GUT Scale Threshold Corrections in a Complete Supersymmetric SO(10) Model: $α_s(M_Z)$ vs. Proton Lifetime

31 Pages·0.25 MB·English

by  Vincent Lucas

OHSTPY-HEP-T-95-028
GUT Scale Threshold Corrections
in a Complete Supersymmetric SO(10) Model:
α (M ) vs. Proton Lifetime
s Z
6
Vincent Lucas1 and Stuart Raby1,2
9
9
1 1Department of Physics, The Ohio State University, 174 W. 18th Ave., Columbus, OH 43210

n
a [email protected]
J

9 (December 27, 1995)
1

1
v
3
0 Abstract
3
1
0
6
9
/ WeshowthatoneloopGUTscalethresholdcorrectionstogaugecouplings
h
p
- are a significant constraint on the GUT symmetry breaking sector of the the-
p
e
h ory. The one loop threshold corrections relate the prediction for αs(MZ) to
:
v
the proton lifetime. We have calculated these corrections in a new complete
i
X
r SO(10) SUSY GUT. The results are consistent with the low energy measure-
a
ment of α (M ). We have also calculated the proton lifetime and branching
s Z
ratios in this model. We show that proton decay rates provide a powerful test
for theories of fermion masses.
PACS numbers: 12.15.Ff, 12.15.Hh, 12.60.Jv
Typeset using REVTEX
1
I. INTRODUCTION
It has been known for some time now that Supersymmetric [SUSY] Grand Unified The-
ories [GUTs] [1]
• suppress the non-SUSY GUT prediction for proton decay, via heavy gauge boson
exchange, by increasing the GUT scale, and
• lead to a 10% increase (for one pair of Higgs doublets) in the GUT prediction for the
weak mixing angle, sin2θ , compared to non-SUSY GUTs.
W
Now, since sin2θ is known to better than 1% accuracy, one naturally uses the values
W
of the two low energy parameters, the fine structure constant, α, and sin2θ as input to
W
determine the GUT coupling, α , and the GUT scale, M , and predicts the value of the
G G
strong coupling, α . [Note, all low energy parameters are typically evaluated at the scale,
s
M .] This prediction is in excellent agreement with the low energy data [2–4]. The global
Z
fit to α from measurements at all energies [2] gives α = 0.117 ± 0.005 with low (high)
s s
energy measurements favoring values of α ≤ (≥) 0.12. A global fit to all electroweak
s
data [3] gives α = 0.127±0.005±0.002±0.001. In comparison, SUSY GUTs predict [4]
s
α = 0.127±0.005±0.002, where the first error is the uncertainty in the low energy sparticle
s
spectrum, and the second, the top and Higgs masses. Note that the non-supersymmetric
GUT prediction [5] α = 0.073±0.001±0.001 is clearly ruled out.
s
In addition, SUSY GUTs contain new sources of proton decay (dimension 5 operators)
mediated by color triplet Higgs fermions [6,7]. They typically lead to the dominant decay
mode p → K+ν¯ with the lifetime, τ ∝ M˜2, where M˜ is an effective color triplet mass.
p t t
These results are valid for any grand unified symmetry group, as long as SU(3)× SU(2)×
U(1) is embedded in an unbroken simple subgroup above the GUT scale. Consideration of
fermion masses, however, leads us to the special grand unified symmetry group SO(10) [8].
It is the unique group which combines all the fermions of one family {u, d, e, ν} into one
irreducible 16 dimensional representation with the addition of only one electroweak singlet
state, ν¯, the so-called right-handed neutrino. Consequently, SO(10) Clebschs can be used
2
to relate all charged fermion mass matrices. In this context, a paper by Anderson et al.
[9][Paper I] showed how all charged fermion masses and mixing angles can be reasonably
described in terms of four dominant effective mass operators at M . Hall and one of us
G
(S.R.) [10] [Paper II] then showed how to extend the effective theory at M to a complete
G
SO(10) SUSY GUT valid up to the Planck or string scales, M. That theory included several
sectors necessary for – (1) GUT symmetry breaking, (2) doublet-triplet splitting (Higgs
sector), (3) charged fermion masses (fermion mass sector) and also (4) giving a large mass
to the electroweak singlet neutrinos (neutrino mass sector). It contained sufficient symmetry
to be “natural.” Hence, it included all operators consistent with the symmetries and only
contained the SO(10) states1 —
(n +3) 16+n 1¯6+n 10+n 45+n 54+n 1 (1)
16 16 10 45 54 1
with n ,n ,n ,n ,n specific integers. Finally, the effective mass operators of paper I
16 10 45 54 1
were recovered upon integrating out states with mass greater than M .
G
In any complete GUT, the prediction for α receives corrections at one loop from thresh-
s
olds at the weak scale, including the sparticle spectrum, and at the GUT scale from states
with mass of order M . The dominant contribution at M comes from states in the sym-
G G
metry breaking and Higgs sectors.
The Higgs sector, on the other hand, also affects the proton decay rate. The effective
color triplet Higgs fermion mass, M˜ , must be significantly larger than M in order to
t G
increase the proton lifetime. The Higgs doublets on the other hand must remain massless
at the GUT scale.
In this paper,
• we show that the threshold corrections due to doublet-triplet splitting, in the simplest
models, increases the predicted value for α (M ). Thus, while the GUT predictions for α
s Z s
and τ are unrelated at the tree level, at one loop the two effects are coupled. One cannot
p
1These two features are consistent with a stringy origin to such a model [11].
3
suppress proton decay without at the same moment increasing the prediction for α . We
s
then show that the model of paper II is ruled out by this effect.
• We also present a new complete SUSY GUT which appears to be consistent with all
low energy data at the 1σ level [12]. This model has in fact a much simpler GUT symmetry
breaking sector than that in paper II with fewer states. We present some typical results for
proton decay rates in this model. In a future paper [13], we will present a more detailed
study.
II. ONE LOOP THRESHOLD CORRECTIONS AT M .
G
One loop threshold corrections at M for gauge couplings are given by [14]
G
α−1(M ) = α−1 −∆ (2)
i G G i
where ∆ is the leading log threshold correction to α . Furthermore,
i i
1 M
∆ = bζlog ζ (3)
i 2π i (cid:12)M (cid:12)
Xζ (cid:12) G(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where the sum is over all superheavy particles and bζ is the contribution the superheavy
i
particle ζ would make to the beta function coefficient b if the particle were not integrated
i
out at M .
G
At one loop the definition of the GUT scale is somewhat arbitrary. In order to avoid
large logarithms it must certainly be at the geometric mean of the heavy masses; otherwise
we are free to choose its value. A particularly convenient choice is to define M as the scale
G
where the two gauge couplings, α , i = 1,2, meet. At this point ∆ = ∆ and we define
i 1 2
α˜ ≡ α (M ) = α (M ). (4)
G 1 G 2 G
Then define
ǫ ≡ (α (M )−α˜ )/α˜ , (5)
3 3 G G G
i.e. the relative shift in α at M . In general, a value of ǫ ∼ −(2−3%) is needed to obtain
3 G 3
α ∼ 0.12.
s
4
A. Formula for ǫ in a general SO(10) theory
3
The most general SO(10) SUSY GUT we consider includes the complete multiplets listed
in equation (1) with arbitraryvalues ofn ,n ,n ,n ,n . These states can be decomposed
16 10 45 54 1
into their SU(3) × SU(2) × U(1) content by first considering the decomposition to SU(5)
[see table I] and then the rest of the way [15].
We have derived a general formula for ǫ in this case (see appendix 1) given by the
3
expression (valid to lowest order in α˜ )
G
α˜ 1 M
ǫ = G [(bγ −bγ)− (bγ −bγ)]log det′ γ (6)
3 2π 3 2 2 2 1 (cid:12) M (cid:12)
Xγ (cid:12) G(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where
detM if γ = t,g,w,s,σ
γ

det′Mγ =  (MdgγeatuMginγo)4 if γ = q,u,e,x (7)
detM′ if γ = d
where M′ is defined as the doublet mass md atrix M with the massless Higgs doublets pro-
d d
jected out.2 Note, ǫ only depends on the number of states in the theory through the mass
3
matrices M in each charge sector γ. Moreover, the effective determinants, defined above,
γ
explicitly take into account the special contributions of vector multiplets to the threshold
corrections.
Putting in the values of bγ for i = 1,2,3 into eqn. (6) we find
i
α˜ 3 M 3 M
ǫ = G log det′ g − log det′ w
3 π (cid:18)2 (cid:12) M (cid:12) 2 (cid:12) M (cid:12)
(cid:12) G(cid:12) (cid:12) G(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
33 M 21 M (cid:12)9 (cid:12) M (cid:12) 3 (cid:12) M 6 M
+ log det′ s − log det′ σ + log det′ u + log det′ e − log det′ q
10 (cid:12) M (cid:12) 10 (cid:12) M (cid:12) 10 (cid:12) M (cid:12) 10 (cid:12) M (cid:12) 5 (cid:12) M (cid:12)
(cid:12) G(cid:12) (cid:12) G(cid:12) (cid:12) G(cid:12) (cid:12) G(cid:12) (cid:12) G(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12)3 (cid:12) M 3(cid:12) (cid:12) M (cid:12) (cid:12) (cid:12) (cid:12)
− log det′ d + log det′ t . (8)
5 (cid:12) M (cid:12) 5 (cid:12) M (cid:12)(cid:19)
(cid:12) G(cid:12) (cid:12) G(cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
The dominant contribution to ǫ comes from the GUT symmetry breaking and elec-
3
troweak Higgs sectors with superspace potential W and W , respectively. An
symbreaking Higgs
2Our notation for the states discussed above is found in table II.
5
additional small contribution comes from the fermion mass sector. These sectors are by
reasons of “naturalness” necessarily invariant under several U(1) symmetries (including an
R symmetry). In appendix 2 we show that these symmetries impose stringent constraints
on the form of ǫ . In brief, ǫ is implicitly a function of the vacuum expectation values
3 3
[vevs] of fields in the symmetry breaking sector. These vevs transform under the U(1) and
R symmetries. As a consequence of the invariance, we find
¯
3α˜ detM
G t
ǫ = f(ζ ,...,ζ )+ log +··· (9)
3 1 m 5π (cid:12)(cid:12)(cid:12)MGdetM¯d′(cid:12)(cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where the first term represents the contributions from W . It is only a function
symbreaking
of U(1) and R invariant products of powers of vevs {ζ }. The second term, coming from
i
the Higgs sector, is discussed further in the next section and the ellipsis refers to the small
additional contribution arising from the fermion mass sector. Note, M¯ , M¯′ only include
t d
those states, from 5s and ¯5s of SU(5) contained in W and W , which mix with
symbreaking Higgs
the Higgs sector (see the next section).
B. The dependence of ǫ on the Higgs sector
3
We assume the Higgs sector includes any number of 10s with, for simplicity, only one
of these, say 101, coupling to light fermions. We also include the doublet-triplet splitting
mechanism introduced by Dimopoulos-Wilczek [16]. Accordingly, the terms in the super-
space potential relevant for doublet - triplet splitting are of the form
3 3
W = ¯5 [a (B − L)] 5 −¯5 [a (B − L)] 5 + M ¯5 5 (10)
d−t 2 1 1 1 1 2 ij i j
2 2
iX,j≥2
where a 3 (B − L) is the vev of the field A in the 45 dimensional representation and
1 2 1
(B − L) is the [Baryon - Lepton number] charge matrix (see eqn. (14)). Thus, since
the doublets in 5 ,¯5 ⊂ 10 have zero B − L, they remain massless. [Note, some of the
1 1 1
¯5 ,5 states may come from 16 and 16 representations, respectively, in W . We
i j symbreaking
include, however, only those states which mix with 10 .] With this superpotential, we have
1
6
M¯′ ≡ M[d], and the triplet mass matrix, M¯ , with non-vanishing determinant, includes the
d t
terms which mix states in 10 with ¯5 , 5 , for i,j ≥ 2 and the sub-matrix M[t], where the
1 i j
matrix M[d], (M[t]) is given by M with Clebschs appropriate for doublets (triplets).
Proton decay in this model is mediated by the Higgses in 10 . Hence, the coefficient of
1
the resulting effective dimension 5 baryon violating operators [6,7] is given by the inverse of
the triplet mass matrix in the 11 direction, i.e.
detM[t] detM¯′
(M−1) ≡ = d ×g
t 11 detM¯ detM¯
t t
where g ≡ detM[t]. We show, in appendix 2, that g = g(ζ ,···,ζ ) is a holomorphic function
detM[d] 1 m
of the set of U(1) and R invariant products of powers of vevs. If we now define the effective
triplet mass by M˜ −1 ≡ (M−1) , we obtain the final form for the factor in eqn. (9) –
t t 11
detM¯t = M˜ g(ζ ,···,ζ ). As a consequence, eqn. (9) becomes
detM¯′ t 1 m
d
3α˜ M˜
G t
ǫ = F(ζ ,...,ζ )+ log +··· (11)
3 1 m (cid:12) (cid:12)
5π (cid:12)MG(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
where f and g are absorbed in the function F. We thus find that suppressing the pro-
ton decay rate by increasing the ratio M˜t has the effect of increasing the value of ǫ and
MG 3
consequently increasing the value of α .3
s
C. An Example : ǫ in the model of paper II
3
As an example, for the particular model of paper II, there is only one independent
invariant ratio of the GUT scale vevs given by ζ = vv . In principle, ǫ could depend on an
a1a2 3
arbitrary function f(ζ). However, when we put the effective determinants for this model in
equation (8) we find
3α˜ M˜
G t
ǫ = 21 log(2)+log . (12)
3 (cid:12) (cid:12)
5π (cid:26) (cid:12)MG(cid:12)(cid:27)
(cid:12) (cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
3Thisobservation in thecontext of minimalSU(5), wherein thatcase M˜ is thecolor tripletHiggs
t
mass, was discussed earlier by Hisano et al. [7].
7
Remarkably, all dependence of ǫ on the GUT scale vevs has dropped out except for the
3
dependence on M˜ . Unfortunately, the large positive constant that appears in the expres-
t
sion means that in order to get ǫ negative, the proton’s lifetime would be many orders
3
of magnitude lower than the experimental bound. In addition, changing the Yukawa cou-
pling coefficients of the terms of W cannot cure this problem. For the most part,
symbreaking
changing the Yukawa coupling constants of one of the terms of W has the result
symbreaking
of multiplying the effective determinants of the mass matrices M in eqn. (6), for states γ
γ
in a complete SU(5) multiplet, all by the same amount. Thus, this multiplicative factor has
no effect on ǫ . Hence, as a consequence of (12), we find that the symmetry breaking sector
3
of the model of paper II is ruled out by the dual constraints coming from the low energy
measurement of α and the proton lifetime.
s
Is it possible to find a symmetry breaking sector which has all the U(1) symmetries
required for the naturalness of the theory and is consistent with the constraints of α and τ ?
s p
In the next section we describe a new SO(10) SUSY GUT which satisfies all our criteria. In
fact it may contain the minimal symmetry breaking sector consistent with the requirements
of (1) obtaining the effective fermion mass operators of paper I, (2) “naturalness,” and (3)
retaining only those states at energies below M , which either have trivial SM charge or are
G
contained in the minimal supersymmetric standard model [MSSM].
III. A COMPLETE SO(10) SUSY GUT
A. The GUT symmetry breaking and Higgs sectors
For this model,
W = 1 A′(A3 +S SA +S A A ) (13)
symbreaking M 1 1 3 1 4 1 2
+A (ψψ +S A˜)+SA˜2
2 1
+S′(SS +A A˜)+S S′2
2 1 3
8
where the fields {A , A , A˜, A′}, {S, S′}, ψ, ψ¯, {S ,···,S } are in the 45, 54, 16, 1¯6 and
1 2 1 1 4
1 representations, respectively. [Note, traces and contractions of indices are implicit.] The
supersymmetric minimization condition ∂W = 0 gives four discrete choices for the direction
∂A′
1
of the vev of A . We will assume that nature chooses the B - L direction. The second term
1
gives A˜ a vev in the X direction, A a vev in the Y direction, and ψ and ψ vevs in the
2
right-handed neutrino-like direction (see eqn. (14) below). The third and fourth terms, and
the S subterm of the first term, were added to give mass to all non-MSSM fields which are
4
not in a singlet representation of the standard model gauge group. The last term was added
in accordance with our “naturalness” criterion, namely that the theory should be the most
general one consistent with the symmetries. The term S S′2 is consistent with the U(1) and
3
R symmetries of the symmetry breaking sector of the theory that will be discussed below,
and we are aware of no other additional symmetry that might exclude this term. Therefore
the term must be included.
The above vevs are given by
1
 
 1 
 
  3
 
hA1i = a1 1 ⊗η ≡ a12(B −L) (14)
 
 
 0 
 
 
 
 0
 
 
1
 
 1 
 
  3
 
hA2i = a2 1 ⊗η ≡ a22Y
 
 
 −3/2 
 
 
 
 −3/2 
 
 
9
1
 
 1 
 
  1
hA˜i = a˜ 1 ⊗η ≡ a˜ X
  2
 
 
 1 
 
 
 
 1 
 
 
1
 
 1 
 
 
 
hSi = s 1 ⊗1
 
 
 
 −3/2 
 
 
 
 −3/2 
 
 
hψi = v |SU(5) singleti
hψi = v |SU(5) singleti
where
0 −i 1 0
   
η = ,1 = .
 i 0   0 1 
   
   
The vacuum minimization conditions are explicitly
1
a2 = −sS , S a˜ = vv (15)
1 3 1 4
2
sS + a a˜ = 0, a S +2sa˜ = 0
2 1 2 1
5
Using these equations, the set {a ,a ,a˜,v,v,S } form a complete set of independent vari-
1 2 4
ables.
Two caveats –
• Note at tree level the GUT symmetry breaking vevs are undetermined since the po-
tential in these directions are both supersymmetric and flat. We will not discuss the process
10