Table Of ContentOne Loop Predictions of the Finely Tuned SSM
Asimina Arvanitaki, Chad Davis, Peter W. Graham, and Jay G. Wacker
Institute for Theoretical Physics
Department of Physics
Stanford University
Stanford, CA 94305 USA
5
0
0 Abstract
2
n
We study the finely tuned SSM, recently proposed by Arkani-Hamed and Dimopoulos, at the one loop level. The
a
runnings of the four gaugino Yukawa couplings, the µ term, the gaugino masses, and the Higgs quartic coupling
J
are computed. The Higgs mass is found to be 130 – 170 GeV for M > 106 GeV. Measuring the Yukawa coupling
7 s
2 constantsatthe 10%levelcanbegintoconstrainthe SUSY breakingscale. Measuringthe relationshipsbetweenthe
couplings will provide a striking signal for this model.
3
v
4
1 Introduction
3 these states having mass in the 100GeV to 3 TeV range.
0 There is a universal form of the low energy effective
6
actionforthefinelytunedSSMthatpreservesgaugecou-
0 Recently there has been interest in studying a version of
pling unification and dark matter and has five relevant
4 the Supersymmetric Standard Model (SSM) where nat-
0 uralness is no longer a guiding principle [1]. This comes interactions – four Yukawa couplings from the gauginos
/ and the Higgs quartic coupling. These are predicted by
h atatimeofseveralgrowingproblemsassociatedwiththe
p standard implementation of naturalness [2]. The most high energy supersymmetry from four parameters: the
- pressing naturalness issue is the cosmological constant, Standard Model gauge couplings g1 and g2, tanβ, and
p
the scale of the scalar masses, M . At the LHC or NLC
e the experimental value of which appears to be fine-tuned s
h to one part in 10120 and completely dwarfs the standard it may be possible to measure five new couplings and ex-
v: hierarchy problem. While it is conceivable that these plain them from only two new parameters.
i two separate fine tunings are divorced, they could also Inthisnote,wecalculatetheoneloopbetafunctions
X
be linked with weak anthropicism [3]. There are other of these five couplings, as well as those of the µ term
r problemswiththeSSMdirectlyrelatedtoparticlephysics and the gaugino masses. We then run these couplings
a
issues,suchasthenon-discoveryofsuperpartnersatLEP from their SUSY values at Ms down to the top mass mt
or Fermilab, the lack of FCNCs, the non-discoveryof the [7, 8, 9, 10]. We do not compute threshold corrections
Higgs,andthenon-discoveryofprotondecay. Allofthese because they are subdominant to the large logarithms.
increase the fine-tuning required in the SSM. Every one We define two different effective tanβ that are relatedto
ofthesephenomenologicalproblemsisamelioratedbyde- the gaugino-Higgsino Yukawa coupling. By RG evolving
coupling the scalars [1, 4]. thesetoahigherscaleitispossibletodeterminethescale
of SUSY breaking.
The two major successes of the SSM [5] are gauge
coupling unification [6] and a viable dark matter candi-
date. However, removing the scalars of the SSM does
2 One loop beta functions
not significantly alter either of these predictions. If one
is willing to ignore the original motivation for the SSM
anddecoupleallbuttheonescalarHiggsdoubletrequired The tree level Lagrangiancontains the terms
forelectroweaksymmetrybreaking,thenoneimmediately
hasaphenomenologicallyviablemodelwithoutthe usual L ⊃ B˜(κ′1h†H˜1+κ′2hH˜2)
concernsofthe SSM. The existence oflightgauginosand +W˜a(κ h†τaH˜ +κ H˜ τah) λh4 (1)
1 1 2 2
Higgsinos is inferred indirectly through gauge coupling − | |
1
unification and evidence for dark matter, which point to µH˜1H˜2 (M1B˜B˜+M2W˜W˜ +M3g˜g˜).
− − 2
1
At the SUSY breaking scale the following relations are 117700
satisfied: 116600
LL
VV
κ′1 =r130g1sinβ κ′2 =r130g1cosβ HHsGesGe111145450000
ss
κ1 =√2g2sinβ κ2 =√2g2cosβ MaMa113300
λ= 35g12+g22 cos22β. (2) iggsiggs112200
8 HH
111100
However, these couplings run in a non-supersymmetric
fashion from the SUSY breaking scale down to low ener- 22 44 66 88 1100 1122 1144
LLoogg1100HHMMss(cid:144)(cid:144)GGeeVVLL
gies.
Allofthe followingresultsaregivenwithSU(5)nor-
Figure 1: The Higgs mass as a function of the SUSY
malization of the hypercharge. The beta function for the
breaking scale log (M /GeV). The upper bands are for
10 s
Higgs quartic coupling is
tanβ(M ) = 50 and the lower ones are tanβ(M ) = 1.
s s
16π2β = +24λ2 6y4 +12λy2 The width of each grey band is the experimental uncer-
λ − top top tainty, mainly due to m . The width of each black band
t
27 9
+ g4+ g2g2 is the uncertainty when expected improvements from a
200 1 20 1 2
future linear collider are taken into account.
9 9
+ g4 λg2 9λg2
8 2− 5 1 − 2
5 1
−8(κ41+κ42)− 4κ21κ22 andsimilarlyforκ2 afterchangingκ1 ↔κ2 andκ′1 ↔κ′2.
The beta function for the µ term is
2(κ′2+κ′2)2 (κ κ′ +κ κ′)2
− 1 2 − 1 1 2 2
+3λ(κ21+κ22)+4λ(κ′12+κ′22). (3) 16π2β = µ( 9 g2+ 9g2)
µ − 10 1 2 2
The beta function for the top Yukawa coupling is 3
+ κ κ M +2κ′κ′M
2 1 2 2 1 2 1
9 17 9
16π2β = y3 y ( g2+ g2+8g2) 3 1
ytop 2 top− top 20 1 4 2 3 + µ(κ2+κ2)+ µ(κ′2+κ′2). (7)
8 1 2 2 1 2
3
+ y (κ2+κ2)
4 top 1 2 The beta functions for the gaugino masses are
+y (κ′2+κ′2). (4)
top 1 2 16π2β = 8µκ′κ′ +2M (κ′2+κ′2) (8)
M1 1 2 1 1 2
As acheck,whenallκ’saresettozero,these β functions 16π2β = 12g2M +2µκ κ (9)
reproduce those of the Standard Model [11]. The beta M2 − 2 2 1 2
1
function for the bino Yukawa coupling is +2M2(κ21+κ22)
16π2βκ′1 = 3κ′1yt2op−κ′1(290g12+ 49g22) 16π2βM3 = −18g32M3. (10)
5 9 In the following sections we run the Yukawa cou-
+2κ′13+4κ′1κ′22+ 8κ′1κ21 plings and the mass terms fromthe SUSY breaking scale
3 3 down to the low scale. We examine the behavior of the
+4κ′1κ22+ 2κ1κ2κ′2 (5) various parameters at the low scale as a function of Ms.
andsimilarlyforκ′ afterchangingκ κ andκ′ κ′.
2 1 ↔ 2 1 ↔ 2 3 Higgs Mass
The beta function for the wino Yukawa coupling is
16π2β = 3y2 κ κ ( 9 g2+ 33g2) TheHiggsquarticcouplingatMs dependsonlyoncos2β
κ1 top 1− 1 20 1 4 2 and M and can easily be run down with the beta func-
s
+11κ3+ 3κ κ′2+ 1κ κ2 tions of the previous section. We find that the Higgs
8 1 2 1 1 2 1 2 is heavier than in the usual SSM with low-scale SUSY
+κ κ′2+2κ′κ κ′ (6) breaking [10, 12]. The dimensionful A-terms and µ term
1 2 1 2 2
2
are around the weak/dark matter scale and are small in
1.2
comparison to the SUSY breaking scale. They give fi-
nite threshold effects to the Higgs quartic coupling that 1.1
are O(A2/Ms2) and can be neglected in this model. We Ls 1
have used a top mass of 178.0 4.3 GeV [13]. The MS M
H
± Κ
rtoelpatYiounka[1w0a, 1co1u]pling was set to yt = 0.99±0.02 by the L(cid:144)mt0.9 Κ
H0.8 1
Κ Κ
2
mt =ytv(1+ 13616gπ322 −216yπt22). (11) 0.7 ΚΚ21''
2 4 6 8 10 12 14
For a SUSY breaking scale of 109 GeV, we find that Log10HMs(cid:144)GeVL
theHiggsmassvariesfrom140to165GeVascos2β goes
from0to1atthehighscale. TheHiggsmassasafunction Figure2: Theratioκ(m )/κ(M )asafunctionofM for
t s s
of Ms is shown in Fig. 1 for tanβ = 1 and tanβ = 50. fixed tanβ(Ms)=5.
For values of tanβ between 1 and 50, the Higgs mass is
between the bounds shown. The Higgs quartic coupling
is insensitive to tanβ for large tanβ.
Experimental uncertainties in yt and g3 lead to an 6.25
uncertainty in the prediction of the Higgs mass as shown
6
by the wide bands in Fig. 1. The error in the top mass
5.75
dominates while the uncertainty due to g is approxi-
3
mately one tenth as large. As a test of the theoretical 5.5
uncertainty,each5 ¯5fermionaddedinattheTeVscale 5.25
⊕
increases the Higgs mass by 0.2% for Ms =109 GeV. 5
A future linear collider may be able to measure the
4.75
Higgs mass to a precision of 100 MeV, the top mass to
4.5
200 MeV, and α to 1% [14]. The narrow bands in Fig.
s 2 4 6 8 10 12 14
1 show the uncertainty in the Higgs mass prediction us- Log10HMs(cid:144)GeVL
ing these more precise measurements and assuming the
current central value. The small error on the Higgs mass Figure 3: The solid line shows tanβ (m ) as a func-
low t
measurementcould allow the most precise determination tion of M . The dashed line is for tanβ′ (m ). Here
s low t
of the SUSY breaking scale within the context of this tanβ(M )=5.
s
model. If tanβ is measured to 50%, M will be known
s
to within an order of magnitude. Although the bands in
Fig. 1 asymptote at high scales making M difficult to
s
determine from the Higgs mass, we do not expect M to
be greater than 1013 GeV [1]. s 700 MM21
500 M3
Μ
4 Yukawa Couplings and Mass
V 300
e
G
Terms
200
150
The gaugino couplings are set at M by Eq. (2) and
s
RG evolved to mt. There are two separate low energy 100
definitions of tanβ, 4 6 8 10 12 14
Log10HMs(cid:144)GeVL
κ (m) κ′(m)
tanβ (m)= 1 tanβ′ (m)= 1 , (12)
low κ2(m) low κ′2(m) Figure 4: The gaugino masses and µ evaluated at mt as
a function of M for fixed tanβ(M )=5.
s s
that run from equal values at the SUSY breaking scale.
Running up from the weak scale to the point where they
3
unify provides a clear determination of the SUSY break- nomenal signal of high scale supersymmetry.
ing scale (Fig.3). If the couplings could be measured
to 10% at a future LC [15] this would determine M
s
Acknowledgements
to within a few orders of magnitude. Note that there
are fixed points in the evolution of some of the tanβ’s
at tanβ = 0,1, . However, the gaugino couplings do We would like to thank N. Arkani-Hamed, S. Dimopou-
changeasM isc∞hangedandthereforecanprovideause- los,S.Martin,A.Pierce,andS.Thomasforusefuldiscus-
s
ful measure of M even when tanβ does not change sions. Special thanks to G.F. Giudice and A. Romanino
s low
significantly with M . forpointingouterrorsinourcalculation[16]. Ourresults
s
arenowinagreement. C.D.istheMellamFamilyGradu-
The Yukawa couplings run significantly from their
ate Fellow. P.W.G. is supportedby the NationalDefense
supersymmetric values (Fig.2). We find that, for tanβ &
Science and Engineering Graduate Fellowship. J.G.W. is
5, the ratios κ(m )/κ(M ) are relatively unaffected by
t s
supported by NSF grant PHY-9870115and the Stanford
changes in tanβ. The four Yukawa couplings and the
Institute for Theoretical Physics.
Higgs quartic arefive independently measurableparame-
ters that are determined by the scale of SUSY breaking
and tanβ. Thus, this model predicts that these five cou-
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