素粒子物理学2素粒子物理学序論B

2010年度講義第11回

今回の目次

エネルギーフロンティア実験の紹介ヒッグス探索実験装置としてLHC

超対称性粒子・事象の探索

2

超対称性Supersymmetry (SUSY)

階層性問題と Fine Tuning

GUT scale (~1015 GeV)だとしたら電弱スケール(246GeV)との間に大きな隔たり ⇐ 不自然階層性問題と呼ばれる

これを受け入れても別の問題ヒッグス質量の放射補正エネルギースケールΛまで放射補正すると補正量は

4

δm2 ∼ λΛ2

µ2phys = µ2 − λΛ2

f

f

• •H H • • •

W, Z, HW, Z, H

Figure 1.20: Feynman diagrams for the one–loop corrections to the SM Higgs boson mass.

Cutting off the loop integral momenta at a scale Λ, and keeping only the dominant

contribution in this scale, one obtains

M2H = (M0

H)2 +3Λ2

8π2v2

[M2

H + 2M2W + M2

Z − 4m2t

](1.183)

where M0H is the bare mass contained in the unrenormalized Lagrangian and where we

retained only the contribution of the top heavy quark for the fermion loops. This is a

completely new situation in the SM: we have a quadratic divergence rather than the usual

logarithmic ones. If the cut–off Λ is very large, for instance of the order of the Grand

Unification scale ∼ 1016 GeV, one needs a very fine arrangement of 16 digits between the

bare Higgs mass and the radiative corrections to have a physical Higgs boson mass in the

range of the electroweak symmetry breaking scale, MH ∼ 100 GeV to 1 TeV, as is required

for the consistency of the SM. This is the naturalness of fine–tuning problem18.

However, following Veltman [137], one can note that by choosing the Higgs mass to be

M2H = 4m2

t − 2M2W − M2

Z ∼ (320 GeV)2 (1.184)

the quadratic divergences can be canceled and this would be even a prediction for the Higgs

boson mass. But the condition above was given only at the one–loop level and at higher

orders, the general form of the correction to the Higgs mass squared reads [138, 139]

Λ2∞∑

n=0

cn(λi) logn(Λ/Q) (1.185)

where (16π2)c0 = (3/2v2)(M2H + 2M2

W + M2Z − 4m2

t )2 and the remaining coefficients cn

can be calculated recursively from the requirement that M2H should not depend on the

renormalization scale Q. For instance, for the two–loop coefficient, one finds [138]

(16π2)2c1 = λ(114λ − 54g22 − 18g2

1 + 72λt)2 + λ2

t (27g22 + 17g2

1 + 96g2s − 90λ2

t )

−15

2g42 +

25

2g41 +

9

2g21g

22 (1.186)

18Note, however that the SM is a renormalizable theory and this cancellation can occur in a mathematicallyconsistent way by choosing a similarly divergent counterterm. Nevertheless, one would like to give a physicalmeaning to this scale Λ and view it as the scale up to which the SM is valid.

70

観測にかかる物理的な質量

裸の質量 補正量

O(1019)の補正からO(102)GeVの質量を作り出さないとならない⇐ Fine Tuning Problem

Fine Tuning 問題を避ける手段

　　　　　 の補正はボソンに特有（2次発散）ヒッグス”ボソン”のパートナーとなるフェルミオンがもし存在したら…

ボソンの2次発散をキャンセルできるフェルミオンループによる放射補正はボソンループと符号が反対

質量の縮退したフェルミオン・ボソンペアがあれば、Fine Tuning Problem は回避 ⇒ 超対称性ゲージボソン・フェルミオンは大丈夫ゲージ対称性とカイラル対称性のおかげ

5

δm2 ∼ λΛ2

f

f

• •H H • • •

W, Z, HW, Z, H

Figure 1.20: Feynman diagrams for the one–loop corrections to the SM Higgs boson mass.

Cutting off the loop integral momenta at a scale Λ, and keeping only the dominant

contribution in this scale, one obtains

M2H = (M0

H)2 +3Λ2

8π2v2

[M2

H + 2M2W + M2

Z − 4m2t

](1.183)

where M0H is the bare mass contained in the unrenormalized Lagrangian and where we

retained only the contribution of the top heavy quark for the fermion loops. This is a

completely new situation in the SM: we have a quadratic divergence rather than the usual

logarithmic ones. If the cut–off Λ is very large, for instance of the order of the Grand

Unification scale ∼ 1016 GeV, one needs a very fine arrangement of 16 digits between the

bare Higgs mass and the radiative corrections to have a physical Higgs boson mass in the

range of the electroweak symmetry breaking scale, MH ∼ 100 GeV to 1 TeV, as is required

for the consistency of the SM. This is the naturalness of fine–tuning problem18.

However, following Veltman [137], one can note that by choosing the Higgs mass to be

M2H = 4m2

t − 2M2W − M2

Z ∼ (320 GeV)2 (1.184)

the quadratic divergences can be canceled and this would be even a prediction for the Higgs

boson mass. But the condition above was given only at the one–loop level and at higher

orders, the general form of the correction to the Higgs mass squared reads [138, 139]

Λ2∞∑

n=0

cn(λi) logn(Λ/Q) (1.185)

where (16π2)c0 = (3/2v2)(M2H + 2M2

W + M2Z − 4m2

t )2 and the remaining coefficients cn

can be calculated recursively from the requirement that M2H should not depend on the

renormalization scale Q. For instance, for the two–loop coefficient, one finds [138]

(16π2)2c1 = λ(114λ − 54g22 − 18g2

1 + 72λt)2 + λ2

t (27g22 + 17g2

1 + 96g2s − 90λ2

t )

−15

2g42 +

25

2g41 +

9

2g21g

22 (1.186)

18Note, however that the SM is a renormalizable theory and this cancellation can occur in a mathematicallyconsistent way by choosing a similarly divergent counterterm. Nevertheless, one would like to give a physicalmeaning to this scale Λ and view it as the scale up to which the SM is valid.

70

−Y 2f Λ2

超対称性

ボソンとフェルミオンを交換する対称性

Qはスピノールの演算子fermionic operator なので以下の反交換関係を持つ

① 2回演算すると時空における平行移動時空の演算を含んでいる

②　　　　　　と交換する ⇒ 質量を変えない既知の全ての粒子に質量の同じパートナー（超粒子）が存在する

6

supersymmetry will be accomplished in Sec. 3d. In the following subsection it will be shownthat this Lagrangian also leads to the cancellation of quadratic divergencies from one–loop gaugecontributions to the Higgs two–point function πφφ(0), thereby extending the result of the previoussection. Finally, soft SUSY breaking is treated in Sec. 3f.

Many excellent reviews of and introductions to the material covered here already exist [6, 7]; Iwill therefore be quite brief. My notation will mostly follow that of Nilles [7].

3a. The SUSY Algebra

We saw in Sec. 2 how contributions to the Higgs two–point function πφφ(0) coming from the knownSM fermions can be cancelled exactly, if we introduce new bosonic fields with judiciously chosencouplings. This strongly indicates that a new symmetry is at work here, which can protect theHiggs mass from large (quadratically divergent) radiative corrections, something that the SM isunable to do.2 We are thus looking for a symmetry that can enforce eqs.(7) and (10) (as wellas their generalizations to gauge interactions). In particular, we need equal numbers of physical(propagating) bosonic and fermionic degrees of freedom, eq.(7a). In addition, we need relationsbetween various terms in the Lagrangian involving different combinations of bosonic and fermionicfields, eqs.(7b) and (10).

It is quite clear from these considerations that the symmetry we are looking for must connectbosons and fermions. In other words, the generators Q of this symmetry must turn a bosonic stateinto a fermionic one, and vice versa. This in turn implies that the generators themselves carryhalf–integer spin, i.e. are fermionic. This is to be contrasted with the generators of the Lorentzgroup, or with gauge group generators, all of which are bosonic. In order to emphasize the newquality of this new symmetry, which mixes bosons and fermions, it is called supersymmetry (SUSY).

The simplest choice of SUSY generators is a 2–component (Weyl) spinor Q and its conjugateQ. Since these generators are fermionic, their algebra can most easily be written in terms of anti–commutators:

Qα, Qβ =Qα, Qβ

= 0; (12a)

Qα, Qβ

= 2σµ

αβPµ; [Qα, Pµ] = 0. (12b)

Here the indices α, β of Q and α, β of Q take values 1 or 2, σµ = (1, σi) with σi being the Paulimatrices, and Pµ is the translation generator (momentum); it must appear in eq.(12b) for the SUSYalgebra to be consistent with Lorentz covariance [10].

For a compact description of SUSY transformations, it will prove convenient to introduce“fermionic coordinates” θ, θ. These are anti–commuting, “Grassmann” variables:

θ, θ =θ, θ

=

θ, θ

= 0. (13)

A “finite” SUSY transformation can then be written as exp[i(θQ + Qθ − xµP µ)

]; this is to be

compared with a non–abelian gauge transformation exp (iϕaT a), with T a being the group generators.2In principle one can cancel the one–loop quadratic divergencies in the SM without introducing new fields, by

explicitly cancelling bosonic and fermionic contributions; this leads to a relation between the Higgs and top masses[8]. However, such a cancellation would be purely “accidental”, not enforced by a symmetry. It is therefore notsurprising that this kind of cancellation cannot be achieved once corrections from two or more loops are included [9].

7

①　　　　　②

m2 = PµPµ

Q|boson ∼ |fermion Q|fermion ∼ |boson

一大革命

場の理論で許される対称性は、並進対称(Pμ)、ローレンツ対称(Mμν)、ゲージ対称(Bl)

Coleman-Mandulaの定理交換関係を満たすgeneratorを使う場合、ポアンカレ群と内部対称性に関する群とを混ぜることができない⇒ 重力まで含めた統一理論を作れない！重力は時空に関する対称性⇐ポアンカレ群に属する

Haag-Loupuszanski-Sohniusの定理反交換関係を含めると超対称代数＝Pμ⊗Mμν⊗Bl⊗Qμν⇒ 重力まで拡張できる

7

ポアンカレ群（時空に関する対称性） 内部対称性

超粒子

8

gravitino: GS=3/2Graviton: GS=2

Higgsino: H01, H

02, H

+-S=1/2Higgs: h, H,A, H +-S=0

Bino : B0

Wino : W+-, W0

gluino: g

S=1/2

photon : ! (B0 and W0)

Weak Boson : W+-, Z

gluon: g

S=1

charged scalar lepton: e, ",#

scalar neutrino: $, $, $

scalar quark: u, c, t

d, s, b

S=0

charged lepton: e, ",#

neutrino: $, $, $

quark: u, c, t

!!! !d, s, b

S=1/2

!!!!!"#$%&'!!!!()*&'~ ~ ~

~ ~

~

~

~~~

~

~

~

~ ~~

~

~

~~

!"#$%&'()&*+,-./,01234/,56789:;<=>

6?89@ABC

6D89EFGH

IJKL'MN

OP;QR

STUV*1W.

XMYZ[\UV

;]^_`abcd

NY.`;

1234*

ef-.g'h

ijk;lmn;

_opqH

1234!"#$%#&'()*+,-./0

超粒子の性質

フェルミオンには右巻きと左巻きがあるので、対応するパートーナー(squark, slepton)にも右巻きと左巻き相互作用に関してはパートナーと同じ量子数（ハイパーチャージ、カラー荷、電荷など）本来は（超対称が破れていなかったら）質量もパートナーと同じ多くのモデルで R parity (= (-1)2S+3B-L)が保存既知（標準理論で扱う）粒子はR>0、超粒子はR<0既知の粒子の反応で超粒子が生成されるときは対生成超粒子は超粒子１個プラス既知の粒子に崩壊⇒ 最も軽い超粒子（LSP Lightest Super Paticle)はそれよりもさらに崩壊できないので安定

9

SUSYのご利益

Fine Tuning Problem を解決できるヒッグスが素粒子(ボソン)として存在しないと意味ない繰り込み可能

重力まで含めた統一の可能性SU(3), SU(2), U(1) に基づく3つの力を統一できそうダークマターの最有力候補２つのヒッグスのパートナー、光子、Z が混合してニュートラリーノ( )と呼ばれる質量固有状態一番軽い が候補

10

Introduction

73%

Dark Matter23% Atom

4%

Dark Matter (DM) density

of the universe

ΩDMh2

= 0.113 ± 0.009 (WMAP)

What is the DM?

There is no DM candidate in

the standard model

Very precious !!DM＜１０％

•neutral, stable particle

SUSY SU(5)

χ0

χ01

三角異常項

バリオン数保存、レプトン数保存は真の対称性か？時空に関する対称性でもゲージ対称性でもないクォークとレプトンも同じ仲間の可能性

フェルミオンによる三角異常項

繰り込み可能（正しい理論である）のためにはフェルミオン同士でキャンセルしなければならない

1つの世代の中でクォーク二重項とレプトン二重項はペアとなっている ⇐ 偶然とは考えにくいクォークとレプトンを統合する統一理論は自然

11

図 9: Z0, π0 → 2γ反応に寄与する３角異常項。fはフェルミオン (クォ－クとレプトン)を示す。このループの存在により軸性カレントが保存しない。

しない。軸性カレントはカイラルゲージ対称性の結果として生じるが、このゲージ対称性が大局的であれば問題はない。実際、π0 → 2γ反応にはこの異常項が寄与していることが実験と比較して確かめられている。また、陽子や中性子の質量も大部分は量子異常項の寄与であることが知られている (表 1.1のクォーク質量と陽子質量を比較せよ)。しかし、局所ゲージ対称性の場合はゲージカレントが保存しないことになり、繰り込み不可能という重大事が発生する。Z0 → 2γはその例である。しかし、標準理論の場合は、各フェルミオンのこの図への寄与は電荷に比例するので、全てのフェルミオンについて和を採ると、

３色 × (Qu + Qd) + Qν + Qe = 3 ×2

3+ 3 ×

(−13

)+ 0 + (−1) = 0 (36)

となって、三角異常項の寄与の総和はゼロとなり、繰り込み可能性が保証されるのである。レプトンとクォークの寄与が相殺して量子異常が消滅することは、両者の連携が不可欠なことを意味し、同じ家族の一員と見なすのが妥当であることを示す。レプトン-クォーク対応にはこうした重大な意味が隠されていたのであり、レプトンとクォークを同じ多重項に入れる大統一理論の理論的根拠を与える。正当な理論には、量子異常が存在してはならないという要請は、新理論を構成するときの重要な条件である。現代の最先端理論である超紐の理論が、10次元時空のみで成立するという帰結も、量子異常の議論から導かれたものである。

1.13 世代の謎世代の謎はどのように理解されるのであろうか？　アイデアはいくつかあるが定説はないので、数例の文献紹介にとどめる。歴史的に常に有効であった考え方は、クォークレプトンを素粒子ではなく、より基本的な粒子プレオンの複合粒子であると見なすことである 15) 。標準理論が実験事実と良く合うことから、クォークやレプトンが複合粒子であるにしても、その束縛エネルギースケール

21

jµ5 = ψγµγ5ψ

QfNc = 3(Qu + Qd) + (Qν + Qe) = 0

T3LQ2

fNc = 0 ⇒

超対称性ヒッグス

ヒッグス二重項は最低2個必要ヒッグスのsuperpartner(higgsino)はフェルミオンなので三角異常項を作る⇒ 標準理論同様、キャンセルするために2個以上超対称性の構造cf.

Minimal Model では2個

( ) は down(up) type quark と結合ヒッグスボソンは5(=8-3)個

12

LHdlR = LHulR

φ ≡

φ+

φ0

→ φc =

φ∗

0

−φ+

φ1 =

0v1

φ2 =

0v2

v2 = v2

1 + v22

tanβ =v2

v1

φ1 φ2

h0, H

0, A

0, H

±

CP even, odd

SUSYのモデル

MSSM (Minimal Super-symmetric Standard Model)超対称性を導入するだけで、特別の仮定を入れてない仮定がないのである意味正しいことが保証されてる何の仮定もないので、標準理論同様、非常に多くのフリーパラメータが必要（124個！）全ての粒子に対する質量（湯川結合定数）粒子間の結合定数...などなど

“尤もらしい”仮定を入れてパラメータの数を減らす仮定の仕方により様々なモデル

13

超対称性の破れ

SUSYが破れていなかったら、0.5MeVの　　などなどパートナーであるべき超粒子が見つかっているはず発見されていない ⇒ SUSYは破れている

単に手で質量を入れるのではなく、自発的に超対称性も破れて欲しいSUSYの破れ方には様々なモデルいずれもSUSYの破れは hidden sectorSupergravity (SUGRA)Gauge Mediated Symmetry Breaking (GMSB)Anomaly Mediate Symmetry Breaking (AMSB)

14

e

L = LSUSY + δL

Supergravity (SUGRA)

Hidden sector と重力を通じてのみ繋がっている尤もらしい”仮説”によって、パラメータは5個m0: GUT scaleでのカイラルスカラーの共通質量

m1/2: GUT scaleでのゲージーノの共通質量

A: GUT scaleで共通のtrilinear coupling (単位 GeV)

μ: ヒッグス、ヒッグシーノの質量項

b: ヒッグスのmixing parameter⇒ 最終的には　　　　　　　　　　　　　の5つ

15

(m0,m1/2, A, tanβ, sign(µ))

m2(q) = m2(l) = m20

M(g) = M(W±) = M(B) = m1/2

YttLφtRAtYt˜tLφtR

bµHuHd

fermion) pair is also proportional to the weak hypercharge Y as given in Table 1.1. The interactionsshown in Figure 5.3 provide, for example, for decays q → qg and q → W q′ and q → Bq when the finalstates are kinematically allowed to be on-shell. However, a complication is that the W and B statesare not mass eigenstates, because of splitting and mixing due to electroweak symmetry breaking, aswe will see in section 7.2.

There are also various scalar quartic interactions in the MSSM that are uniquely determined bygauge invariance and supersymmetry, according to the last term in eq. (3.75), as illustrated in Fig-ure 3.3i. Among them are (Higgs)4 terms proportional to g2 and g′2 in the scalar potential. These arethe direct generalization of the last term in the Standard Model Higgs potential, eq. (1.1), to the caseof the MSSM. We will have occasion to identify them explicitly when we discuss the minimization ofthe MSSM Higgs potential in section 7.1.

The dimensionful couplings in the supersymmetric part of the MSSM Lagrangian are all dependenton µ. Using the general result of eq. (3.51), µ provides for higgsino fermion mass terms

− Lhiggsino mass = µ(H+u H−

d − H0uH0

d ) + c.c., (5.4)

as well as Higgs squared-mass terms in the scalar potential

− Lsupersymmetric Higgs mass = |µ|2(|H0u|2 + |H+

u |2 + |H0d |2 + |H−

d |2). (5.5)

Since eq. (5.5) is non-negative with a minimum at H0u = H0

d = 0, we cannot understand electroweaksymmetry breaking without including a negative supersymmetry-breaking squared-mass soft term forthe Higgs scalars. An explicit treatment of the Higgs scalar potential will therefore have to waituntil we have introduced the soft terms for the MSSM. However, we can already see a puzzle: weexpect that µ should be roughly of order 102 or 103 GeV, in order to allow a Higgs VEV of order174 GeV without too much miraculous cancellation between |µ|2 and the negative soft squared-massterms that we have not written down yet. But why should |µ|2 be so small compared to, say, M2

P,and in particular why should it be roughly of the same order as m2

soft? The scalar potential of theMSSM seems to depend on two types of dimensionful parameters that are conceptually quite distinct,namely the supersymmetry-respecting mass µ and the supersymmetry-breaking soft mass terms. Yetthe observed value for the electroweak breaking scale suggests that without miraculous cancellations,both of these apparently unrelated mass scales should be within an order of magnitude or so of 100GeV. This puzzle is called “the µ problem”. Several different solutions to the µ problem have beenproposed, involving extensions of the MSSM of varying intricacy. They all work in roughly the sameway; the µ term is required or assumed to be absent at tree-level before symmetry breaking, and thenit arises from the VEV(s) of some new field(s). These VEVs are in turn determined by minimizing apotential that depends on soft supersymmetry-breaking terms. In this way, the value of the effectiveparameter µ is no longer conceptually distinct from the mechanism of supersymmetry breaking; if wecan explain why msoft # MP, we will also be able to understand why µ is of the same order. In section10.2 we will study one such mechanism. Some other attractive solutions for the µ problem are proposedin refs. [57]-[59]. From the point of view of the MSSM, however, we can just treat µ as an independentparameter.

The µ-term and the Yukawa couplings in the superpotential eq. (5.1) combine to yield (scalar)3

couplings [see the second and third terms on the right-hand side of eq. (3.50)] of the form

Lsupersymmetric (scalar)3 = µ∗(uyuuH0∗d + dyddH0∗

u + eyeeH0∗u

+uyudH−∗d + dyduH+∗

u + eyeνH+∗u ) + c.c. (5.6)

Figure 5.4 shows some of these couplings, proportional to µ∗yt, µ∗yb, and µ∗yτ respectively. These playan important role in determining the mixing of top squarks, bottom squarks, and tau sleptons, as wewill see in section 7.4.

33

µ ∼ M2X

Mplank

質量スペクトラム

超粒子の質量はSUSYの各モデルの中にある”幾つかの”パラメータで決まる標準理論のように完全なフリーパラメータではない

超粒子は標準理論枠内の粒子よりだいぶ重いそれゆえ今まで発見されなかった⇒ 加速器のエネルギーを上げてより重い粒子を探したい

16

10 Supersymmetry Parameter Analysis: SPA Convention and Project

0

100

200

300

400

500

600

700

m [GeV]SPS1a′ mass spectrum

lR

lLνl

τ1

τ2ντ

χ01

χ02

χ03

χ04

χ±1

χ±2

qR

qL

g

t1

t2

b1

b2

h0

H0, A0 H±

Particle Mass [GeV] Particle Mass [GeV]

h0 116.0 τ1 107.9

H0 425.0 τ2 194.9

A0 424.9 ντ 170.5

H+ 432.7 uR 547.2

χ01 97.7 uL 564.7

χ02 183.9 dR 546.9

χ03 400.5 dL 570.1

χ04 413.9 t1 366.5

χ+1 183.7 t2 585.5

χ+2 415.4 b1 506.3

eR 125.3 b2 545.7

eL 189.9 g 607.1

νe 172.5

Table 5. Mass spectrum of supersymmetric particles [56] and Higgs bosons [58] in the reference point SPS1a′. Themasses in the second generation coincide with the first generation.

Particle Mass “LHC” “ILC” “LHC+ILC”

h0 116.0 0.25 0.05 0.05

H0 425.0 1.5 1.5

χ01 97.7 4.8 0.05 0.05

χ02 183.9 4.7 1.2 0.08

χ04 413.9 5.1 3 − 5 2.5

χ±1 183.7 0.55 0.55

eR 125.3 4.8 0.05 0.05

eL 189.9 5.0 0.18 0.18

τ1 107.9 5 − 8 0.24 0.24

qR 547.2 7 − 12 − 5 − 11

qL 564.7 8.7 − 4.9

t1 366.5 1.9 1.9

b1 506.3 7.5 − 5.7

g 607.1 8.0 − 6.5

Table 6. Accuracies for representative mass measurementsof SUSY particles in individual LHC, ILC and coherent“LHC+ILC” analyses for the reference point SPS1a′ [massunits in GeV]. qR and qL represent the flavors q = u, d, c, s.[Errors presently extrapolated from SPS1a simulations.]

While the picture so far had been based on evaluat-ing the experimental observables channel by channel,global analysis programs have become available [67,68] in which the whole set of data, masses, cross sec-tions, branching ratios, etc. is exploited coherently toextract the Lagrangian parameters in the optimal wayafter including the available radiative corrections formasses and cross sections. With increasing numbers ofobservables the analyses can be expanded and refinedin a systematic way. The present quality of such an

analysis [68] can be judged from the results shown inTable 7. These errors are purely experimental and donot include the theoretical counterpart which must beimproved considerably before matching the experimen-tal standards.

Extrapolation to the GUT scale

Based on the parameters extracted at the scale M , wecan approach the reconstruction of the fundamental su-persymmetric theory and the related microscopic pic-ture of the mechanism breaking supersymmetry. Theexperimental information is exploited to the maximumextent possible in the bottom-up approach [12] in whichthe extrapolation from M to the GUT/Planck scaleis performed by the renormalization group evolutionfor all parameters, with the GUT scale defined by theunification point of the two electroweak couplings. Inthis approach the calculation of loops and β functionsgoverning the extrapolation to the high scale is basedon nothing but experimentally measured parameters.Typical examples for the evolution of the gaugino andscalar mass parameters are presented in Fig. 1. Whilethe determination of the high-scale parameters in thegaugino/higgsino sector, as well as in the non-coloredslepton sector, is very precise, the picture of the col-ored scalar and Higgs sectors is still coarse, and strongefforts should be made to refine it considerably.

On the other hand, if the structure of the theory atthe high scale was known a priori and merely the ex-perimental determination of the high-scale parameterswere lacking, then the top-down approach would leadto a very precise parametric picture at the high scale.This is apparent from the fit of the mSUGRA parame-ters in SPS1a′ displayed in Table 8 [67]. A high-qualityfit of the parameters is a necessary condition, of course,

これはあくまでも一例

ダークマターとの関連

超対称性粒子のうちどれがダークマター候補なのかはモデルに依存SUGRAならcold dark matter として有力

GMSBなら軽すぎて hot dark matter になってしまいそう

観測されたダークマター密度から予想される質量

⇒ にぴったり（弱い相互作用でちょうどよい）　17

dn

dt+ 3Hn = −σAv(n2 − n

2EQ)

σAv ∼ 1 pb ∼ α2weak/(150 GeV)2

N1

N1

f

f

f

(a)

N1

N1

A0 (h0,H0)

b, t, τ−, . . .

b, t, τ+, . . .

(b)

N1

N1

Ci

W+

W−

(c)

Figure 9.13: Contributions to the annihilation cross-section for neutralino dark matter LSPs from (a)t-channel slepton and squark exchange, (b) near-resonant annihilation through a Higgs boson (s-wavefor A0, and p-wave for h0, H0), and (c) t-channel chargino exchange.

N2

N1

Z

f

f C1

N1

W

f

f ′ C1

N1

W

W

γ, Z

Figure 9.14: Some contributions to the co-annihilation of dark matter N1 LSPs with slightly heavierN2 and C1. All three diagrams are particularly important if the LSP is higgsino-like, and the last twodiagrams are important if the LSP is wino-like.

f

N1

f

f

γ, Z f

N1

f

f

γ, Z f

f

Ni

f

f

Figure 9.15: Some contributions to the co-annihilation of dark matter N1 LSPs with slightly heaviersfermions, which in popular models are most plausibly staus (or perhaps top squarks).

If N1 is mostly higgsino or mostly wino, then the the annihilation diagram fig. 9.13c and the co-annihilation mechanisms provided by fig. 9.14 are typically much too efficient [271, 272, 273] to providethe full required cold dark matter density, unless the LSP is very heavy, of order 1 TeV or more. Thisis often considered to be somewhat at odds with the idea that supersymmetry is the solution to thehierarchy problem. However, for lighter higgsino-like or wino-like LSPs, non-thermal mechanisms canbe invoked to provide the right dark matter abundance [176, 274].

A recurring feature of many models of supersymmetry breaking is that the lightest neutralino ismostly bino. It turns out that in much of the parameter space not already ruled out by LEP with abino-like N1, the predicted relic density is too high, either because the LSP couplings are too small, orthe sparticles are too heavy, or both, leading to an annihilation cross-section that is too low. To avoidthis, there must be significant contributions to 〈σv〉. The possibilities can be classified qualitatively interms of the diagrams that contribute most strongly to the annihilation.

First, if at least one sfermion is not too heavy, the diagram of fig. 9.13a is effective in reducingthe dark matter density. In models with a bino-like N1, the most important such contribution usuallycomes from eR, µR, and τ1 slepton exchange. The region of parameter space where this works out rightis often referred to by the jargon “bulk region”, because it corresponded to the main allowed regionwith dark matter density less than the critical density, before ΩDMh2 was accurately known and beforethe highest energy LEP searches had happened. However, the diagram of fig. 9.13a is subject to ap-wave suppression, and so sleptons that are light enough to reduce the relic density sufficiently are,in many models, also light enough to be excluded by LEP, or correspond to light Higgs bosons thatare excluded by LEP, or have difficulties with other indirect constraints. In the minimal supergravity

100

χ01

χ01

G

探索における信号

LSPは安定、かつ弱い相互作用しかしない⇒ ニュートリノ同様検出不能（専用検出器が必要）運動量が保存していないようにみえる大きな missing ET 　　　 がSUSY事象の特徴

18

p pg

q

q

χ20

χ10

χ10

χ1−

q

q

q

e−νe

µ+

µ−

≡ −

i=1

−→pTi

LHCでの探索能力

SUSYの動機として自然な領域はほぼカバー⇒ SUSYの厳格なテストが可能ダークマターがSUSYのLSPだったら、非常に少ない統計量（1週間程度のデータ収集）で発見可能

19

「対称性」についてコメント

自由場時空に対する対称性を考えると運動方程式がわかる[例] 並進対称 ⇒ 運動量保存

相互作用局所ゲージ対称性（内部空間における対称性）を考えるとどういう形をしてるのかわかる[例] U(1) ⇒ QED

対称性が宇宙の法則を支配してるように見える

20

宇宙の歴史と対称性

宇宙の歴史は対称性の破れの歴史相転移が起こるたびに劇的な変化

21

重力の誕生

強い力の誕生⇒レプトンとクォークの違いが生まれる

電磁気力と弱い相互作用の分化、質量の生成クォークからハドロンが形成される

量子重力の世界

原子の形成

宇宙の歴史と対称性

宇宙の歴史は対称性の破れの歴史相転移が起こるたびに劇的な変化

21

重力の誕生

強い力の誕生⇒レプトンとクォークの違いが生まれる

電磁気力と弱い相互作用の分化、質量の生成クォークからハドロンが形成される

量子重力の世界

原子の形成

←超対称性の破れ？

今回のまとめ

超対称性は標準理論を超える新しい物理の最有力重力まで統一可能色々な御利益正しい理論ならLHCで見つかるはず

素粒子物理学は対称性の研究高エネルギー状態は高い対称性を持った状態

22