ 14 ] by resorting to t h e lightest s u p e r s y m m e t r i c particle. F u r t h e r m o r e a decrease in the Z w i d t h with respect to the SM v a l u e c o u l d n o t be e x p l a i n e d in this m o d e l . T h e stringent b o u n d s on the Z ' p a r a m e t e r s in ( 8 ) are such that one seriously d o u b t s that such a Z ' c o u l d actually exist. M a k i n g the VR h e a v y ( t h r o u g h s o m e sort o f see-saw m e c h a n i s m ) will get us o f f the h o o k with respect to p r i m o r d i a l nucleosynthesis. H o w ever, m o r e a c c u r a t e direct m w a n d m~ m e a s u r e m e n t s will e v e n t u a l l y h a v e the s a m e effect. Finally, we e x p e c t the results in this p a p e r will be s i m i l a r to t h o s e in o t h e r m o d e l s [6,7] w i t h Z - Z ' m i x i n g a n d Z ' c o u p l i n g s to light r i g h t - h a n d e d neutrinos. We t h a n k A. Faraggi a n d K. E n q v i s t for useful discussions. 396

[ 1 ] See e.g.J. Hewett and T. Rizzo, Phys. Rep. 183 ( 1989 ) 193, and references therein. [2] E. Cohen, J. Ellis, K. Enqvist and D.V. Nanopoulos, Phys. Lett. B 165 (1985)76; J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Mod. Phys. Lett. A 1 (1986) 57. [3] J. Ellis, K. Enqvist, D.V. Nanopoulos and F. Zwirner, Nucl. Phys. B 276 (1986) 14. [4] G. Costa et al., Nucl. Phys. B 297 (1988) 244. [5] U. Amaldi et al., Phys. Rev. D 36 (1987) 1385. [6] V. Barger, J. Hewett and T. Rizzo, University of Wisconsin preprint MAD/PH / 514; T. Rizzo, Phys. Rev. D 40 (1989) 3035. [7 ] M. Gonzalez-Garcia and J. Valle, preprint SLAC-PUB-5076 (1989). [8 ] J. Ellis, K. Enqvist, D.V. Nanopoulos and S. Sarkar, Phys. Lett. B 167 (1986)457. [9 ] G. Steigman, K. Olive, D. Schramm and M. Turner, Phys. Lett. B 176 (1986) 33. [1o1 G. Abrams et al., Phys.Rev. Lett. 63 (1989) 724; 2173. [11 ] F. Abe et al., Phys. Rev. Lett. 63 (1989) 720. [121 L3 Collab., B. Adeva et al., Phys. Lett. B 231 (1989) 509; ALEPH Collab., D. Decamp et al., Phys. Lett. B 231 (1989) 519; OPAL Collab., M.Z. Akrawy et al., Phys. Lett. B 231 ( 1989 ) 530; DELPHI Collab., P. Aarnio et al., Phys. Lett. B 231 ( 1989 ) 539. [13 ]ALEPH Collab., J. Steinberger, CERN Seminar (5 December 1989). [141 J.L. Lopez and D.V. Nanopoulos, Texas A&M University preprint TAMU-CTP-62/89. [15 ] See e.g.S. Weinberg, Gravitation and cosmology (Wiley, New York, 1972) Ch. 15. [161 J. Yang et al., Astrophys. J. 281 (1984) 493. [171 K. Olive, D. Schramm and G. Steigman, Nucl. Phys. B 180 [FS2] (1981) 497. [181 D. Dicus et al., Phys. Rev. D 26 (1982) 2694. [191 S. Riley and J. Irvine, Manchester University preprint M/ C-TH 8928 (1989). [201 M. Gonzalez-Garcia and J. Valle, Phys. Lett. B 240 (1990) 163.