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Predictive Supersymmetry from Criticality

a r X i v :h e p -p h /0611249v 2 4 D e c 2006UCB-PTH-06/20

LBNL-61989

Predictive Supersymmetry from Criticality

Yasunori Nomura and David Poland Department of Physics,University of California,Berkeley,CA 94720Theoretical Physics Group,Lawrence Berkeley National Laboratory,Berkeley,CA 94720Abstract Motivated by the absence of any direct signal of new physics so far,we present a simple supersymmetric model in which the up-type Higgs mass-squared parameter m 2H u crosses zero at a scale close to the weak scale.Such a theory may be motivated either by the conventional naturalness picture or by the landscape picture with certain assumptions on prior probability distributions of parameters.The model arises from a simple higher dimensional setup in which the gauge and Higgs ?elds propagate in the bulk while the matter ?elds are on a brane.The soft supersymmetry breaking parameters receive contributions from both moduli and anomaly mediations,and their weak scale values can be analytically solved for in terms of a single overall mass scale M .The expected size for M depends on whether one adopts the naturalness or landscape pictures,allowing for the possibility of distinguishing between these two cases.We also present possible variations of the model,and discuss more general implications of the landscape picture in this context.

1Introduction

Weak scale supersymmetry is an extremely attractive idea.It is based on a beautiful theoretical construction of enlarging the spacetime structure to anticommuting variables,and is supported indirectly by the successful uni?cation of gauge couplings at high energies[1].It also stabilizes the large hierarchy between the weak and the Planck scales due to a cancellation between the standard model and its superpartner contributions to the Higgs potential.In fact,this latter property has been one of the strongest motivations for weak scale supersymmetry.

From the experimental point of view,the most exciting aspect of weak scale supersymmetry is the existence of various superpartners at the TeV scale.Can we predict the spectrum of these superparticles?We already know,from the absence of a large new contribution to?avor changing neutral current and CP-violating processes,that the superparticle spectrum must have a certain special structure,such as?avor universality.Moreover,non-discovery of both superparticles and a light Higgs boson at LEP II puts strong constraints on the spectrum.This typically leads to?ne-tuning of order a few percent in reproducing the correct scale for electroweak symmetry breaking,and is called the supersymmetric?ne-tuning problem(for a recent analysis,see[2]).It seems plausible that successfully addressing this problem provides a key to the correct theory at the TeV scale,and to a fundamental mechanism or principle behind it.

There are two di?erent approaches towards the supersymmetric?ne-tuning problem.A con-ventional approach is to search for a model that is“natural.”In the context of the minimal supersymmetric standard model(MSSM),this amounts to looking for a model in which the su-persymmetry breaking mass-squared parameter for the up-type Higgs?eld,m2H

,is somehow

u suppressed at the weak scale,since the electroweak scale is determined approximately by

M2Higgs

the cosmological constant[6],and partly by the suggestion that string theory has an exponentially

large number of discrete nonsupersymmetric vacua[7],it has become increasingly plausible that

our universe is only one among a tremendous number of various universes,in which physical

constants can take vastly di?erent values.This“landscape”hypothesis may lead to a signi?cant

change in our notion of naturalness,and it is reasonable to consider the supersymmetric?ne-

tuning problem in this context.It has recently been argued that the landscape picture may lead to

a small hierarchy between the Higgs mass-squared parameter and the scale of superparticle masses

?m under certain assumptions on the probability distributions of various couplings and?m[8].

Speci?cally,under the existence of statistical“pressures”pushing?m towards larger values,the

relation v2~?m2/8π2may be obtained from environmental selection,where v is the electroweak scale.1Moreover,if the parameterμalso scans independently with?m and if the holomorphic

supersymmetry breaking Higgs mass-squared parameter,μB,is su?ciently small at a high scale,

then we obtain v2~|μ|2~|m2H u|~?m2/8π2.

It is interesting that the two di?erent pictures described above can both lead to a scenario in

crosses zero at a scale not much di?erent from which the supersymmetry breaking parameter m2H

u

the weak scale.In fact,the two pictures may not be totally unrelated.Suppose,for example,

that the ultraviolet theory at the gravitational or uni?cation scale gives universal scalar squared

masses m20(>0),as in the minimal supergravity scenario[11].In this case,the parameter m2H

u crosses zero at a renormalization scale of order the weak scale,as long as the gaugino masses are

small compared with|m20|1/2.This phenomenon is known as focus point behavior,and this class of theories was claimed to be natural[3],since|m2H

u|is relatively small at the weak scale and thus no strong cancellation is required between the two terms in the right-hand-side of Eq.(1).An

immediate criticism of this argument,based on the conventional viewpoint,is that if the value

of the top Yukawa coupling,y t,were di?erent,then the property of|m2H

u|being small at the weak scale would be destroyed—in other words,the fractional sensitivity of the weak scale,v,

to a variation of the top Yukawa coupling,?ln v2/?ln y t,is very large.This criticism,however,

is not appropriate if the property of|m2H

u|??m2at the weak scale is a result of environmental selection.In this case,if y t were changed,the scale of supersymmetry breaking masses,?m,would

also be changed in such a way that|m2H

u|~?m2/8π2??m2at the“new”weak scale~|m H u|. As a result,we always?nd|m2H

u|??m2at the“weak scale”regardless of the value of y t.The observed value of y t will then be determined as a result of(another)environmental selection,

presumably a combination of the consideration in[12]and others.

From the point of view of model-building,i.e.searching for the model describing physics above the TeV scale,we may then be motivated to look for a model in which|m2H

u|is suppressed compared with?m2at the weak scale,i.e.|m2H

u|crosses zero at a scale close to the weak scale. If this property arises without a strong cancellation between the“tree-level”and“radiative”

contributions to|m2H

u|,then we can consider that the model is natural in the conventional sense. Even if it arises due to a strong cancellation,however,the model may still be interesting since

it can arise as a result of environmental selection under certain circumstances.Note that the requirement of|m2H

u|being suppressed at the weak scale is di?erent from the one that the Higgs mass-squared parameter,|m2h|?|m2H u+|μ|2|,is suppressed at the weak scale,which should

always be the case.We are requiring that the cancellation(if any)must take place“inside”m2H

,

u and|μ|2.

and not between m2H

u

Since the condition of|m2H

u|??m2at the weak scale gives only one constraint on the large number of soft supersymmetry breaking masses,we clearly need other guiding principles to

narrow down the possibilities and obtain predictions on the superparticle masses.Without having a detailed knowledge of physics at the gravitational or uni?cation scale,we simply take the viewpoint that the physics at that scale should be“simple”–su?ciently simple that the resulting supersymmetry breaking masses also take a simple form.This clearly makes sense if we take the conventional“universe”picture,and may also be supported by the absence of large supersymmetric?avor-changing and CP-violating contributions(which would arise if the superparticle masses were chaotic).In the context of the“multiverse”(or landscape)picture, we merely hope that such a“simple”model is statistically preferred by the vacuum counting in the fundamental theory.In practice,if a su?ciently“simple”model de?ned at the high energy scale gives|m2H

u|??m2at the weak scale,we consider it interesting regardless of the level of

.

cancellation occurring in m2H

u

In this paper we present an example of such models.The model is very simple,and arises as a low-energy e?ective theory of higher dimensional theories in which the standard model gauge and Higgs?elds propagate in the bulk while matter?elds are con?ned on a(3+1)-dimensional brane.The compacti?cation scale is of the order of the uni?cation scale,and the low-energy e?ective theory below this scale is simply the MSSM.Upon stabilizing a volume modulus by a simple gaugino condensation superpotential,the superparticle masses in the low-energy theory receive contributions from both moduli and anomaly mediations.We?nd that

at a scale(very)close to the weak scale,satisfying the criterion this model gives vanishing m2H

u

described above.All the supersymmetry breaking parameters,except for the holomorphic Higgs mass-squared parameter,are predicted(essentially)in terms of a single overall mass parameter M,with the resulting spectrum showing a pattern distinct from conventional supergravity and gauge mediation models.This model gives a“non-hierarchical”spectrum of Mλ~m?f(=O(M)),

where Mλand m?

represent generic gaugino and sfermion masses,although variations of the f

model giving the“hierarchical”spectrum of Mλ~m?f/4π(~|μ|)may also be considered.The scale of the overall mass parameter M depends on which of the naturalness or landscape pictures we take,but will be generally in the range between O(v)and a multi-TeV scale.For the Higgs sector,we simply assume that the required structures for theμandμB parameters are prepared, presumably by statistical preference in the case that the landscape picture is adopted.

The paper is organized as follows.In the next section we present our model and derive predictions on the supersymmetry breaking masses which are independent of the picture adopted. In section3we discuss the implications of the model in both the“universe”and“multiverse”pictures,and argue that the di?erence can appear in the size of the overall mass scale for the superparticle masses.In section4we conclude by giving discussions on the issue of obtaining predictions for the superparticle masses in the landscape picture.In particular,we present several possible scenarios arising from a landscape of vacua in the“vicinity”of the particular model in section2,and elucidate under what conditions,or with what additional assumptions,the setup can give strong predictions on the superparticle spectrum.

2Model

In this section we present a simple model that has the property that the soft Higgs mass squared is vanishing at a scale close to the weak scale.We consider that physics above the uni?cation scale is higher dimensional,and that the standard model gauge and Higgs?elds propagate in the bulk while the matter?elds are localized on a(3+1)-dimensional brane.The low-energy e?ective theory is then given by the following4D supergravity action:

S= d4x√

T W aαW aα+C3W +h.c. ,(2)

4

where C is the chiral compensator super?eld,T is the moduli super?eld parameterizing the volume of the compact dimensions,and gμνis the metric in the superconformal frame.The super?elds H and M collectively represent the Higgs and matter?elds of the MSSM,i.e.H= H u,H d and M=Q i,U i,D i,L i,E i with i the generation index,and the superpotential W contains the usual MSSM Yukawa couplings W Yukawa.This setup naturally arises,for example,if grand uni?cation is realized in higher dimensions above the compacti?cation scale[13].2In Eq.(2),we

have assumed that moduli?elds other than T,e.g.ones parameterizing the shape of the compact dimensions,(if any)are absent in the low-energy theory.We have also assumed that higher order terms,e.g.terms involving powers of1/(T+T?),are su?ciently suppressed,which is technically natural since the theory is weakly coupled at the compacti?cation scale.

To obtain realistic phenomenology at low energies,the moduli?eld T must be stabilized.We assume that the stabilization superpotential for T takes the simple form arising from a single gaugino condensation.The superpotential W is then given by

W=W Yukawa+Ae?aT+c,(3)

where a is a real constant.The parameters A and c are constants of order unity and the gravitino mass(?1),respectively(in units of the4D gravitational constant M Pl?1018GeV, which is taken to be1).These parameters can be taken real in the presence of an approximate shift symmetry for Im T.Since the superpotential of Eq.(3)stabilizes the modulus T at a supersymmetry preserving anti-de Sitter vacuum,with T+T? ?2a?1ln(a/c),we need an uplifting(supersymmetry breaking)potential,which we take to be independent of T in the superconformal basis:

δS=? d4x√

D3branes,located at the bottom of a warped throat.(The con?guration of the gauge,Higgs and matter?elds described before corresponds to identifying them as D7-,D7-and D3-brane?elds,respectively.)

The minimization of the potential,derived from Eqs.(2–4),leads to supersymmetry breaking (F-term)expectation values for the compensator C and the modulus T:

F C

=m3/2,(5)

(T+T?)3/2

F T

m3/2≡M0,(6)

a(T+T?)

where m3/2is the gravitino mass.This implies that there is a little hierarchy between the sizes of F C and F T:

F C/C

(T+T?)=ln M Pl

2

localized strongly towards the SU(5)-violating brane.This reproduces the action of Eq.(2)at low energies while preserving the SU(5)understanding of the matter quantum numbers.

so that the supersymmetry breaking parameters in the MSSM receive comparable contributions from both moduli and anomaly mediations[15].Here,we have recovered the gravitational con-stant M Pl in the right-hand-side of Eq.(7).Note that the above Eqs.(5–7)are valid up to corrections of O(1/8π2)=O(1/ln(M Pl/m3/2)).

The supersymmetry breaking masses in the present model show the behavior of a reduced e?ective messenger scale,M mess,due to an interplay between the moduli and anomaly mediated contributions[16](for a simple proof,see[2]).By solving renormalization group equations at the one-loop level,the soft supersymmetry breaking masses at an arbitrary renormalization scaleμR are given by

M a(μR)=M0 1?b aμR ,(8) m2I(μR)=M20 r I?4 γI(μR)?1d lnμR ln M messμR ,(9) A IJK(μR)=M0 ?(r I+r J+r K)+2 γI(μR)+γJ(μR)+γK(μR) ln M mess

(M Pl/m3/2)1/2

,(11) where M U represents the compacti?cation scale,which is of the order of the uni?cation scale ≈1016GeV,and f is an O(1)coe?cient depending,e.g.,on A in Eq.(3).The parameter M0is de?ned in Eq.(6)and represents the overall mass scale for the supersymmetry breaking parameters.

The expressions of Eqs.(8–10)show that the supersymmetry breaking masses in this model take a very simple form:

M1=M2=M3=M0,(12)

m2?

Q i =m2?

U i

=m2?

D i

=m2?

L i

=m2?

E i

=0,m2H

u

=m2H

d

=M20,(13)

A u=A d=A e=?M0,(14)

-1000

100

200

300

400

500

600

700

800

1001000100001000001e+061e+071e+081e+091e+10

m a s s e s (G e V )μR (GeV )Supersymmetry breaking masses

Figure 1:Evolutions of soft supersymmetry breaking masses below M mess =5×109GeV for M 0=400GeV and tan β=10.Solid lines represent the gaugino masses (M 3,M 2and M 1from the top),dashed lines the ?rst two generation sfermion masses (m ?Q ,m ?U ,m ?D ,m ?L and m ?E from the top),and dotted lines the Higgs mass parameter (m H d and m H u from the top).Here,m Φ(Φ=?Q,?U,?D,?L,?E,H u ,H d )is de?ned by m Φ≡sgn(m 2Φ)|m 2Φ|1/2.The pole mass for the top quark is chosen to be the central value of the recently reported range m t =171.4±2.1GeV [19].at the e?ective messenger scale

M mess ?

4

Approximate ?avor universality for these corrections must be assumed in the case that M 0is not much larger than a TeV.

for illustrative purposes.In the?gure,we have taken the supersymmetry breaking masses of

Eqs.(12–14)at the scale M mess,and evolved them down using the one-loop renormalization

group equations of the MSSM.(The two-loop renormalization group equations have been used for

the supersymmetric parameters.)Note that while the soft supersymmetry breaking parameters

are depicted only forμR≤M mess,it should be understood that they are,in fact,generated at a scale of order M U.(The squark and slepton squared masses are negative at scales above M mess,

but this does not cause a problem since our vacuum is metastable at the time scale of the age of

the universe.)Below,we will choose M0and M mess to be free parameters of our analysis,since

these parameters have O(1)uncertainties that cannot be determined from the low-energy data

alone.The value of tanβis determined by the Higgs sector parameters,μandμB,whose origin

we leave unspeci?ed.5

A remarkable feature of the superparticle masses in Fig.1is that the up-type Higgs mass-

squared parameter crosses zero at the superparticle mass scale:

m2H

u

(μC)=0atμC?M0.(16)

While the precise value ofμC–the scale where m2H

u

crosses zero–depends on the values of

M mess and tanβ,it is of order M0for a wide range of these parameters.Note thatμC does not

depend on M0,since the renormalization group equations are homogeneous in M0.(If we take

M U to be a free parameter,instead of M mess,thenμC depends slightly on M0for a?xed M U,

through a weak dependence of M mess on M0.)In the example of M0=400GeV in Fig.1,the

value ofμC is within a factor of2from M0for M mess≈(109~1010)GeV for tanβ≈(5~30).(In fact,a value of M mess givingμC within a factor of2from M0can be found for tanβ≈(3~50). The mass squared for the right-handed stau,however,becomes negative at the weak scale for tanβ>~30.)These results do not change signi?cantly by including higher order e?ects,e.g.the two-loop renormalization group e?ects,or by varying the top quark mass within a2σrange of the recently reported value,m t=171.4±2.1GeV[19].At the leading order,we?nd from Eq.(9) that the scaleμC is given by

μC≈M mess exp 8π2 6y2t?3g22?

32g23y2t?36y4t+18g22y2t+3g42

≈10?7M mess,(17) where the top Yukawa coupling,y t,and the SU(3)C and SU(2)L gauge couplings,g3and g2,are evaluated at the scaleμR?μC,and we have neglected the small e?ects from the bottom Yukawa

0200400600800

100051015202530M a , m F ~, m H d (G e V )

tan βSupersymmetry breaking parameters (M a , m F ~, m H d

)0

200400600800100051015202530

?A t ,b ,τ, m F ~3

(G e V )

tan βSupersymmetry breaking parameters (?A t,b,τ, m F ~3

)Figure 2:Predictions for the soft supersymmetry breaking parameters as a function of tan βfor M 0=400GeV.The left panel shows the predictions for the gaugino masses (solid;M 3,M 2and M 1from the top),the ?rst two generation sfermion masses (dashed;m ?Q ,m ?U ,m ?D ,m ?L and m ?E from the top),and the down-type Higgs boson mass m H d (dotted).The right panel shows those

for the third generation scalar trilinear interaction parameters (solid;A b ,A t and A τfrom the top)and the third generation sfermion masses (dashed;m ?D 3,m ?Q 3,m ?U 3,m ?L 3and m ?E 3from the top).For A t ,A b and A τ,which are negative,the absolute values are plotted.The scalar trilinear interaction parameters for the ?rst two generations,A u ,A d and A e ,are not shown.

coupling,y b ,and the U (1)Y gauge coupling,g 1.To obtain μC ?M 0,a larger M 0requires a larger M mess ∝M 0.For fundamental parameters of the theory,this implies f ∝M 0(see Eq.(11)).

Since the superparticle mass scale M 0is close to μC ,we can evaluate the soft supersymmetry breaking parameters at the superparticle mass scale M 0approximately by substituting Eq.(17)into Eqs.(12–14).This gives predictions for all the supersymmetry breaking masses,except for the holomorphic Higgs mass-squared parameter μB (and m 2H u ),in terms of the overall mass

scale M 0and the running gauge and Yukawa couplings at that scale.Note that we even do not have to know the value of M mess –for given values of M 0and tan β,which we need to obtain the values of the Yukawa couplings,we can predict all the supersymmetry breaking parameters with the assumption of Eq.(16).

In Fig.2,we present the predicted values of the supersymmetry breaking parameters for M 0=400GeV as a function of tan β.The left panel shows the predictions of the gaugino masses,M a ,the ?rst two generation sfermion masses,m ?F ,and the down-type Higgs boson mass,m H d .The right panel shows the third generation scalar trilinear interaction parameters,A t,b,τ,and the third generation sfermion masses,m ?F 3.The scalar trilinear interaction parameters for

the ?rst two generations,A u,d,e ,are not shown.The predictions for M a ,m ?F ,m ?Q 3,m ?U 3,m ?L 3,

A t(and A u,which is not shown)are rather insensitive to the value of tanβ,while those for

m H

d ,m?

D3

,m?

E3

,A b,Aτ(and A d,A e)have weak sensitivities to tanβ.(The sensitivity is strong

for m?

E3

for tanβ>~30where it approaches zero.)For tanβ~10,the predicted ratios among the soft supersymmetry breaking parameters(including the?rst two generation scalar trilinear interaction parameters)are given by

M1:M2:M3:m?

Q :m?

U

:m?

D

:m?

L

:m?

E

:m?

Q3

:m?

U3

:m?

D3

:m?

L3

:m?

E3

:

m H

d

:?A u:?A d:?A e:?A t:?A b:?Aτ

?0.71:0.91:1.8:1.5:1.4:1.4:0.52:0.30:1.3:1.1:1.4:0.51:0.28:

1.1:

2.2:2.6:1.3:1.7:2.5:1.

3.(18) Here,we have presented the numbers in units of M0.Note that these numbers are subject to errors of O(10%),coming from“higher order”e?ects,for quantities associated with the colored superparticles.(The errors for quantities that are not associated with the colored superparticles are smaller.)In the case that we take the“universe”picture,these e?ects include the fact that the superparticle mass scale M0does not“coincide”withμC,although the two are of the same order.This source of errors does not exist if we adopt the“multiverse”picture,where M0and μC are very close.

The predictions of Eq.(18)have a sensitivity to the value of M0,but only through the running of the gauge and Yukawa couplings.As a result,these predictions,and the predictions for the ratios of the supersymmetry breaking masses obtained from Fig.2,are valid in a wide range of M0with only small corrections.In the case that M0is in a multi-TeV region(as will be the case in the“multiverse”picture;see the next section),the corrections are still smaller than about 10%.For example,the predictions of Eq.(18)change for M0=3TeV to

M1:M2:M3:m?

Q :m?

U

:m?

D

:m?

L

:m?

E

:m?

Q3

:m?

U3

:m?

D3

:m?

L3

:m?

E3

:

m H

d

:?A u:?A d:?A e:?A t:?A b:?Aτ

?0.63:0.90:1.8:1.5:1.4:1.4:0.56:0.33:1.3:1.0:1.4:0.56:0.31:

1.1:

2.2:2.7:1.4:1.8:2.5:1.4,(19) but these are not much di?erent from the ones in Eq.(18).

We?nally discuss the Higgs sector of the model.To have the correct electroweak sym-metry breaking phenomenology,theμandμB parameters must be of order the weak scale. In particular,the classical contribution to B≡μB/μof order the gravitino mass must be suppressed.Here we simply assume that the value of B is su?ciently suppressed,for exam-ple the case that B is somehow dominated by the quantum(anomalous)contribution:B= 2M0{γH u(μR)+γH d(μR)}ln(M mess/μR).(This expression for B is,in fact,a solution to the

one-loop renormalization group equation.)We may also consider the case that μis generated by the expectation value of a singlet ?eld through W =λSH u H d (at least in the context of the “universe”picture),whose e?ect on the evolutions of the Higgs soft masses are suppressed if the value of λis su?ciently small.

3Implications

We have seen that the model given by Eqs.(2,3,4)provides the predictions of Eq.(18),which depend only very weakly on the values of tan βand M 0(see Fig.2and Eq.(19)).The expected range for the overall scale M 0,however,di?ers depending on the scenario we consider.In this section we discuss this issue,as well as other phenomenological implications of the model.Let us ?rst take the conventional “universe”picture,i.e.the overall scale M 0does not e?ectively “scan.”In this case,our guiding principle will be “naturalness,”i.e.the observed scale of electroweak symmetry breaking,v ?174GeV,should be a “typical”value in the parameter space of the model.For ?xed values of the supersymmetric couplings,this is rather clear in our model because of the suppression of m 2H u relative to the other soft masses at the weak

scale.(We assume that the Higgs sector is arranged such that there is no large μB term of order μm 3/2.)Naturalness of the model becomes clearer when compared with other,typical supersymmetry breaking models.Consider,for example,a model in which the supersymmetry breaking parameters of Eqs.(12–14)are generated at the uni?cation scale,M U ≈1016GeV,as in the pure moduli mediated model of [18].In this case,the size of the up-type Higgs mass squared |m 2H u |,relative to the other soft masses,is much larger at the weak scale.(The evolutions

of soft masses in the two models are depicted in Fig.3.)An important point is that while |m 2H u |keeps increasing towards the infrared from the scale μC where m 2H u crosses zero,dragged by

increasing M 3through g 3and y t ,the right-handed slepton masses m ?E stay almost constant,as they receive only small contributions through g 1.As a consequence,if the crossing scale μC is

much larger than the weak scale,we would obtain a hierarchy |m 2H u |/m 2?E ?1at the weak scale (see Fig.3b),leading to ?ne-tuning between the m 2H u and |μ|2terms in Eq.(1)under the LEP II

constraint of m ?E >~100GeV.Our model avoids this because μC is close to the weak scale (see Fig.3a).

Since there is no particular reason that μC is extremely close to the scale of superparticle masses,|μC ?M 0|/M 0?1,we expect that there is some discrepancy between the two quantities,e.g.|ln(μC /M 0)|=O (1).The value of m 2H u at the weak scale is then not much smaller than m 2?E ,so that the overall scale M 0is not much larger than the weak scale to avoid ?ne-tuning in Eq.(1).We typically expect 400GeV <~M 0<~1TeV,where the lower bound comes from m ?E >~100GeV.With these values of M 0,the physical mass for the lightest neutral Higgs boson,M Higgs ,can

-400-200

020*******

800100100001e+061e+081e+101e+121e+141e+16m a s s e s (G e V )μR (GeV )(a)-600-400

-200

200

400

600

800

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1200

100100001e+061e+081e+101e+121e+141e+16

m a s s e s (G e V )μR (GeV )(b)Figure 3:Evolutions of soft supersymmetry breaking masses in the model of section 2(the left panel),and in the model where the supersymmetry breaking parameters of Eqs.(12–14)are given at the uni?cation scale,M U ≈1016GeV (the right panel).Here,we have taken M 0=400GeV and tan β=10in both cases.Solid lines represent the gaugino masses (M 3,M 2and M 1from the top),dashed lines the ?rst two generation sfermion masses (m ?Q ,m ?U ,m ?D ,m ?L and m ?E from the top),and dotted lines the Higgs mass parameter (m H d and m H u from the top).

easily exceed the experimental lower bound of M Higgs >~114GeV [20].This is because our model provides a relatively large value of A t at the weak scale,so that we can avoid the Higgs-boson mass bound with relatively small top squark masses.(The importance of large A t in reducing ?ne-tuning was particularly emphasized in Refs.[5,2].)In Fig.4we plot M Higgs ,calculated using FeynHiggs 2.4.1[21],as a function of M 0for tan β=5(dotted line)and tan β=10(solid line).The μparameter is chosen to be μ=150GeV arbitrarily,but the dependence of the result on μis very weak.From the ?gure,we expect that M Higgs <~120GeV.The value of B

is given by B ≈M 20/μtan β,so that the preferred tan βrange of 5<~tan β<~20requires a

somewhat small value of B of order 0.1M 20/μ.The sensitivity of the weak scale to variations of

the supersymmetric parameters is also not so large in this model,since there is no superparticle that has a particularly large mass compared with others.The lightest supersymmetric particle (LSP)is either (a neutral component of)the Higgsino or the right-handed stau.In the former case the LSP may be the dark matter of the universe if it is produced nonthermally.In the latter case it will have to decay into some neutral particle,e.g.the axino –the supersymmetric partner of the axion,which may compose the dark matter.

Let us now turn to the case that we adopt the “multiverse,”or the “landscape,”picture.More precisely,we now assume that the overall supersymmetry breaking parameter M 0has di?erent values in di?erent “parts”of the multiverse,or in di?erent vacua of the theory,with larger

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122

124

126400600800100012001400160018002000

M H i g g s (G e V )M 0 (GeV )The lightest Higgs boson mass

Figure 4:Physical Higgs boson mass M Higgs as a function of M 0for tan β=5(dotted line)and tan β=10(solid line).The μparameter is chosen to be μ=150GeV.The horizontal dashed line represents the experimental lower bound of M Higgs ?114GeV.

values preferred by some positive power n :dP ∝dM n 0,where P is the probability distribution

function.In general the distribution of M 0depends on the structure of the supersymmetry breaking (uplifting)sector and the sector that produces the constant term c in the superpotential,

and the assumption of dP ∝dM n 0corresponds typically to tree-level supersymmetry breaking

(since tree-level supersymmetry breaking naturally prefers larger breaking scales).Note that since environmental selection “locks”the value of d in Eq.(4)as d ≈|c |2,through the condition for the cosmological constant being small [6],all the supersymmetry breaking parameters (including the ones generated through direct interactions with the uplifting sector,if any)scale in a similar way.As discussed in Ref.[8],this leads to a small hierarchy between the weak scale and the scale

of the sfermion masses.With the statistical pressure of dP ∝dM n 0,the sfermion masses ?m are

pushed towards larger values,but not beyond the scale where the Higgs mass-squared parameter m 2h crosses zero,since a larger ?m

would lead to the recovery of electroweak symmetry in the Higgs sector,a situation hostile to the existence of observers.This leads to |m 2h |=O (?m 2/8π2)??m 2,since m 2h becomes zero around the scale ?m

.Moreover,if the parameter μscans independently,and if the parameter B is su?ciently small at a high scale,then we obtain |μ|2=O (?m 2/8π2)??m 2

and thus also |m 2H u |=O (?m

2/8π2)??m 2—the property we found in our model (see Eq.(16)).In order for this argument to be signi?cant,the model must satisfy the conditions for μand B given above,which we assume to be the case.An implication of this picture is then that the overall

scale parameter M0is expected to be somewhat larger than the weak scale:M20/8π2~|m2H u|~|μ|2~v2.The precise hierarchy depends on the strength of the pressure n,but we generically expect M0to be in a multi-TeV region.This implies that in this picture all the superparticles, other than the Higgsinos,as well as all the CP-odd,the heavier CP-even,and the charged Higgs bosons,A0,H0and H±,have masses in this region(~M0>~1TeV).The ratios between the superparticle masses are still given by Eq.(19),and the three Higgs boson masses are given by

.

m A0~m H0~m H±~m H

d

The spectrum just described can lead to quite distinct phenomenology.For example,if M0 is somewhat large,e.g.M0>~2TeV,all the superparticles and heavier Higgs bosons are beyond the discovery reach of the LHC,except for the Higgsinos.Thus,the LHC will e?ectively see the (one Higgs doublet)standard model,plus possibly the Higgsinos.Discoveries of superparticles, however,may be possible if M0is lower.The LSP is the neutral component of the Higgsinos, which may be the dark matter of the universe.For example,if M0?3TeV,the gravitino mass is m3/2?100TeV,and the moduli?eld mass is m T=O(1000~10000TeV).The moduli ?eld is expected to dominate the energy density of the early universe,and then it decays into the superparticles and gravitinos,which in turn decay into the LSP.With these masses,the constraint from big-bang nucleosynthesis can be avoided(see e.g.[22])and the LSP may compose the dark matter,presumably with some(small)amount of dilution of its energy density.Alternatively, the LSP may decay into a lighter particle,e.g.the axino.

4Discussions:Predictions from the Landscape?

Since it has been di?cult to?nd ways of experimentally“testing”the landscape picture,it is important to consider what implications it can have on the low-energy spectrum and what predic-tions we can get from it when combined with additional assumptions.In this paper we discussed a framework which may either arise from the naturalness consideration in the conventional pic-ture or from the landscape picture under certain circumstances,and presented an example model which leads to strong predictions of the superparticle masses.The essential ingredients of the framework were

(i)The up-type Higgs mass-squared parameter m2H

crosses zero at a scale close to the super-

u

particle mass scale.

(ii)The structure of the theory at the uni?cation(or compacti?cation)scale is“simple”as far as the observable sector is concerned.

The reason that this can lead to strong predictions,despite the fact that each ingredient is not necessarily giving a very strong constraint,is that a generic theory satisfying(ii)does not typically lead to the property of(i),so that the combination of these two criteria can be a very strong

constraint on models.The model we presented has a fairly simple structure at the uni?cation scale,arising from a simple setup in higher dimensions,and yet gives a vanishing m2H

u

at a scale close to the weak scale.All the supersymmetry breaking parameters,except for the holomorphic Higgs mass-squared parameterμB(and m2H

u

),are predicted in terms of a single overall mass scale M0(and tanβ= H u / H d ).The parameter M0is expected to be in a multi-TeV region if it scans with a preference towards larger values.

On the other hand,it is clear that the speci?c model discussed above is not a unique model satisfying(i)and(ii).For example,we can consider the situation in which the scalar masses are approximately universal at a high scale,with somewhat suppressed gaugino masses.As observed in[3],this leads to suppressed m2H

u

at the weak scale.In fact,this situation can be realized in the setup of Eqs.(3,4)if the supersymmetry breaking(uplifting)sector gives universal scalar masses through direct interactions with the observable sector?elds.6In the context of the landscape picture,with a statistical pressure acting towards larger values for the supersymmetry breaking masses,this can lead to relatively degenerate scalar masses in a multi-TeV region and gaugino masses in a few hundred GeV region,with the relative gaugino masses still given by Eq.(18).The Higgsino masses are comparable to the gaugino masses,and the value of tanβwill be relatively large of O(10),for an unsuppressed B parameter.(The possibility of a relatively unsuppressed B parameter is an advantage of unsuppressed scalar masses.)In either of these models,the spectrum of superparticles is special such that it leads to a suppressed value of m2H

u

at the weak scale,which appears to us as a result of an accidental cancellation due to the speci?c values of the observed gauge and Yukawa couplings.

While the conditions of(i)and(ii)are keys to obtain strong predictions for the superparticle spectrum,neither is a necessary consequence of the landscape picture.Indeed,it is possible that environmental selection leads to the Higgs mass-squared parameter being small due to

cancellation between m2H

u andμ2,and not just inside m2H

u

,in which case(i)is not necessarily

satis?ed.Moreover,a simple ultraviolet structure may not be preferred by the statistics of landscape vacua,and the condition(ii)may also be violated.In these cases we lose a strong constraint on the superparticle spectrum,reducing predictivity,but may still get an interesting pattern for the spectrum.For example,landscape statistics may prefer the case in which the supersymmetry breaking(uplifting)sector gives somewhat random scalar masses through direct interactions with the observable sector,in the setup of Eqs.(3,4).(Flavor universality may still have to be assumed unless these masses are very large.)With the statistical pressure of pushing the overall mass scale to larger values,we?nd the Higgs mass-squared parameter somewhat suppressed compared with the scalar superparticle masses.The spectrum will then contain the

scalar superparticles and the Higgsinos in a multi-TeV region,whose masses do not obey simple relations.The gaugino masses,however,may still be of order a few hundred GeV and obey Eq.(18)in the case that the direct e?ect from the supersymmetry breaking sector is suppressed in the gauge kinetic functions.Deviations from Eq.(18),however,can also occur,e.g.,if the moduli-stabilization and uplifting sectors deviate from the minimal form of Eqs.(3,4),in which case the gaugino masses unify at a scale that is not necessarily the intermediate scale of Eq.(15), or if the direct e?ect is not suppressed in the gauge kinetic functions,in which case the gaugino masses are of order a multi-TeV.The value of tanβwill generically be of O(1)for an unsuppressed value for the B parameter.

To summarize,we have argued that both the conventional naturalness picture and the land-

crosses zero scape picture(with certain assumptions)may point to a scenario in which m2H

u

near the weak https://www.sodocs.net/doc/7015172756.html,bining this constraint with a simple ultraviolet structure can lead to a highly predictive superparticle spectrum,an example being the model presented in section2.The model predicts all the supersymmetry breaking parameters,except for the holomorphic Higgs

),in terms of a single overall mass scale M0(with a weak mass-squared parameterμB(and m2H

u

dependence on tanβ).This parameter is expected to be of order a few hundred GeV if it does not scan but in a multi-TeV region if it does scan,allowing for the possibility of experimentally distinguishing between these two cases.We have also discussed implications of the landscape picture on the supersymmetry breaking masses in a general setup arising from Eqs.(3,4)with possible additional interactions.Depending on the form of these interactions,strong predictions on the entire superparticle masses may be lost,but some predictions,such as those on the gaug-ino masses,may still be preserved.It is interesting that the experimental observation of one of these spectra may hint at possible statistical pressures acting on parameters of the theory,and thus the gross structure of vacua in the“vicinity”of our own one.

Acknowledgments

We thank Lawrence Hall for discussions.This work was supported in part by the Director,O?ce of Science,O?ce of High Energy and Nuclear Physics,of the US Department of Energy under Contract DE-AC02-05CH11231.The work of Y.N.was also supported by the National Science Foundation under grant PHY-0555661,by a DOE Outstanding Junior Investigator award,and by an Alfred P.Sloan Research Fellowship.

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工程测试技术试题及答案

工程测试技术试题及答案Last revision on 21 December 2020

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控制软件说明书

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一.系统介绍―――――――――――――――――――――――――――――2二.软件的安装――――――――――――――――――――――――――――2 三.基本信息设置―――――――――――――――――――――――――――2 1)部门班组设置―――――――――――――――――――――――――3 2)人员资料管理―――――――――――――――――――――――――3 3)数据库维护――――――――――――――――――――――――――3 4)用户管理―――――――――――――――――――――――――――3 四.门禁管理―――――――――――――――――――――――――――――4 1)通迅端口设置―――――――――――――――――――――――――42)控制器管理――――――――――――――――――――――――――43)控制器设置――――――――――――――――――――――――――64)卡片资料管理―――――――――――――――――――――――――11 5)卡片领用注册―――――――――――――――――――――――――126)实时监控―――――――――――――――――――――――――――13 五.数据采集与事件查询――――――――――――――――――――――――13 六.考勤管理―――――――――――――――――――――――――――――14 1)班次信息设置――――――――――――――――――――――――――14 2)考勤参数设置――――――――――――――――――――――――――15 3)考勤排班――――――――――――――――――――――――――――15 4)节假日登记―――――――――――――――――――――――――――16 5)调休日期登记――――――――――――――――――――――――――16 6)请假/待料登记―――――――――――――――――――――――――17 7)原始数据修改――――――――――――――――――――――――――17 8)考勤数据处理分析――――――――――――――――――――――――17 9)考勤数据汇总―――――――—――――――――――――――――――18 10)考勤明细表—―――――――――――――――――――――――――18 11)考勤汇总表――――――――――――――――――――――――――18 12)日打卡查询――――――――――――――――――――――――――18 13)补卡记录查询—――――――――――――――――――――――――19

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1.

2.对于一阶传感器系统,当其时间常数(或τ) 越小,其频率响应特性越好。 3.激波管标定系统中,激波管的作用是一种动态 标定设备,能产生阶跃压力信号输出。 4.金属电阻应变片的规格一般以面积(或长× 宽)和初始阻值表示。 5.用电阻应变片测量构件的变形,影响电阻应变 片电阻变化的因素有:应变片的灵敏度和初始阻值、被测构件的应变量、以及应变片沿构件的粘贴方向。(因为:△R=KεR,K为灵敏度,R为应变片初始阻值,ε被测构件的应变量) 6.将电阻丝绕成应变片后,由于存在横向效应, 其灵敏系数一般会减小。 7.在电桥测量中,由于电桥接法不同,输出电 压的灵敏度也不同,全桥接法可以得到最大灵敏度输出。 8.应变片的温度误差补偿方法通常可分为:桥 路补偿法、应变片自补偿法。

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