素粒子物理学2 素粒子物理学序論B 2010年度講義第11回

素粒子物理学2

素粒子物理学序論B

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

階層性問題と Fine Tuning

GUT scale ( 10

15

_{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

δm

2

_{∼ λΛ}

2

µ

2_phys

= µ

2

_{− λΛ}

2 f ¯ f

•

•

H H

•

•

•

W, Z, H W, 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

M_H2 = (M_H0 )2 + 3Λ

2

8π2_v2

!

M_H2 + 2M_W2 + M_Z2 _{− 4m}2_t " (1.183) where M_H0 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 M_H2 = 4m2_{t − 2MW}2 _{− MZ}2 _{∼ (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

c_n(λi) logn(Λ/Q) (1.185)

where (16π2)c0 = (3/2v2)(M_H2 + 2M_W2 + M_Z2 − 4m2_t)2 and the remaining coefficients cn

can be calculated recursively from the requirement that M_H2 should not depend on the renormalization scale Q. For instance, for the two–loop coefficient, one finds [138]

(16π2)2c_{1 = λ(114λ − 54g2}2 _{− 18g1}2 + 72λt)2 + λ2_t (27g₂2 + 17g₁2 + 96g_s2 − 90λ2_t) −15 2 g 4 2 + 25 2 g 4 1 + 9 2g 2 1g22 (1.186)

18_{Note, however that the SM is a renormalizable theory and this cancellation can occur in a mathematically}

consistent way by choosing a similarly divergent counterterm. Nevertheless, one would like to give a physical meaning 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(10

19

_{)の補正からO(10}

2

_{)GeVの質量を作り出さないとならない}

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

δm

2

_{∼ λΛ}

2 f ¯ f • • H H • • • W, Z, H W, 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

M_H2 = (M_H0)2 + 3Λ

2

8π2_v2

!

M_H2 + 2M_W2 + M_Z2 _{− 4m}2_t" (1.183) where M_H0 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 M_H2 = 4m2_{t − 2MW}2 _{− MZ}2 _{∼ (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)(M_H2 + 2M_W2 + M_Z2 − 4m2_t)2 and the remaining coefficients cn

can be calculated recursively from the requirement that M_H2 should not depend on the renormalization scale Q. For instance, for the two–loop coefficient, one finds [138]

(16π2)2c1 = λ(114λ − 54g22 − 18g12 + 72λt)2 + λ2t(27g22 + 17g12 + 96gs2 − 90λ2t) −15₂ g₂4 + 25 2 g 4 1 + 9 2g 2 1g22 (1.186)

18_{Note, however that the SM is a renormalizable theory and this cancellation can occur in a mathematically}

consistent way by choosing a similarly divergent counterterm. Nevertheless, one would like to give a physical meaning to this scale Λ and view it as the scale up to which the SM is valid.

70

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 shown

that this Lagrangian also leads to the cancellation of quadratic divergencies from one–loop gauge

contributions to the Higgs two–point function π

_φφ

(0), thereby extending the result of the previous

section. Finally, soft SUSY breaking is treated in Sec. 3f.

Many excellent reviews of and introductions to the material covered here already exist [6, 7]; I

will 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 π

_φφ

(0) coming from the known

SM fermions can be cancelled exactly, if we introduce new bosonic fields with judiciously chosen

couplings. This strongly indicates that a new symmetry is at work here, which can protect the

Higgs mass from large (quadratically divergent) radiative corrections, something that the SM is

unable to do.

2

We are thus looking for a symmetry that can enforce eqs.(7) and (10) (as well

as 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 relations

between various terms in the Lagrangian involving different combinations of bosonic and fermionic

fields, eqs.(7b) and (10).

It is quite clear from these considerations that the symmetry we are looking for must connect

bosons and fermions. In other words, the generators Q of this symmetry must turn a bosonic state

into a fermionic one, and vice versa. This in turn implies that the generators themselves carry

half–integer spin, i.e. are fermionic. This is to be contrasted with the generators of the Lorentz

group, or with gauge group generators, all of which are bosonic. In order to emphasize the new

quality 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 conjugate

Q. Since these generators are fermionic, their algebra can most easily be written in terms of anti–

commutators:

{Q

α

, Q

β

} =

!

Q

_α˙

, Q

_β˙ "

= 0;

(12a)

!

Q

α

, Q

_β˙ "

= 2σ

_{α ˙}µ_β

P

µ

;

[Q

α

, P

µ

] = 0.

(12b)

Here the indices α, β of Q and ˙α, ˙

β of Q take values 1 or 2, σ

µ

= (1, σ

i

) with σ

i

being the Pauli

matrices, and P

_µ

is the translation generator (momentum); it must appear in eq.(12b) for the SUSY

algebra 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 − x}

µ

P

µ

)

$

; this is to be

compared with a non–abelian gauge transformation exp (iϕ

a

T

a

), with T

a

being the group generators.

2_{In 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 not surprising that this kind of cancellation cannot be achieved once corrections from two or more loops are included [9].

7

①     ②

m

2

= P

_µ

P

µ

一大革命

場の理論で許される対称性は、

並進対称(P

μ

)、ローレンツ対称(M

μν

)、ゲージ対称(B

l

)

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＝P

μ

M

μν

B

l

Q

μν

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: G S=3/2 Graviton: G S=2 Higgsino: H0 1, H02, H +-S=1/2 Higgs: h, H,A, H +-S=0 Bino : B0 Wino : W+-_{, W}0 gluino: g S=1/2 photon : ! (B0 _{and W}0₎ 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/,5 6789:;<=> 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

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 Matter 23% Atom 4%

Dark Matter (DM) density

of the universe

Ω_DMh2 _{= 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)

˜

χ

0

˜

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_{, π}0 _{→ 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_{× (Q}u +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

j

₅µ

= ¯

ψγ

µ

γ

5

ψ

� Q_fN_c = 3(Q_u + Q_d) + (Q_ν + Q_e) = 0 � T_3LQ2_fN_c = 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

¯

LH

_d

l

_R

_{�= ¯}

LH

_u

l

_R φ _≡ � φ₊ φ0 � → φc = � φ∗₀ −φ+ �

�φ

1

� =

�

0

v

₁

�

�φ

2

� =

�

0

v

₂

�

v

2

= v

₁2

+ v

₂2

tan β =

v

2

v

₁

φ

₁

φ

₂

h

0

, 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 sector

Supergravity (SUGRA)

Gauge Mediated Symmetry Breaking (GMSB)

Anomaly Mediate Symmetry Breaking (AMSB)

14

˜e

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

m

0

: GUT scaleでのカイラルスカラーの共通質量

m

1/2

: 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

(m

₀

, m

_1/2

, A, tan β, sign(µ))

m

2

(˜q) = m

2

(˜l) = m

2₀

M (˜

g) = M ( ˜

W

±

) = M( ˜

B) = m

_1/2

Y

_t

t

¯

_L

φt

_R

A

_t

Y

_t

t

˜¯

_L

φ ˜

t

_R

bµH

_u

H

_d

fermion) pair is also proportional to the weak hypercharge Y as given in Table 1.1. The interactions shown in Figure 5.3 provide, for example, for decays q_{! → q}g and_! q_{! →} W q" ! and q_{! →} Bq when the final! states are kinematically allowed to be on-shell. However, a complication is that the W and" B states! are not mass eigenstates, because of splitting and mixing due to electroweak symmetry breaking, as we will see in section 7.2.

There are also various scalar quartic interactions in the MSSM that are uniquely determined by gauge 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 are the direct generalization of the last term in the Standard Model Higgs potential, eq. (1.1), to the case of the MSSM. We will have occasion to identify them explicitly when we discuss the minimization of the MSSM Higgs potential in section 7.1.

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

− Lhiggsino mass = µ(H!_u+H!_d− ₋ H!_u0H!_d0) + c.c., (5.4) as well as Higgs squared-mass terms in the scalar potential

− Lsupersymmetric Higgs mass = |µ|2(_|Hu0_|2 + _|Hu+_|2 + _|Hd0_|2 + _|Hd−_|2). (5.5) Since eq. (5.5) is non-negative with a minimum at H_u0 = H_d0 = 0, we cannot understand electroweak symmetry breaking without including a negative supersymmetry-breaking squared-mass soft term for the Higgs scalars. An explicit treatment of the Higgs scalar potential will therefore have to wait until we have introduced the soft terms for the MSSM. However, we can already see a puzzle: we expect that µ should be roughly of order 102 or 103 GeV, in order to allow a Higgs VEV of order 174 GeV without too much miraculous cancellation between _|µ|2 and the negative soft squared-mass terms that we have not written down yet. But why should _|µ|2 be so small compared to, say, M_P2, and in particular why should it be roughly of the same order as m2_soft? The scalar potential of the MSSM 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. Yet the 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 100 GeV. This puzzle is called “the µ problem”. Several different solutions to the µ problem have been proposed, involving extensions of the MSSM of varying intricacy. They all work in roughly the same way; the µ term is required or assumed to be absent at tree-level before symmetry breaking, and then it arises from the VEV(s) of some new field(s). These VEVs are in turn determined by minimizing a potential that depends on soft supersymmetry-breaking terms. In this way, the value of the effective parameter µ is no longer conceptually distinct from the mechanism of supersymmetry breaking; if we can explain why m_{soft # MP}, we will also be able to understand why µ is of the same order. In section 10.2 we will study one such mechanism. Some other attractive solutions for the µ problem are proposed in refs. [57]-[59]. From the point of view of the MSSM, however, we can just treat µ as an independent parameter.

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∗(uy! uuH! _d0∗ + dy! ddH! _u0∗ + !eyeeH! _u0∗

+uy! _udH! _d−∗ + dy! _duH_{! u}+∗ + !ey_eνH_{! u}+∗) + c.c. (5.6) Figure 5.4 shows some of these couplings, proportional to µ∗y_t, µ∗y_b, and µ∗y_τ respectively. These play an important role in determining the mixing of top squarks, bottom squarks, and tau sleptons, as we will see in section 7.4.

33

µ

_∼

M

2 X

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 ˜lL ˜ νl ˜ τ1 ˜ τ2 ˜ ντ ˜ 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 τ˜1 107.9 H0 425.0 τ˜2 194.9 A0 424.9 ν˜τ 170.5 H+ 432.7 u˜R 547.2 ˜ χ01 97.7 u˜L 564.7 ˜ χ02 183.9 d˜R 546.9 ˜ χ03 400.5 d˜L 570.1 ˜ χ04 413.9 t˜1 366.5 ˜ χ+₁ 183.7 t˜2 585.5 ˜ 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!. The masses 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 measurements of SUSY particles in individual LHC, ILC and coherent “LHC+ILC” analyses for the reference point SPS1a! [mass units 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 to extract the Lagrangian parameters in the optimal way after including the available radiative corrections for masses and cross sections. With increasing numbers of observables the analyses can be expanded and refined in a systematic way. The present quality of such an

analysis [68] can be judged from the results shown in Table 7. These errors are purely experimental and do not include the theoretical counterpart which must be improved considerably before matching the experimen-tal standards.

Extrapolation to the GUT scale

Based on the parameters extracted at the scale ˜M , we can approach the reconstruction of the fundamental su-persymmetric theory and the related microscopic pic-ture of the mechanism breaking supersymmetry. The experimental information is exploited to the maximum extent possible in the bottom-up approach [12] in which the extrapolation from ˜M to the GUT/Planck scale is performed by the renormalization group evolution for all parameters, with the GUT scale defined by the unification point of the two electroweak couplings. In this approach the calculation of loops and β functions governing the extrapolation to the high scale is based on nothing but experimentally measured parameters. Typical examples for the evolution of the gaugino and scalar mass parameters are presented in Fig. 1. While the determination of the high-scale parameters in the gaugino/higgsino sector, as well as in the non-colored slepton sector, is very precise, the picture of the col-ored scalar and Higgs sectors is still coarse, and strong efforts should be made to refine it considerably.

On the other hand, if the structure of the theory at the high scale was known a priori and merely the ex-perimental determination of the high-scale parameters were lacking, then the top-down approach would lead to 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-quality fit of the parameters is a necessary condition, of course,

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 = −�σ

A

v

�(n

2

− n

2EQ

)

�σ

A

v

� ∼ 1 pb ∼ α

2_weak

/(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-wave for A0, and p-wave for h0, H0), and (c) t-channel chargino exchange.

! N2 ! N1 Z f f C!1 ! N1 W f f" _C!₁ ! N1 W W γ, Z Figure 9.14: Some contributions to the co-annihilation of dark matter N!1 LSPs with slightly heavier

!

N2 and C!1. All three diagrams are particularly important if the LSP is higgsino-like, and the last two

diagrams are important if the LSP is wino-like.

! f ! N1 f f γ, Z f! ! N1 ! f f γ, Z _f! ! f ! N_i f f Figure 9.15: Some contributions to the co-annihilation of dark matter N!1 LSPs with slightly heavier

sfermions, which in popular models are most plausibly staus (or perhaps top squarks).

If N!1 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 provide the full required cold dark matter density, unless the LSP is very heavy, of order 1 TeV or more. This is often considered to be somewhat at odds with the idea that supersymmetry is the solution to the hierarchy problem. However, for lighter higgsino-like or wino-like LSPs, non-thermal mechanisms can be invoked to provide the right dark matter abundance [176, 274].

A recurring feature of many models of supersymmetry breaking is that the lightest neutralino is mostly bino. It turns out that in much of the parameter space not already ruled out by LEP with a bino-like N!1, the predicted relic density is too high, either because the LSP couplings are too small, or

the sparticles are too heavy, or both, leading to an annihilation cross-section that is too low. To avoid this, there must be significant contributions to _{!σv". The possibilities can be classified qualitatively in} terms 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 reducing the dark matter density. In models with a bino-like N!1, the most important such contribution usually

comes from e!R, µ!R, and τ!1 slepton exchange. The region of parameter space where this works out right

is often referred to by the jargon “bulk region”, because it corresponded to the main allowed region with dark matter density less than the critical density, before ΩDMh2 was accurately known and before

the highest energy LEP searches had happened. However, the diagram of fig. 9.13a is subject to a p-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 that are excluded by LEP, or have difficulties with other indirect constraints. In the minimal supergravity

100

˜

χ

0₁

˜

χ

0₁

˜

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 E

T

    がSUSY事象の特徴

18 p

�g

p

�

q

�

q

� χ₂0 � χ₁0 � χ10 � χ₁− q q q e− νe µ+ µ−

≡ −

�

i=1

−→

p

T i

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