Going beyond the theta function approximation we discuss supersymmetric
unification of gauge couplings with exact decoupling of light and heavy
particles at energy scales below their masses. We find that the Minimal SUSY
model is strongly disfavored while the Missing Doublet Model survives with GUT
scale masses rising into the $10^{18}$ GeV region.Comment: 12 pages, revtex, no figure

Clavelli, L.

Coulter, P. W.

English

arXiv

Going beyond the theta function approximation we discuss supersymmetric unification of gauge couplings with exact decoupling of light and heavy particles at energy scales below their masses. We find that the Minimal SUSY model is strongly disfavored while the Missing Doublet Model survives with GUT scale masses rising into the 10^{18} GeV region

Clavelli, L

Coulter, P W

CERN Document Server

hep-ph@xxx/9507261 UAHEP954
SUSY Unication with Smooth Threshold Behavior
L. Clavelli and P.W. Coulter
Department of Physics and Astronomy
University of Alabama
Tuscaloosa, Alabama 35487
Abstract
Going beyond the theta function approximation we discuss supersymmetric
unication of gauge couplings with exact decoupling of light and heavy par-
ticles at energy scales below their masses. We nd that the Minimal SUSY
model is strongly disfavored while the Missing Doublet Model survives with
GUT scale masses rising into the 1018 GeV region.
11.30.Pb,14.80.Ly
Typeset using REVTEX
1
Although there are tantalizing indications that the elementary forces become approx-
imately supersymmetric in the TeV region and grand unied at a scale above 1016 GeV,
there remain two possible problems that have recently received some attention. Problem
number one is the following. The strong coupling constant at the Z scale, 3(MZ), predicted
in supersymmetric (SUSY) grand unied theory (GUT), while agreeing with the apparent
value at the Z resonance, disagrees markedly with the value expected from low energy mea-
surements which lies 12% to 20% lower (3 to 5 ). The dierence between the apparent
value at the Z and the actual value is one of the possible indications of a light gluino [1]
and can be correlated with the excess in Z ! bb [2{4]. Problem number two lies in the fact
that the scale at which the three gauge couplings appear to unify,  2  1016 GeV, lies about
two orders of magnitude below what would be expected in a more fundamental picture such
as string theory. It is possible that the solution to both problems lies in physics at energy
scales far above that of present accelerators. GUT scale eects are at present highly model
dependent. One approach has been to assume that, at the scale at which the U(1) and
SU(2) couplings meet, the SU(3) coupling deviates from these by a fractional amount  as
a parametrization of gravitational or mass-splitting eects among the GUT scale particles.
In such an approach the three couplings would not be seen to meet. A ve percent eect
at the GUT scale can lead to a ten percent dierence in the predicted 3(MZ) [5{7]. A
suggested approach to the second problem [8{10], is to seek an intermediate scale at which
new particles lie which could redirect the gauge couplings to a unication nearer the Planck
scale.
In order to explore alternative solutions to the above problems, we have, in a recent
paper [11], proposed the following point of view. We assume that gravitational eects can be
neglected. With this assumption of negligible gravitational eects, the model dependence of
GUT scale eects lies in the unknown degeneracy splitting among GUT scale particles as well
as the possibility of higher GUT scale Higgs representations. In the minimal supersymmetric
model (MSSM), the GUT scale masses are MV for the GUT scale gauge boson supermultiplet
and M and MD for the GUT scale Higgs supermultiplets. If the theory is unied far above
2
these masses, it will remain unied (in the theta function approximation) down to the
maximum GUT scale mass below which calculable deviations from unication will occur
depending on the three GUT scale masses. The full range of possible GUT scale eects
in this model is then determined by running over all possible values for MV , M, and MD
consistent with proton decay and other experimental constraints. In the Missing Doublet
Model (MDM) [12] which has a richer GUT scale Higgs spectrum, there are four masses
to be varied, MV , M, MD, and M. Above the highest GUT scale particle, the MSSM
is asymptotically free while the unied coupling constant in the MDM grows with energy.
This leads to a possible new constraint in the MDM, namely that the gauge coupling not
become grossly non-perturbative as one approaches the Planck scale. It might be considered
an appealing feature of the MDM that the coupling constant grows above the GUT scale
becoming strong near the Planck scale where unication with gravity might then be natural.
The result of this exercise [11] was that GUT scale degeneracy splitting in the MSSM does
not solve either of the two problems mentioned above. On the other hand the MDM does
lead to a value of 3(MZ) in good agreement with expectations from low energy data. This
latter result is suggested in the earlier work of [13] and is also noted in [7]. For deniteness
we adopt a low energy value for 3(MZ) spanning the range between two recent low energy
analyses [14,15]. Thus we take
3(MZ) =
8><>>:
:104 :005 (heavy gluino case)
:119 :005 (light gluino case)
(1)
The lower extremes of this range correspond to the analysis of [14] while the upper extremes
correspond to that of [15]. Many other low energy analyses are consistent with these ranges
and virtually all (except for some from  decay) lie within two  which is sucient for our
present considerations. Another result in the MDM was that the GUT scale, dened as the
maximum mass of GUT scale particles, can in fact increase into the 1017 GeV region [11].
In the current work we wish to explore the further renement of smooth threshold behav-
ior at the SUSY and GUT scales. Recently these smooth threshold eects have been given
3
some renewed attention [7,16]. The rst of these considers smooth thresholds at the SUSY
scale only, relying on an arbitrary shift of 3 to parameterize GUT scale eects. The second
does not discuss the solution ranges for the GUT scale masses. In the region of overlap our
results are consistent with theirs and we extend their conclusions.
It is well known that the variation in a gauge coupling (q) due to the one-loop propagator
correction from a fermion of mass mi in the Euclidean region is given [17] by
4q
d
dq
−1(q) = −2binf (q=mi) (2)
where
nf(q=m) = 6
Z 1
0
dxx2(1− x)2
x(1− x) +m2=q2
= 1−
3
2
(w2 − 1)
"
1−
w2 − 1
2w
ln

w + 1
w − 1
#
(3)
with
w =
q
1 + 4m2=q2: (4)
For a bosonic loop one would have
4q
d
dq
−1(q) = −2binb(q=mi) (5)
with
nb(q=m) = 1 + 3(w
2 − 1)

1−
w
2
ln

w + 1
w − 1

: (6)
nf (q=m) and nb(q=m) exhibit the decoupling behavior,
nf;b(q=m)!
8>><>:
1 q  m
0 q  m
(7)
So that, as an approximation, one can write for the contribution from a particle of mass m
as
nf;b(q=m) = (q −m) (theta function approximation): (8)
The n’s can be written analytically as perfect derivatives
4
nf;b(q=m) = q
d
dq
ff;b(q=m) (9)
with
ff(q=m) =
w2
2
"
1−
w2 − 3
2w
ln

w + 1
w − 1
#
(10)
fb(q=m) =
w2
2

−1 + w ln

w + 1
w − 1

(11)
Thus if one integrates eq.2 from some q0 to q1 the result is
4−1(q0) = 4
−1(q1)− 2b [ff (q0=m)− ff (q1=m)] (12)
with the corresponding result, changing ff for fb, in the case of a contribution from a boson.
If q0=m 1 q1=m , this becomes
4−1(q0) = 4
−1(q1)− 2b
"
− ln(q1=m) +
5
6
+
q20
10m2
−
5
2
m2
q21
+O

(q0=m)
4; (m=q1)
4 ln(q1=m)
i
(13)
The theta function approximation is equivalent to keeping only the rst term in the square
bracket. The constant term can be taken into account in the theta function approximation
by imposing a discrete shift (matching condition) in the couplings at q = m but in practice
this shift is generally neglected. If there are particles in the vicinity of q0 or q1 the power
series in eq.13 is slowly convergent and the theta function approximation becomes poor. Of
course, the theta function approximation could be dened to be exact if the non-logarithmic
eects were properly incorporated elsewhere as in extracting couplings from data [18] but
in practice this is not done and it seems much more economical to include the threshold
mass eects into the running of the couplings. Further discussion of the theoretical basis for
smooth decoupling ("Mass Dependent Subtraction Procedure") is given in [16]. Practically
all current grand unication studies including [19{22,11] have relied heavily on the theta
function approximation to the beta function. A notable exception has been the work of
[24] where the full nf (q=m) was used for the top quark and gaugino contributions together
5
with the full nb(q=m) for the SUSY scalars. However even in this work the eect of smooth
thresholds at the GUT scale was neglected. In the current work we extend the smooth
threshold behavior to the GUT scale particles and to the low-lying quarks, leptons, and
gauge bosons. Our purpose is to investigate the dierential eect on the gauge unication
solutions when one goes from theta function to smooth decoupling. The eect on the b=
mass ratio, which is not part of the current study, would be expected to be especially
signicant since the b Yukawa which, in the theta function approximation, is rising most
rapidly in the low energy region begins to be strongly suppressed as one approaches the b
scale if one imposes a smooth decoupling. The GUT scale eects are also expected to be
large. Therefore, we do not attempt to t the top quark mass, tan or the b= mass ratio.
In [11] we considered, in the theta function approximation, the eect of non-degeneracy
among the GUT scale particles in both the minimal supersymmetric model and the missing
doublet model. In such a treatment the GUT scale is considered to be the mass of the
heaviest of the GUT scale particles since if the couplings are unied there they remain
unied at higher energies. With smooth decoupling the couplings can be assumed to be
unied far above the GUT scale masses but will begin to diverge as one approaches the
GUT scale. We somewhat arbitrarily take unied couplings at the Planck mass assuming
that all GUT scale particles have much smaller masses and that perturbation theory is
still at least qualitatively valid there. Gravitational eects, which are beyond the scope of
the present paper, are supplementary to the eects studied here but should not void our
qualitative conclusions. In this paper we restrict our interest to gauge coupling unication.
We integrate the one-loop contributions exactly and analytically including the full threshold
behavior while treating the two loop (including Yukawa) contributions, 2Li , numerically
with a crude (but fully adequate) approximation to smooth decoupling ignoring the two
loop contributions of the GUT scale particles. The two loop contribution, 2L3 is found to be
only 5 to 7 percent of the analytic one loop contribution in the MDM and 9 to 17 percent
in the MSSM and roughly linearly related to 3(MZ) in each case. The b coecients are as
given in [13,11] except that we separate the GUT scale Higgs supermultiplet contributions
6
into separate contributions from bosons and fermions in the ratio of 1:2. Similarly the
contribution of the GUT scale gauge supermultiplet separates into bosons and fermions in
the ratio 11:(-2). The top Yukawa, t(MP ), at the gauge unication point is taken to be
between 0:1 and 0:9. Due to the xed point behavior large values of the top Yukawa rapidly
evolve down to values of order 3 so that the gauge couplings are relatively insensitive to
the GUT scale behavior and values of the Yukawa couplings. Then
4−1i (MZ) = 4
−1
i (MP )− 2
X
j
bi(j) [fj(MZ ;mj)− fj(MP ;mj)] + 
2L
i : (14)
At the Planck mass, 1:22  1019 GeV, we take the unication condition
4−13 (MP )− 1 = 4
−1
2 (MP )−
2
3
= 4−11 (MP )  4
−1
0 : (15)
We choose the Planck scale gauge coupling, 0, and the Planck scale top Yukawa, t(MP ), at
random as we do for the various GUT scale masses, MV ;MD;M;M, and the SUSY scales,
m0, and m1=2. For perturbative consistency, however, we require that 0 < 1=4, t(MP ) < 1,
and mj=MP < 1=5. The SUSY masses are chosen with 100 GeV < m0 < 1 TeV and 50
GeV < m1=2 < 330 GeV. Splitting between partners of left and right handed fermions is
neglected. The squark and slepton masses are dened in terms of m0 and m1=2 as in [16].
That is: m2~q = m
2
0 + 7m
2
1=2 , m
2
l;L = m
2
0 + 0:5m
2
1=2, and m
2
l;R = m
2
0 + 0:15m
2
1=2. We discard
as non-solutions values of these parameters inconsistent with the renormalization group
running and the experimental values −1(MZ) = 127:9  :2 and sin
2 W = :2320  :0008.
Further technical details on our \top-down" approach may be found in [11]. By such Monte
Carlo methods it is possible to completely determine the multidimensional solution space.
In [11] it was found that the MSSM, with or without GUT scale degeneracy breaking, was
inconsistent with the low energy data on 3 if proton decay constraints and a theoretically
desirable SUSY scale below 1 TeV were imposed. This was also noted in the degenerate
case by [25] and has been emphasized more recently by [4]. The lower limits on 3(MZ) in
the MSSM with theta function decoupling are consistent with those found by other authors
[26,23]. If one requires only agreement with the LEP values of 3 the inconsistency is
7
not apparent. One of the conclusions of [11] was that this inconsistency disappears if the
MSSM is replaced by the missing doublet model. This result was also noted in [7]. These
considerations are independent of whether or not the gluino is light (in the GeV region) as
is still not experimentally ruled out. In table 1 we compare the smooth threshold results
for 3(MZ) and the GUT scale masses with the results of [11] where GUT scale degeneracy
was broken but sharp (theta function) decoupling was used. Slight dierences between our
current requirements and those of [11] with regard to sin2 W and the b= mass ratio do not
aect the clear, qualitative conclusions that can be drawn from the comparison in table 1.
The numbers in table 1 are shown in the heavy gluino scenario but are not sensitive to this
choice. For example, with smooth decoupling, if one puts m1=2 = 0 (light gluino option)
the minimum 3(MZ) drops only to 0:113 in the MDM and only to 0:169 in the MSSM.
On the other hand Ref. [7] nds some preference in the MSSM for m1=2  m0 implying at
least a relatively light gluino. However, our results indicate that with smooth decoupling
at the GUT scale the MSSM cannot be saved by this mechanism. In the MDM we discard
solutions with 3(MZ) above 0:135 since these seem of no phenomenological interest. The
same requirement in the MSSM would eliminate all solutions leading to our conclusion that
the MSSM is no longer viable when smooth decoupling is taken into account.
To summarize the conclusions of this study we may say the following. In [11] we noted
that the MSSM with theta function threshold behavior predicted an 3(MZ) inconsistent
with extrapolations from low energy data. Our current results strongly reinforce this con-
clusion and disfavor the MSSM even if the higher LEP values of 3(MZ) are used. Because
of the very large values of 3 predicted in the MSSM with smooth thresholds, current esti-
mates of gravitational eects cannot salvage the situation without calling into question the
successful prediction of sin2 W . For this reason it is our opinion that the MSSM is highly
unlikely to be realized in nature. The low energy measurements of 3 and sin
2 W , therefore,
strongly suggest a richer GUT scale Higgs structure such as that given in the MDM. The
phenomenological superiority with respect to grand unication of the MDM over the MSSM
was rst pointed out in [11]. This model, when smooth threshold behavior is taken into
8
account, also contains unication solutions with the heaviest GUT scale particles in the 1018
GeV region as suggested by string theory. If we require 3(MZ) < 0:12 we nd, in fact,
that all the solutions in the MDM have M > 5  1017 GeV. All the MDM solutions have
the leptoquark gauge boson supermultiplet in the 1016 GeV region or below suggesting that
proton decay could be dominated by the ep decay modes expected in non-supersymmetric
SU(5). If we compare the unication lower limits on 3(MZ) from table 1 with the results
from low energy analyses given in eq.1 we see that the light gluino option is somewhat fa-
vored. However, if there are 10% eects from gravity or other sources this preference might
be eliminated.
The authors acknowledge useful comments on this work from Professor P.H. Cox of Texas
A&M University-Kingsville. This work was supported in part by the Department of Energy
under grant DE-FG05-84ER40141.
9
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10
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11
TABLES
TABLE I. Minimum and maximum values of 3(MZ) and the GUT scale masses in the uni-
cation solution space of the MSSM and MDM with either sharp or smooth decoupling. Underlined
values are upper or lower limits imposed for phenomenological reasons discussed in the text.
MSSM MSSM MDM MDM
sharp smooth sharp smooth
3(MZ) (:117; :133) (:174; :24) (:095; :114) (:116; :135)
MV (GeV) (0:7; 82)1015 (1:8; 130)1015 (2:3; 21)1015 (1:0; 6:4)1015
MD (GeV) (1:0; 24)10
16 (:01; 2:4)1018 (1:0; 18)1016 (1:0; 24)1016
M (GeV) (0:3; 15)1016 (:01; 2:4)1018 (1:2; 6:7)1016 (:021; 2:4)1018
M (GeV) - - (1:0; 7:8)1016 (0:21; 2:4)1018
12


SUSY Unification with Smooth Threshold Behavior

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