We apply the full one-loop corrections to the masses, gauge couplings, and
Yukawa couplings in the minimal supersymmetric standard model. We focus on
predictions for the strong coupling and the bottom-quark pole mass in the
context of SU(5) supersymmetric grand unification. We discuss our results in
both the small and large $\tan\beta$ regimes. We demonstrate that the finite
(non-logarithmic) corrections to the weak mixing angle are essential in
determining $\alpha_s$ and $m_b$ when some superpartner masses are light.
Minimal SU(5) predicts acceptable $\alpha_s$ and $m_b$ only at small
$\tan\beta$ with SUSY masses of $\cal O$(TeV). The missing doublet model
accommodates gauge and Yukawa coupling unification for small or large
$\tan\beta$, and for any SUSY mass scale. In the large $\tan\beta$ case, the
bottom mass is acceptable only if the Higgsino mass parameter $\mu$ is
positive.Comment: 10 pages, epsf, 7 figures included. Talk given at the PASCOS
  Symposium/Johns Hopkins Workshop, Baltimore, MD, March 22-25, 199

Pierce, Damien

English

arXiv

We apply the full one-loop corrections to the masses, gauge couplings, and Yukawa couplings in the minimal supersymmetric standard model. We focus on predictions for the strong coupling and the bottom-quark pole mass in the context of SU(5) supersymmetric grand unification. We discuss our results in both the small and large \tan\beta regimes. We demonstrate that the finite (non-logarithmic) corrections to the weak mixing angle are essential in determining \alpha_s and m_b when some superpartner masses are light. Minimal SU(5) predicts acceptable \alpha_s and m_b only at small \tan\beta with SUSY masses of \cal O(TeV). The missing doublet model accommodates gauge and Yukawa coupling unification for small or large \tan\beta, and for any SUSY mass scale. In the large \tan\beta case, the bottom mass is acceptable only if the Higgsino mass parameter \mu is positive

Pierce, D

CERN Document Server

JHU-TIPAC-95022
hep-ph/9508219
THE STRONG COUPLING AND THE BOTTOM MASS
IN SUPERSYMMETRIC GRAND UNIFIED THEORIES

DAMIEN PIERCE
Department of Physics and Astronomy
Johns Hopkins University
3400 N. Charles Street
Baltimore, Maryland 21218, USA
E-mail: pierce@planck.pha.jhu.edu
We apply the full one-loop corrections to the masses, gauge couplings, and
Yukawa couplings in the minimal supersymmetric standard model. We focus
on predictions for the strong coupling and the bottom-quark pole mass in the
context of SU(5) supersymmetric grand unication. We discuss our results in
both the small and large tan  regimes. We demonstrate that the nite (non-
logarithmic) corrections to the weak mixing angle are essential in determining

s
and m
b
when some superpartner masses are light. Minimal SU(5) predicts
acceptable 
s
and m
b
only at small tan  with SUSY masses of O(TeV). The
missing doublet model accommodates gauge and Yukawa coupling unication
for small or large tan, and for any SUSY mass scale. In the large tan  case,
the bottom mass is acceptable only if the Higgsino mass parameter  is positive.
1. Introduction
This talk is organized as follows. First, we briey discuss the supersymmetric
standard model, in order to introduce the parameter space we are considering. Next,
we discuss the prediction for the strong coupling constant and the bottom pole mass
with and without including GUT threshold eects, in the small tan  case. Lastly we
discuss our results in the large tan  regime.
The minimal supersymmetric model is attractive in that it solves the hierarchy
problem. It does this, however, at the expense of doubling the number of degrees
of freedom of the standard model. This is problematic, as supersymmetry must be
broken and hence the number of new parameters needed to describe the model is in
general quite large. An organizing principle is needed in order to reduce the number
of parameters. In a minimal supergravity scenario the number of supersymmetry
breaking parameters needed to describe the supersymmetric model are few: a uni-
versal scalar mass M
0
, a universal gaugino mass M
1=2
, a universal trilinear scalar

Talk given at the PASCOS Symposium/Johns Hopkins Workshop, Baltimore, MD, March 22-25,
1995.
1
coupling A
0
, and a bilinear scalar coupling B. In addition there is a supersymmetric
Higgs mass term, . Given values for these ve parameters at the GUT scale, we
use the renormalization group equations
1
(RGE's) to determine the various parti-
cle masses and couplings at the weak scale. For a large top-quark mass, one of the
Higgs boson masses is driven negative, and the radiative breaking of SU(2)  U(1)
symmetry becomes manifest. The Higgs bosons obtain vev's v
1
and v
2
, and we can
determine the Z-boson mass. In practice it is convenient to assume electroweak sym-
metry breaks radiatively and to take M
Z
as an input parameter, as well as the ratio
of vev's tan  v
2
=v
1
. We then determine 
2
and B from the symmetry breaking
conditions. To summarize, then, the supersymmetric model is parametrized by
M
0
; M
1=2
; A
0
; tan ; sign():
The exact one-loop corrections to the masses, gauge couplings and Yukawa cou-
plings of the minimal supersymmetric model are described in Ref. [2]. These correc-
tions are essential ingredients for accurate tests of grand unication. They allow one
to extract the underlying DR parameters from a given set of measured observables.
The DR parameters can then be run up to a high scale to explore the consequences
of dierent unication hypotheses.
Alternatively, the radiative corrections can be used to translate various limits into
excluded regions of the DR parameter space. This is illustrated in Fig. 1, where we
show the excluded region of the M
0
; M
1=2
parameter space at the one-loop level, from
current experimental constraints.
In the following, we treat all supersymmetric threshold corrections in a complete
one-loop analysis.
a
Our work stands in contrast to most previous studies, which are
based on the \leading logarithm approximation." For the gauge coupling threshold
corrections, this approximation involves taking the standard-model value of sin
2

W
and adding the logarithmic parts of the SUSY threshold corrections. The approxima-
tion works well if all of the SUSY particle masses are much greater than M
Z
, in which
case the decoupling theorem implies that the nite eects of the SUSY particles are
negligible for all low-energy observables.
However, in realistic models it is not unusual for the supersymmetric spectrum
to contain light particles of order the Z-mass. In this case the leading logarithm
approximation breaks down. This is illustrated in Fig. 2, where we compare the value
of 
s
and m
b
in the leading logarithm approximation (LLA) with the value obtained
in the full calculation. In this talk we use the full set of one-loop radiative corrections
to evaluate the DR gauge and Yukawa couplings. The DR couplings serve as the
boundary conditions for the two-loop gauge and Yukawa coupling renormalization
group equations, which determine the couplings at very high scales.
In what follows, we have converted the strong coupling to the MS scheme so that
by 
s
we refer to the standard MS value evaluated at the scale M
Z
. By m
b
we refer
a
See Chankowski et al.
3
for a similar treatment of nite corrections to sin
2

W
.
2
Figure 1: Excluded region (shaded) of the M
0
; M
1=2
plane, for tan  = 2; m
t
= 175
GeV, A
0
= 0, and 
s
= 0:117. All masses are evaluated at one-loop. The symbols indicate
which experimental constraint is relevant: 
+
) m

+ > 47 GeV; ~g ) m
~g
> 125 GeV;
~ ) m
~
> 42 GeV; h) m
h
> 60 GeV.
to the bottom-quark pole mass.
2. Small tan 
As a reference point, we show in Fig. 3 contours of m
b
and 
s
in the M
0
; M
1=2
plane, with no GUT thresholds, tan  = 2, m
t
= 175 GeV, and A
0
=0. We conne our
attention to the region of the theory which is more natural, i.e. to the region where
the superpartner masses are less than about 1 TeV. We nd the strong coupling is
large (
s
> 0:127) compared to the PDG value
4

s
= 0:117  0:005. Similarly, the
bottom mass is quite large (m
b
> 6 GeV), far outside the preferred region which we
take to be 4:7 < m
b
< 5:2 GeV.
The experimental uncertainty in the determination of 
s
is primarily due to the
uncertainty in determining the electromagnetic coupling at the Z-scale. We use the
value recently determined by Eidelman and Jegerlehner
5
. The one-sigma uncertainty
in our input 
EM
(M
Z
) results in a one-sigma uncertainty in our output 
s
of about
0:001, and an uncertainty of typically 0.07 GeV in our bottom mass prediction.
Martin and Zeppenfeld
6
and Swartz
7
have also performed analyses to determine

EM
(M
Z
). The central value for 
s
increases by about 0.001 if we use the value of

EM
(M
Z
) as determined by Martin and Zeppenfeld, and it increases by about 0.002
if we use the value of 
EM
(M
Z
) from Swartz.
The bottom mass is reduced if we go further into the small tan  region, due
3
Figure 2: Comparison of (a) 
s
and (b) m
b
in the leading logarithm approximation (LLA)
versus the full one-loop calculation, for M
0
= 60 GeV, tan  = 2; m
t
= 175 GeV, A
0
= 0,
and  > 0. The dashed line shows the result if the non-universal corrections are neglected.
to the large top-quark Yukawa coupling in the bottom mass renormalization group
equation. We consider the case where the top Yukawa is as large as possible; we set

t
(M
GUT
) = 3, which is on the verge of the nonperturbative regime. In this case we
will obtain the smallest possible bottom mass. As seen in Fig. 4, for m
t
= 180 GeV
the bottom mass is less than 5.2 GeV and the strong coupling is less than 0.127 only
if the squark masses are in the TeV region.
If we consider particular GUT models, we can determine whether the GUT thresh-
old corrections to the gauge and Yukawa couplings can help improve the situation.
We parametrize the GUT threshold corrections by "
g
and "
b
, where
g
3
(M
GUT
) = g
GUT
(M
GUT
) (1 + "
g
) ;

b
(M
GUT
) = 

(M
GUT
) (1 + "
b
) ;
where 
b
and 

are the b- and  -Yukawa couplings, and M
GUT
is dened as the scale
at which g
1
and g
2
meet, g
GUT
 g
1
(M
GUT
) = g
2
(M
GUT
). A smaller value of 
s
requires "
g
< 0. In the small tan  region the bottom mass is tightly correlated with
the value of the strong coupling. Hence, setting "
g
< 0 reduces both 
s
and m
b
. In
fact, the bottom mass is an order of magnitude more sensitive to "
g
than to "
b
. In
what follows we examine the GUT corrections in two SU(5) GUT models.
In the minimal SU(5) model
8
, the gauge coupling threshold correction "
0
g
is given
by
9
"
0
g
=
3g
2
GUT
40
2
log

M
H
3
M
GUT

; (1)
4
.14 
.127 
.13 
 .127 
.13 
.137 
.134 
6.1 
6.3 
6.8 
6.3 
6.9 
6.2 
6.5 
Figure 3: Contours of 
s
(solid lines) and m
b
(dashed) in the M
0
, M
1=2
plane, with
tan  = 2, m
t
= 175 GeV, and A
0
= 0. The upper left hand corners are excluded on
cosmological grounds (charged LSP) and the lower shaded regions are excluded by particle
searches. The m
b
contours are labeled in GeV.
where M
H
3
is the mass of the color-triplet Higgs particle that mediates nucleon decay.
From this expression, we see that "
0
g
< 0 whenever M
H
3
< M
GUT
. However, M
H
3
is bounded from below by proton decay experiments. The M
H
3
mass limit is of the
form
10
M
H
3
>M
j1 + y
tK
j
sin 2
f( ~w;
~
d; ~u; ~e)
whereM is a nuclear matrix element, y
tK
parametrizes the amount of third generation
mixing, and f is a function of the wino, squark and slepton masses.
For the conservative choices M = 0:003 GeV
3
and j1 + y
tK
j = 0:4 we nd that
M
min
H
3
> M
GUT
unless M
0
> 500 GeV and M
1=2
M
0
. Thus, in most of the parameter
space, "
0
g
> 0. For this reason, in minimal SU(5), 
s
is typically even larger than
in the case of no GUT thresholds, as illustrated in Fig. 5(a). In order to obtain the
smallest possible 
s
and m
b
, we have set 
t
(M
GUT
) = 3 in Fig. 5(a). Thus we end
up with rather small values for tan  ( 1:3-1.6), and the Higgs mass constraint rules
out a large part of parameter space. Only in the region M
0
M
1=2
, where the proton
decay amplitude is suppressed, is the strong coupling reduced relative to the case with
no GUT thresholds. The smallest value of 
s
occurs in this region, a somewhat large
value of 0.124. The bottom-quark mass is similarly on the high side of the preferred
region. We have applied the most favorable Yukawa correction "
b
given in Wright
11
,
subject to Yukawa coupling perturbativity constraints (see Bagger et al.
12
).
The missing-doublet model is an alternative SU(5) theory in which the heavy
5
Figure 4: The bottom-quark mass and 
s
vs. m
t
for the case of no GUT-scale thresholds,
for various values of tan , with A
0
= 0,  > 0, and 
t
(M
GUT
) = 3. The circles on the right
(solid) leg in each pair of lines corresponds to M
1=2
equal to (from top to bottom) 60, 100,
200, 500 GeV, with M
0
xed at 100 GeV. The circles on the left (dashed) leg corresponds to
M
0
equal to 100, 200, 400 and 1000 GeV, with M
1=2
= 100 GeV. The horizontal dot-dashed
lines indicate m
b
= 5:2 GeV and 
s
= 0:127. The 's mark points with one-loop Higgs
mass m
h
< 60 GeV.
color-triplet Higgs particles are split naturally from the light Higgs doublets
13
. In
this model the GUT gauge threshold correction is given by
14
"
00
g
=
3g
2
GUT
40
2
(
log
 
M
e
H
3
M
GUT
!
 
25
2
log 5 + 15 log 2
)
' "
0
g
  4% : (2)
Thus, for xed M
H
3
, the missing-doublet model has the same threshold correction as
the minimal SU(5) model, minus 4%. In eq. (2), M
e
H
3
is the eective mass that enters
into the proton decay amplitude, so the bounds on M
H
3
in the minimal SU(5) model
also apply to M
e
H
3
in the missing-doublet model.
The large negative correction in eq. (2) is due to the mass splitting in the 75
representation, and gives rise to much smaller values for 
s
and m
b
. This is illustrated
in Fig. 5(b), where we show contours of 
s
and m
b
in the M
0
; M
1=2
plane, with
M
e
H
3
= M
min
H
3
, at tan  = 2. We nd (even without going into the far infrared top
Yukawa xed point region) values of both the strong coupling and the bottom mass
6
.128 
.131 
 
.105 
.108 
.114 
5.2 
4.4 
4.9 
4.6 5.1 
5.3 
5.5 
.101 
Figure 5: Contours of the smallest possible 
s
(solid lines) and m
b
(dashed) consistent
with nucleon decay in (a) minimal SU(5), and (b) missing doublet SU(5), with m
t
= 175
GeV, A
0
= 0 and  > 0. In (a) 
t
(M
GUT
) = 3 (tan  ' 1:4) and in (b) tan  = 2. The
shaded regions are phenomenologically excluded. The m
b
contours are labeled in GeV.
near their central values.
3. Large tan
Again, as a reference point, we show in Fig. 6 the strong coupling and bottom mass
prediction with no GUT corrections, for A
0
= 0; m
t
= 175 GeV, at tan  = 30. The

s
predictions are not signicantly dierent from the small tan  case. The bottom
mass is signicantly inuenced by the large nite corrections
15
,
m
b
m
b
  
 tan 
16
2
m
2
~q

8
3
g
2
3
m
~g
+ 
2
t
A
t

;
so much so, that for the case  < 0 the bottom mass is hopelessly large. However,
these help to reduce m
b
when  > 0. Even so, we see in Fig. 6(a) that m
b
> 5:5 GeV.
GUT threshold corrections are needed to reduce both 
s
and m
b
further.
In minimal SU(5) the proton decay rate is enhanced at large tan , so the triplet
Higgs mass M
H
3
is forced to be quite large. Hence, we have a large and positive GUT
correction "
0
g
(Eq. (1)) which forces the strong coupling to unacceptably large values
(
>

0.14). Hence we conclude that minimal SU(5) is ruled out at large tan .
In the missing doublet model the large triplet Higgs mass correction is adequately
compensated by the constant  4% correction of Eq. (2). As shown in Fig. 7, this
cancellation results in near central values for 
s
and m
b
. However,  > 0 is clearly
7
.129 
.132 
 
.137 
6.2 
5.7 7 
7.9 
6.8 
7.4 
5.5 
.129 
.132 
.137 
Figure 6: Contours of 
s
(solid) and m
b
(dashed, labeled in GeV) with no GUT thresholds,
m
t
= 175 GeV, A
0
= 0 and tan  = 30.
required when tan  becomes large, as the large nite corrections have the wrong sign
in the  < 0 case, yielding values of m
b
which are unacceptably large.
4. Conclusion
In this talk we have presented results from a complete calculation of the one-loop
corrections to the masses, gauge, and Yukawa couplings in the MSSM. We have seen
that such a calculation allows us to reliably investigate various unied models to de-
termine whether they are compatible with current experimental data. In particular,
we found that the nite SUSY corrections, which are neglected in the leading loga-
rithm approximation, can substantially increase the prediction for 
s
and m
b
when
some of the SUSY partner masses are lighter than or of order M
Z
.
For small tan , we found that in the minimal SU(5) model, 
s
was somewhat
large (
s
> 0:124 with m
~q
< 1 TeV) and m
b
was larger than 5 GeV. The missing
doublet model gave much lower values of 
s
and m
b
, near their central values.
For large tan, the minimal SU(5) model GUT threshold correction became quite
large and positive, and this resulted in unacceptably large values of the strong cou-
pling (
s
>

0:14). The missing doublet model, on the other hand, had no trouble in
accomodating the central values of the strong coupling and m
b
. The values of m
b
were largely determined by important nite corrections, and we required  > 0 in
order for these corrections to result in acceptably small values of the bottom-quark
8
.112 
 
.112 
.118 
.115 6 
6.5 
5.6 
5 
4.9 
4.8 
.115 
.118 
Figure 7: Contours of 
s
(solid) and m
b
(dashed, labeled in GeV) in missing doublet
SU(5), with m
t
= 175 GeV, A
0
= 0 and tan = 30.
mass.
5. Acknowledgements
I would like to thank my collaborators Jonathan Bagger and Konstantin Matchev.
This work was supported by the U.S. National Science Foundation under grant NSF-
PHY-9404057.
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7. M. Swartz, SLAC preprint SLAC-PUB-6710 (1994).
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10


The Strong Coupling and the Bottom Mass in Supersymmetric Grand Unified Theories

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