Precision Corrections to Supersymmetric Unification

12 Pages·0.24 MB·English

by  Jonathan Bagger

January, 1995 JHU-TIPAC-95001
hep-ph/9501277
Precision Corrections to Supersymmetric Unification
Jonathan Bagger Konstantin Matchev
5
9
9 and Damien Pierce
1

n Department of Physics and Astronomy
a
J The Johns Hopkins University

2 Baltimore, Maryland 21218
1

1
v
7
7 Abstract
2
1 We compute the full set of weak-scale gauge and Yukawa threshold corrections
0
in the minimal supersymmetric standard model, including all finite (non-logarithmic)
5
9 corrections, which we show to be important. We use our results to examine the effects
/
h of unification-scale threshold corrections in the minimal and missing-doublet SU(5)
p
- models. We work in the context of a unified mass spectrum, with scalar mass M0
p
e and gaugino mass M1/2, and find that in minimal SU(5) with squark masses less than
h one TeV, successful gauge and Yukawa coupling unification requires M1/2 M0 and
: ≪
v M0 1 TeV. In contrast, we find that the missing-doublet model permits gauge and
i ≃
X Yukawa unification for a wide range of supersymmetric masses.
r
a
1 Introduction
With the advent of precision measurements at LEP and the observation that the gauge cou-
plings unify in the minimal supersymmetric standard model [1], there has been a resurgence
of interest in supersymmetric grand unified theories. In most analyses the values of the elec-
2 2
tromagneticcoupling, αEM, andthestandard-modelweak mixing angle, sSM sin θSM(MZ),
≡
are taken from experiment, and converted into supersymmetric DR parameters, αˆ and
2 2 ˆ
sˆ sin θ(M ), using the leading-logarithmic contribution to the supersymmetric threshold
Z
≡
corrections. TheDR parametersarethenusedtodeterminetheMS strongcoupling constant
at the Z-scale, α (M ), ignoring all unification-scale threshold corrections. Alternatively,
s Z
the measured value of α (M ) is used to constrain the unification-scale parameter space.
s Z
In this letter we will take a closer look at this procedure, and report on the results of a
complete next-to-leading-order analysis of supersymmetric unification. Our approach is new
in two respects. First, we include all finite corrections. This implies that our weak mixing
2 2
angle sˆ is related to s as follows,
SM
2 2
sˆ = s + leading log + finite, (1)
SM
where the leading logarithms are of the form log(MSUSY/MZ) and the finite corrections are
2 2
of order M /M . If all the superpartner masses are heavier than a few times M , the
Z SUSY Z
finite corrections are negligible, in accord with the decoupling theorem. However, realistic
supersymmetric models typically have light particles with masses near M , so the finite
Z
corrections can be significant.
2 2
Second, we do not use a combined-fit value of s to compute sˆ because the finite term
SM
2
in eq. (1) is different for each observable. Therefore, in our analysis, we calculate sˆ directly
from a given set of inputs, namely, the Fermi constant G , the Z-boson mass M , the
F Z
electromagnetic coupling α , the top-quark mass m , and the parameters that describe
EM t
the supersymmetric model. (See also ref. [2].)
2
We emphasize that a precise evaluation of sˆ is essential because the renormalization
group equations (RGE’s) can naturally amplify small corrections. For example, the one-
3 2
loop renormalization group equations, dgˆ/dt = β gˆ /16π , together with gauge coupling
i i i
unification, imply
β2 β3 β3 β1 β1 β2
− + − + − = 0 , (2)
2 2 2
gˆ (µ) gˆ (µ) gˆ (µ)
1 2 3
2
at any scale µ. Solving for α (M ) and varying the inputs αˆ and sˆ , we find
s Z
2
δα δαˆ δsˆ
s
7.5 . (3)
α (M ) ≃ αˆ − sˆ2
s Z
2
This shows that a 1% error in the determination of sˆ leads to an error in the evaluation of
α (M ) of 7.5%.
s Z
2
In what follows, we will briefly describe our calculation of αˆ and sˆ . We will then
examine our prediction for α (M ) in the supersymmetric parameter space. We will find the
s Z
1
unification-scale threshold corrections that are necessary for gauge coupling unification with
a given supersymmetric spectrum, and compare our results with the threshold corrections
that arise in the minimal and missing-doublet SU(5) models. We will close with a similar
analysis under the additional assumption of bottom-tau Yukawa unification.
2 Gauge Coupling Unification
We start by sketching our calculation of the DR gauge parameters gˆ1(MZ) and gˆ2(MZ). As
−5 −2
discussed above, we take as inputs the Fermi constant G = 1.16639 10 GeV , the Z-
F
×
bosonmassMZ = 91.187GeV,theelectromagneticcouplingαEM = 1/137.036,thetop-quark
mass m , and the supersymmetric parameters. From these we calculate the electromagnetic
t
coupling αˆ and the weak mixing angle sˆ2 in the DR renormalization scheme,
αEM 2 2 παEM
αˆ = , sˆ cˆ = , (4)
1 ∆αˆ √2G M2(1 ∆rˆ)
− F Z −
2 2 ˆ
where cˆ = cos θ(M ),
Z
αEM 7 MW 16 mt 1 mH+
∆αˆ = 0.0685 0.0007 log + log + log (5)
± − 2π (−4 (cid:18)MZ (cid:19) 9 (cid:18)MZ(cid:19) 3 (cid:18) MZ (cid:19)
+ 6 4 log mu˜i + 6 1 log md˜i + 3 1 log me˜i + 2 4 log mχ+i ,
i=1 9 (cid:18)MZ(cid:19) i=1 9 (cid:18)MZ(cid:19) i=1 3 (cid:18)MZ(cid:19) i=1 3 (cid:18) MZ (cid:19))
X X X X
and [3]
Πˆ (0) Πˆ (M )
W Z Z
∆rˆ = ∆αˆ + + vertex + box . (6)
2 2
M − M
W Z
Equation (5) includes the light quark contribution extracted from experimental data [4],
together with the leptonic contribution. It also contains the logarithms of the W-boson,
top-quark, charged-Higgs, squark, slepton, and chargino masses. In eq. (6), the Πˆ denote
the real and transverse parts of the gauge boson self-energies, evaluated in the DR scheme.
Equation (6) also includes the vertex and box contributions that renormalize the Fermi
4
constant, as well as the leading higher-order m and QCD standard-model corrections given
t
in ref. [5]. (For a complete description of our calculation see ref. [6]. Note that in this letter,
all hatted objects are DR quantities, and all masses are pole masses.)
From these results we find the weak gauge couplings gˆ1(MZ) and gˆ2(MZ) using the
DR relations
5 eˆ eˆ
gˆ1(MZ) = , gˆ2(MZ) = , (7)
s3 cˆ sˆ
2
where αˆ = eˆ /4π. These couplings serve as the weak-scale boundary conditions for the two-
loop supersymmetric RGE’s. They implicitly determine the unification scale MGUT through
the condition gˆ1(MGUT) = gˆ2(MGUT) gGUT.
≡
2
We fix the strong coupling gˆ3(MGUT) at the scale MGUT by the unification condition
gˆ3(MGUT) = gGUT(1+ǫg) , (8)
where ǫ parametrizes the model-dependent unification-scale gauge threshold correction. We
g
then run gˆ3(MGUT) back to the Z-scale to find the MS coupling
2
gˆ (M )
3 Z −1
α (M ) = (1 ∆α ) , (9)
s Z s
4π ! −
where ∆α denotes the weak-scale threshold correction
s
2 12
∆α = gˆ3(MZ) 1 + 2 log mt + 1 log mq˜i +2log mg˜ . (10)
s 8π2 −2 3 (cid:18)MZ(cid:19) i=1 6 (cid:18)MZ(cid:19) (cid:18)MZ(cid:19)!
X
This procedure allows us to determine α (M ) for a fixed ǫ and a given supersymmetric
s Z g
spectrum. In what follows, we will make the additional assumption that the supersymmetric
spectrum unifies as well. Therefore we also assume that the three gaugino masses unify to
a common value of M1/2 Mˆ1/2(MGUT), the scalar masses unify to a common scalar mass
≡
M0 Mˆ0(MGUT), andthe soft trilinear scalar coupling parameters unify to A0 Aˆ0(MGUT).
≡ ≡
We evolve the parameters M1/2, M0 and A0 to the weak scale using the two-loop super-
symmetric RGE’s [7]. We require the parameters to be such that electroweak symmetry is
spontaneously broken, as naturally occurs when the top-quark mass is large. Then the Higgs
bosons H1 and H2 acquire expectation values v1 vˆ1(MZ) and v2 vˆ2(MZ); their ratio is
≡ ≡
ˆ
denoted tanβ tanβ(MZ) = v2/v1.
≡
We extract the supersymmetric masses from the running DR parameters and tanβ at
the scale M . (We choose µ > 0, where the superpotential contains the term +µǫ HiHj,
Z ij 1 2
with ǫ12 = +1.) We then substitute these masses into eqs. (4) – (6), and repeat the entire
proceduretofindaself-consistent solutiontotherenormalizationgroupequations. Typically,
the process converges after only a few iterations. It allows us to predict α (M ) consistently,
s Z
including all finite corrections.
As a point of reference, we show in Fig. 1 our results for αs(MZ) in the M0, M1/2 plane,
in the absence of unification threshold corrections, formt = 170 GeV, tanβ = 2, and A0 = 0.
Comparing with the experimental value α (M ) = 0.117 .005 [8], we see that α (M ) is
s Z s Z
±
1
rather large. If, for naturalness, we require the squark masses to be below 1 TeV, we obtain
the lower bound
αs(MZ) > 0.126 (No unification thresholds, mq˜ 1 TeV, mt = 170 GeV),
≤
assuming the validity of perturbation theory, which we take to mean that the DR top-quark
ˆ ˆ
Yukawa coupling λt(MGUT) 3. If we tighten this condition to λt(MGUT) 1, the bound
≤ ≤
increases by 0.002. For smaller m , α (M ) is smaller. For example, if m = 160 GeV,
t s Z t
the limit reduces by 0.002. These variations apply independently of the unification-scale
thresholds. Our numbers agree qualitatively with the results of ref. [2].
1The experimental error in ∆αˆ leads to a 0.001 uncertainty in α (M ).
s Z
±
3
Figure1: Contoursofαs(MZ)intheM0,M1/2 planewithmt = 170GeV,tanβ = 2
and A0 = 0. The region mq˜ < 1 TeV is bounded by M1/2 < 400 GeV and M0 < 1
∼
TeV.
The values of α (M ) quoted here are larger than in many previous analyses for two
s Z
2
reasons. First, during the past few years the central value of s , as determined from
SM
precision electroweak measurements, has been decreasing. (This change is correlated with
2
the increasing best-fit value of m .) A smaller value of s leads to an increase in α (M ),
t SM s Z
2
as can be seen from eq. (3). Second, the finite corrections decrease sˆ , and therefore increase
αs(MZ). The finite corrections are important in the region M1/2<200 GeV where αs(MZ)
∼
is appreciably larger than in the leading logarithmic approximation, as shown in Fig. 2.
Of course, the value of α (M ) can be reduced by a unification-scale threshold correction
s Z
with ǫ < 0. In the minimal SU(5) model [9], the unification-scale gauge threshold correction
g
is [10]
2
3g M
ǫ′ = GUT log H3 , (11)
g 40π2 (cid:18)MGUT(cid:19)
2
where M is the mass of the color-triplet Higgs fermion that mediates nucleon decay. From
H3
′
this expression, we see that ǫg < 0 whenever MH3 < MGUT. However, MH3 is bounded from
′ 3
below by proton decay experiments, so ǫ > 0 throughout most of the parameter space.
g
2The gauge threshold ǫ′ also receives contributions from higher-dimensional Planck-scale operators that
g
we ignore. They lead to a 0.006 uncertainty in α (M ) [11].
s Z
3Weusetheformulaegiv±eninref.[12],andchoosetheconservativevaluesβ =0.003GeV3 and 1+ytK =
| |
0.4.
4
Figure 2: The MS coupling αs(MZ) vs. M1/2. The curve labeled LLA shows the
result if we include only the logarithms of the supersymmetric masses (the leading
logarithmapproximation),whilethesolidlinecorrespondstothefullresultincluding
all finite corrections.
In minimal SU(5) we find that the smallest possible values for α (M ) are typically even
s Z
larger than those in Fig. 1. The only exception occurs for M0 M1/2, where the proton
≫
decay amplitude is suppressed. This determines the lower limit
αs(MZ) > 0.123 (Minimal SU(5), mq˜ 1 TeV, mt = 170 GeV),
≤
ˆ
assuming λt(MGUT) 3. In fact, as long as mq˜ 1 TeV, αs(MZ) < 0.127 can only be
≤ ≤
obtained in the region M0 1 TeV. For example, if M0 500 GeV, αs(MZ) > 0.128.
≃ ≤
The missing-doublet model is an alternative SU(5) theory in which the heavy color-
triplet Higgs particles are split naturally from the light Higgs doublets [13]. In this model
the unification-scale gauge threshold can be written as [14]
3g2 Meff 25
ǫ′′ = GUT log H3 log5+15log2 ǫ′ 4% . (12)
g 40π2 ( MGUT!− 2 ) ≃ g −
Thus, for fixed M , the missing-doublet model has the same threshold correction as the
H3
eff
minimal SU(5) model, minus 4%. In eq. (12), M is the effective mass that enters into the
H3
proton decay amplitude, so the bounds on M in the minimal SU(5) model also apply to
H3
eff
M in the missing-doublet model.
H3
The large negative correction in eq. (12) gives rise to much smaller values for α (M ).
s Z
This is illustrated in Fig. 3, where we show the upper and lower bounds on ǫ in the min-
g
imal and missing-doublet SU(5) models, together with the values of ǫ necessary to obtain
g
5
(b)
(a)
minimal SU(5) minimal SU(5)
missing doublet SU(5) missing doublet SU(5)
Figure 3: Thelightshadedregions indicatetheallowed valuesofthegauge coupling
threshold correction ǫ in the minimal and missing-doubletSU(5) models. Thedark
g
shaded region indicates the range of ǫ necessary to obtain α (M )= 0.117 0.1.
g s Z
±
α (M ) = 0.117 0.01. For both SU(5) models we bound ǫ from below by the limits on
s Z g
±
19
protondecayandfromabovebytherequirement M < 10 GeV.Notethatforthemissing-
H3
doubletmodel, theregionofallowedǫ nearlyoverlapstheregionwithα (M ) = 0.117 0.01,
g s Z
±
regardless of the supersymmetric particle masses.
3 Yukawa Coupling Unification
In typical SU(5) models the bottom and tau Yukawa couplings unify at the scale MGUT.
In the remainder of this letter, we will investigate bottom-tau unification to the same next-
to-leading-order accuracy that we used in our analysis of the gauge couplings. Our anal-
ysis is based on two-loop RGE’s and on the complete one-loop expressions that relate the
DR Yukawa couplings to the physical fermion masses [6].
Let us illustrate our procedure for the case of the tau Yukawa coupling. We start by
determining the DR expectation value vˆ from the Z-boson mass,
1
′2 2 2 2 ˆ 2
(gˆ +gˆ )vˆ = M +Π (M ) . (13)
4 Z Z Z
ˆ
Then, given the pole mass m = 1.777 GeV [8], we find the tau Yukawa coupling λ (M )
τ τ Z
using the DR relation
λˆ (M )vˆcosβ/√2 = m +Σˆ (m ) , (14)
τ Z τ τ τ
6
where Σˆτ(mτ) = Σ1+mτΣγ, and the tau self-energy is Σ1+p/Σγ+γ5(Σ5 +p/Σγ5). We follow
ˆ
a similar procedure to find the top-quark Yukawa coupling λ (M ).
t Z
Once we have the top and tau Yukawa couplings at the scale M , we evolve them to the
Z
scale MGUT using the two-loop supersymmetric RGE’s [7]. At MGUT we fix the DR bottom-
ˆ
quark Yukawa coupling λb(MGUT) through the unification condition
ˆ ˆ
λb(MGUT) = λτ(MGUT)(1+ǫb) , (15)
where ǫ parametrizes the unification-scale Yukawa threshold correction. We then run all
b
the Yukawa couplings back to the Z scale, and iterate the procedure self-consistently to
ˆ
determine λ (M ). Finally, we apply the weak-scale threshold corrections to find the pole
b Z
mass for the bottom quark.
For a top-quark pole mass m = 170 GeV, this procedure typically gives a bottom-quark
t
mass outside the range of experiment (which we take to be 4.7 < m < 5.2 GeV [8]). From
b
previous analyses we know that for m < 200 GeV there are two regions of tanβ where
t
bottom-tau unification might occur: tanβ < 2 and tanβ > 40. Here we focus on the
∼ ∼
small tanβ branch, where the top-quark Yukawa coupling at the unification scale is large,
λˆt(MGUT) > 1.
∼
As a point of reference, we first present our results with no unification-scale threshold
corrections. In Fig. 4 we show mb and αs(MZ) versus mt, for various values of tanβ, M0,
ˆ
and M1/2, with λt(MGUT) = 3. From the figure we see that the bottom-quark pole mass is
generally too large, unless the squark masses are of order 1 TeV.
ˆ
The value λt(MGUT) = 3 was chosen because it lies on the edge of perturbation theory.
ˆ
Smaller values of λt(MGUT) give rise to larger values of mb, so the curves in Fig. 4 can be
interpreted as lower limits on the bottom-quark mass. If we require the squark masses to be
below 1 TeV, we obtain the lower bound
mb > 5.1 GeV (No unification thresholds, mq˜ 1 TeV, mt = 170 GeV).
≤
In the absence of unification-scale threshold corrections, we see that, in the small tanβ
region, bottom-tau Yukawa unification favors large values of m and large values of the
b
supersymmetric mass scale. The above bound depends slightly on the top-quark mass, and
ˆ
sensitively on λt(MGUT). For mt = 160 GeV, the limit reduces by 0.1 GeV. The bound
ˆ ˆ
increases by 0.3 GeV for λt(MGUT) = 2 and by 0.8 GeV for λt(MGUT) = 1. These variations
also apply for ǫ = 0.
b
6
As with α (M ), the picture is altered by unification-scale threshold corrections. To
s Z
understand their effects, we first note the striking similarity between the m and α (M )
b s Z
curves in Fig. 4, which implies that the value of m is tightly correlated with the value of
b
α (M ). This leads us to expect that the gauge threshold correction will have an important
s Z
effect on m .
b
ˆ
In Fig. 5 we show the prediction for mb versus mt, for fixed ǫb = 0, with λt(MGUT) = 2,3
and ǫ chosen to give a fixed value of α (M ). We see that any model which gives α (M )
g s Z s Z
≃
7
Figure4: Thebottom-quarkmassandα (M )vs. m forthecaseofnounification-
s Z t
scale thresholds, for various values of tanβ, with A0 = 0 and λˆt(MGUT) = 3. The
right (solid) leg in each pair of lines corresponds to M1/2 varying from 60 to 1000
GeV,withM0 fixedat60GeV.Theleft(dashed)legcorrespondstoM0 varyingfrom
60 to 1000 GeV, with M1/2 = 100 GeV. On the solid lines the circles mark, from
top to bottom, M1/2 = 60, 100, 200, 400, and 1000 GeV, and on the dashed lines
the circles mark M0 = 60, 200, 400, and 1000 GeV. Note that the lowest point on
each left leg and the second-to-lowest point on each right leg corresponds to mq˜ 1
≃
TeV. The horizontal dashed lines indicate m = 5.2 GeV and α (M )= 0.127. The
b s Z
’s mark points with one-loop Higgs mass m < 60 GeV.
h
×
0.12 also predicts m 5 GeV, with ǫ = 0. Furthermore, any 23% < ǫ < 9% gives an
b b b
≃ −
acceptable value for m in this case. Alternatively, if α (M ) 0.13, a large and negative ǫ
b s Z b
≃
is required to achieve the central value for m , namely 33% < ǫ < 8%. Figure 5 shows
b b
− −
that these conclusions hold independently of the top-quark mass and the supersymmetric
mass scale.
In Fig. 6 we plot the most favorable values of α (M ) and m in the minimal SU(5)
s Z b
model. In this model the Yukawa threshold is given by [15]
1 M 1 M 1
ǫ′b = 16π2 (cid:20)4gG2UT (cid:18)log(cid:18)MGVUT(cid:19)− 2(cid:19) − λˆ2t(MGUT) (cid:18)log(cid:18)MGHU3T(cid:19)− 2(cid:19)(cid:21) , (16)
where M is the mass of the superheavy SU(5) gauge bosons. We define the most favorable
V
value as follows. We first minimize the value of α (M ) by picking the smallest possible
s Z
M consistent with nucleon decay. We then minimize m by choosing the smallest M
H3 b V
allowed by the constraint that the SU(5) model Yukawa couplings remain perturbative,
MV > max(0.3MH3,0.5MGUT). From the figure we see that this brings mb to an acceptable
8
Figure5: Thebottom-quarkmassvs. thetop-quarkmassforfixedvaluesofα (M )
s Z
and various tanβ. The solid lines correspond to λˆt(MGUT) = 3 while the dashed
lines correspond to λˆt(MGUT) = 2. The upper line in each pair corresponds to a
light supersymmetric spectrum with M0 = M1/2 = 80 GeV. The lower line in each
pair corresponds to a heavy spectrum, M0 = 1000 GeV, M1/2 = 500 GeV.
4
range for squark masses of order 1 TeV. Indeed, we find the limit
mb > 5.1 GeV (Minimal SU(5), mq˜ 1 TeV, mt = 170 GeV). (17)
≤
In contrast to minimal SU(5), the missing-doublet model can readily accommodate
bottom-quark masses in the range 4.7 GeV < m < 5.2 GeV, with ǫ 0. For exam-
b b
≃
ple, with mt = 170 GeV and M0 = M1/2 = 100 GeV, minimal SU(5) requires ǫb in the
range 15 to 65%. The missing-doublet model, however, requires ǫ in the range 30 to
b
− − −
+20%. Hence even for small (100 GeV) supersymmetric masses the missing-doublet model
O
naturally accommodates both gauge and Yukawa coupling unification.
4 Conclusion
In this paper we have computed the complete one-loop weak-scale threshold corrections in
the minimal supersymmetric standard model. We used them to study gauge and Yukawa
unification with and without unification-scale threshold corrections. In the absence of such
corrections we find that α (M ) and m are large unless the squark masses are larger than
s Z b
4We couldtrytodecreasem furtherbyincreasingM ,whichdecreasesǫ′. However,thecorresponding
b H3 b
increase in α (M ) compensates for this so that the limit in eq. (17) remains unchanged.
s Z
9