素粒子物理学2 素粒子物理学序論B 2010年度講義第10回

素粒子物理学2

素粒子物理学序論B

2010年度講義第10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

復習 標準理論のラグランジアン

4

514

CHAPTER 9. THE STANDARD MODEL

where s

_R,L

are the values of S = T

₃

_{− sin}

2

θ

_W

Q for the corresponding right-handed

and left-handed fields. Namely, Z

0

couples to the right-handed as well as left-handed

components of ". Since s

_νR

= 0, however, the coupling of Z

0

to ν is always purely

left-handed. This is true regardless of the neutrino mass (massive or not).

Let’s move on to the Higgs-fermion coupling. As we have seen in (9.182), it results

in the fermion masses. If we repeat the derivation without ignoring χ, we obtain the

same result with the replacement a

_{→ a + χ. Namely,}

!

h

_"

( ¯

Lφ)"

_R

+ h

_ν

( ¯

Lφ

_c

)ν

_R

+ c.c.

"

=

₋

(a + χ)h

√

"

2

¯

""

₋

(a + χ)h

√

ν

2

νν .

¯

(9.230)

The fermion masses are identified as

m

_"

=

h

√

"

a

2

,

m

ν

=

h

_ν

a

√

2

,

(9.231)

and in terms of the masses, the Higgs-fermion terms are now

−m

"

""

¯

− m

ν

νν

¯

−

m

_"

a

χ(¯

"")

−

m

_ν

a

χ(¯

νν) .

(9.232)

Note that the coupling of the Higgs (χ) to fermion is proportional to the mass of the

fermion.

Putting everything together, the SU (2)

_{× U(1) Lagrangian (9.190) in the unitary}

gauge with the scalar field expanded around the vacuum φ

V AC

= (0, a/

√

2) is now

L = ¯ν(i∂/ − m

ν

)ν + ¯

"(i∂/

− m

"

)" +

1

2

(∂

µ

χ∂

µ

_χ

− µ

2

χ

2

)

−

1

4

F

i µν

F

iµν

+ m

2W

W

_+µ∗

W

µ +

−

1

4

G

µν

G

µν

₊

m

2Z

2

Z

µ

Z

µ

+ eA

_µ

(¯

"γ

µ

")

₋

√

g

2

!

W

₊µ

(¯

νγ

µ

P

_L

") + c.c.

"

− ¯g Z

µ !

¯

νγ

µ

(s

_νL

P

_L

+ s

_νR

P

_R

)ν + ¯

"γ

µ

(s

_"L

P

_L

+ s

_"R

P

_R

)"

"

+

2aχ + χ

2

4

#

g

2

W

_+µ∗

W

₊µ

+

g

¯

2

2

Z

µ

Z

µ$

−

m

"

a

χ(¯

"")

−

m

_ν

a

χ(¯

νν) ,

(9.233)

with

e = g sin θ

_W

,

g =

¯

%

g

2

_{+ g}

_"2

_,

_{sin θ}

W

=

g

¯

g

,

cos θ

W

=

g

"

¯

g

,

(9.234)

and s

_νR,L

, s

_"R,L

are the values of S = T

₃

_{− sin}

2

θ

_W

for the corresponding fields. When

we extend this Lagrangian to include three doublets of leptons and three doublets of

ヒッグスとの結合

ラグランジアン中のヒッグスと結合している部分

HWW

結合の強さ

HZZ

結合の強さ

Hff

結合の強さ＝湯川結合（by definition）

5

We will see in the course of this review that it will be appropriate to use the Fermi coupling constant Gµ to describe the couplings of the Higgs boson, as some higher–order effects are

effectively absorbed in this way. The Higgs couplings to fermions, massive gauge bosons as well as the self–couplings, are given in Fig. 1.2 using both v and Gµ. This general form of

the couplings will be useful when discussing the Higgs properties in extensions of the SM.

•

H f ¯ f gHf f = mf/v = ( √ 2Gµ)1/2 mf × (i)

•

H Vµ V_ν g_{HV V} = 2M_V2/v = 2(√2G_µ)1/2 M_V2 _{× (−ig}µν)

•

H H Vµ Vν g_{HHV V} = 2M_V2/v2 = 2√2G_µ M_V2 _{× (−ig}µν)

•

H H H gHHH = 3M_H2 /v = 3( √ 2Gµ)1/2 M_H2 × (i)

•

H H H H gHHHH = 3M_H2 /v2 = 3 √ 2Gµ M_H2 × (i)

Figure 1.2: The Higgs boson couplings to fermions and gauge bosons and the Higgs self– couplings in the SM. The normalization factors of the Feynman rules are also displayed.

Note that the propagator of the Higgs boson is simply given, in momentum space, by ∆_HH(q2) = i

q2 _{− M}2

H + i!

(1.49)

22

We will see in the course of this review that it will be appropriate to use the Fermi coupling constant Gµ to describe the couplings of the Higgs boson, as some higher–order effects are

effectively absorbed in this way. The Higgs couplings to fermions, massive gauge bosons as well as the self–couplings, are given in Fig. 1.2 using both v and Gµ. This general form of

the couplings will be useful when discussing the Higgs properties in extensions of the SM.

•

H f ¯ f gHf f = mf/v = ( √ 2Gµ)1/2 mf × (i)

•

H Vµ Vν gHV V = 2M_V2/v = 2( √ 2Gµ)1/2 M_V2 × (−igµν)

•

H H V_µ Vν gHHV V = 2M_V2/v2 = 2 √ 2Gµ M_V2 × (−igµν)

•

H H H g_HHH = 3M_H2 /v = 3(√2G_µ)1/2 M_H2 _{× (i)}

•

H H H H gHHHH = 3M_H2 /v2 = 3 √ 2Gµ M_H2 × (i)

Figure 1.2: The Higgs boson couplings to fermions and gauge bosons and the Higgs self– couplings in the SM. The normalization factors of the Feynman rules are also displayed.

Note that the propagator of the Higgs boson is simply given, in momentum space, by ∆HH(q2) = i q2 _{− M}2 H + i! (1.49) 22

g

2

2

v = gm

W

=

e

sin θ

_W

m

W

¯g

2

4

v =

gm

Z

cos θ

_W

=

2e

sin(2θ

_W

)

m

Z

h

_f

_{≡ Y}

_f

=

√

2

v

m

f

ヒッグスとの結合

ラグランジアン中のヒッグスと結合している部分

HWW

結合の強さ

HZZ

結合の強さ

Hff

結合の強さ＝湯川結合（by definition）

5

We will see in the course of this review that it will be appropriate to use the Fermi coupling constant Gµ to describe the couplings of the Higgs boson, as some higher–order effects are

effectively absorbed in this way. The Higgs couplings to fermions, massive gauge bosons as well as the self–couplings, are given in Fig. 1.2 using both v and Gµ. This general form of

the couplings will be useful when discussing the Higgs properties in extensions of the SM.

•

H f ¯ f gHf f = mf/v = ( √ 2Gµ)1/2 mf × (i)

•

H Vµ V_ν g_{HV V} = 2M_V2/v = 2(√2G_µ)1/2 M_V2 _{× (−ig}µν)

•

H H Vµ Vν g_{HHV V} = 2M_V2/v2 = 2√2G_µ M_V2 _{× (−ig}µν)

•

H H H gHHH = 3M_H2 /v = 3( √ 2Gµ)1/2 M_H2 × (i)

•

H H H H gHHHH = 3M_H2 /v2 = 3 √ 2Gµ M_H2 × (i)

Figure 1.2: The Higgs boson couplings to fermions and gauge bosons and the Higgs self– couplings in the SM. The normalization factors of the Feynman rules are also displayed.

Note that the propagator of the Higgs boson is simply given, in momentum space, by ∆_HH(q2) = i

q2 _{− M}2

H + i!

(1.49)

22

We will see in the course of this review that it will be appropriate to use the Fermi coupling constant Gµ to describe the couplings of the Higgs boson, as some higher–order effects are

effectively absorbed in this way. The Higgs couplings to fermions, massive gauge bosons as well as the self–couplings, are given in Fig. 1.2 using both v and Gµ. This general form of

the couplings will be useful when discussing the Higgs properties in extensions of the SM.

•

H f ¯ f gHf f = mf/v = ( √ 2Gµ)1/2 mf × (i)

•

H Vµ Vν gHV V = 2M_V2/v = 2( √ 2Gµ)1/2 M_V2 × (−igµν)

•

H H V_µ Vν gHHV V = 2M_V2/v2 = 2 √ 2Gµ M_V2 × (−igµν)

•

H H H g_HHH = 3M_H2 /v = 3(√2G_µ)1/2 M_H2 _{× (i)}

•

H H H H gHHHH = 3M_H2 /v2 = 3 √ 2Gµ M_H2 × (i)

Figure 1.2: The Higgs boson couplings to fermions and gauge bosons and the Higgs self– couplings in the SM. The normalization factors of the Feynman rules are also displayed.

Note that the propagator of the Higgs boson is simply given, in momentum space, by ∆HH(q2) = i q2 _{− M}2 H + i! (1.49) 22

g

2

2

v = gm

W

=

e

sin θ

_W

m

W

¯g

2

4

v =

gm

Z

cos θ

_W

=

2e

sin(2θ

_W

)

m

Z

h

_f

_{≡ Y}

_f

=

√

2

v

m

f

結合の強さは質量に比例

質量の起源としてヒッグスが好まれる理由

Gauge cancellation

それぞれのダイアグラムは    で発散

スピン１成分は、ゲージボソンの自己結合（SU(2)の

特徴）によってキャンセルできる

6

σ

_∝

√

s

W W-- _WW++ e e-- _ee++ γ γ W W-- _WW++ e e-- _ee++ Z Z W W-- _WW++ e e-- _ee++ ν ν

spin 1

spin 1

質量の起源としてヒッグスが好まれる理由

Gauge cancellation

それぞれのダイアグラムは    で発散

スピン１成分は、ゲージボソンの自己結合（SU(2)の

特徴）によってキャンセルできる

6

σ

_∝

√

s

W W-- _WW++ e e-- _ee++ γ γ W W-- _WW++ e e-- _ee++ Z Z W W-- _WW++ e e-- _ee++ ν ν

spin 1

spin 1

(陽)電子の wrong helicity

state によるスピン０成分

質量の起源としてヒッグスが好まれる理由

Gauge cancellation

それぞれのダイアグラムは    で発散

スピン１成分は、ゲージボソンの自己結合（SU(2)の

特徴）によってキャンセルできる

6

σ

_∝

√

s

W W-- _WW++ e e-- _ee++ γ γ W W-- _WW++ e e-- _ee++ Z Z W W-- _WW++ e e-- _ee++ ν ν

spin 1

spin 1

(陽)電子の wrong helicity

state によるスピン０成分

spin 0

質量の起源としてヒッグスが好まれる理由

Gauge cancellation

それぞれのダイアグラムは    で発散

スピン１成分は、ゲージボソンの自己結合（SU(2)の

特徴）によってキャンセルできる

6

σ

_∝

√

s

W W-- _WW++ e e-- _ee++ γ γ W W-- _WW++ e e-- _ee++ Z Z W W-- _WW++ e e-- _ee++ ν ν W W-- _WW++ e e-- e_e++ H H

spin 0

spin 1

spin 1

(陽)電子の wrong helicity

state によるスピン０成分

spin 0

質量の起源としてヒッグスが好まれる理由

Gauge cancellation

それぞれのダイアグラムは    で発散

スピン１成分は、ゲージボソンの自己結合（SU(2)の

特徴）によってキャンセルできる

6

σ

_∝

√

s

W W-- _WW++ e e-- _ee++ γ γ W W-- _WW++ e e-- _ee++ Z Z W W-- _WW++ e e-- _ee++ ν ν W W-- _WW++ e e-- e_e++ H H

spin 0

spin 1

spin 1

(陽)電子の wrong helicity

state によるスピン０成分

spin 0

⇐ ヒッグスがキャンセルする

質量の起源としてヒッグスが好まれる理由

Gauge cancellation

それぞれのダイアグラムは    で発散

スピン１成分は、ゲージボソンの自己結合（SU(2)の

特徴）によってキャンセルできる

6

σ

_∝

√

s

W W-- _WW++ e e-- _ee++ γ γ W W-- _WW++ e e-- _ee++ Z Z W W-- _WW++ e e-- _ee++ ν ν W W-- _WW++ e e-- e_e++ H H

spin 0

spin 1

spin 1

wrong helicity state m

e

eeH結合の強さ m

e

(陽)電子の wrong helicity

state によるスピン０成分

spin 0

ヒッグスが質量の起源とは言うが…

陽子質量 938MeV/c

2

_{中性子質量 940MeV/c}

2

この質量差は重要 ⇐ 陽子が安定

u-quark質量 4MeV/c

2

d-quark質量 7MeV/c

2

ヒッグス場により生成された質量はわずか2%程度

残り98%はQCDによるカイラル対称性の破れ

⇐ これも自発的対称性の破れ

7

(cf. n → p + e

-

_{+ ν)}

ヒッグス場により生成された質量

でも、もしヒッグスが存在しなかったら？

クォーク質量、電子質量はゼロのまま

陽子の質量＞中性子の質量（クォークの¦電荷¦の違い）

陽子が崩壊できる

強い相互作用による相転移でπが質量を獲得

πが力の媒介粒子となる（m

π

/m

W

約1/2500）

力の媒介粒子の質量で実効的な相互作用の強さが決まる

急速な陽子の崩壊 (p → n + e

+

_{+ ν)}

陽子が不安定

コンプトン波長無限大

水素原子すら形成されない

現在の宇宙とは全く別の姿になる

8 M = _8mig₂ W (u_νµγ_µ(1 − γ5)u_µ)(u_eγµ(1 − γ5)u_νe)

（閑話休題）等価原理

慣性質量 F = ma

重力質量 F = - GMm/r

2

等価原理 = 慣性質量と重力質量の区別はない

相対性理論

エネルギーと質量の等価性 E = mc

2

質量源によって時空が歪む

重力レンズ：光が重力源によって曲げられる、

というか、歪んだ空間を光は直進する

素粒子の重力質量は測定されていない（？）

9

ヒッグス質量に対する理論的制限

標準理論ではヒッグス質量はパラメータで予言不能

高次の効果を利用して理論的な制約を課すことは可能

上限（ヒッグス重過ぎると）

ユニタリティ

WW散乱断面積が発散

Triviality

ヒッグスの自己結合定数λ(Q

2

_{)がゼロ、}

あるいは >>1（摂動論が破綻）

下限（ヒッグスが軽過ぎると）

真空に対する安定性

最小値の存在が必要

10 where s, t are the Mandelstam variables [the c.m. energy s is the square of the sum of the momenta of the initial or final states, while t is the square of the difference between the momenta of one initial and one final state]. In fact, this contribution is coming from longitudinal W bosons which, at high energy, are equivalent to the would–be Goldstone bosons as discussed in §1.1.3. One can then use the potential of eq. (1.58) which gives the interactions of the Goldstone bosons and write in a very simple way the three individual amplitudes for the scattering of longitudinal W bosons

A(w+w− → w+w−) = − ! 2M 2 H v2 + "_M2 H v #2 1 s − M2 H + "_M2 H v #2 1 t − M2 H $ (1.150)

which after some manipulations, can be cast into the result of eq. (1.149) given previously.

• W− W+ _W− W+ • H • • • H

Figure 1.15: Some Feynman diagrams for the scattering of W bosons at high energy. These amplitudes will lead to cross sections σ(W+_W−_{→ W}+_W−_{) # σ(w}+_w−_{→ w}+_w−₎

which could violate their unitarity bounds. To see this explicitly, we first decompose the scattering amplitude A into partial waves a! of orbital angular momentum "

A = 16π

∞

%

!=0

(2" + 1)P!(cos θ) a! (1.151)

where P!are the Legendre polynomials and θ the scattering angle. Since for a 2 → 2 process,

the cross section is given by dσ/dΩ = |A|2_/(64π2_{s) with dΩ = 2πdcos θ, one obtains}

σ = 8π s ∞ % !=0 ∞ % !!₌₀ (2" + 1)(2"#_{+ 1)a} !a!! & 1 −1

d cos θP!(cos θ)P!!(cos θ)

= 16π s ∞ % !=0 (2" + 1)|a!|2 (1.152)

where the orthogonality property of the Legendre polynomials, ' d cos θP!P!! = δ_!!!, has

been used. The optical theorem tells us also that the cross section is proportional to the imaginary part of the amplitude in the forward direction, and one has the identity

σ = 1 sIm [ A(θ = 0) ] = 16π s ∞ % !=0 (2" + 1)|a!|2 (1.153) 60

ヒッグス質量に対する理論的制限

WW散乱断面積

ヒッグスの自己結合定数

Triviality

真空の安定性

λ>0が必要

11

A(W W → W W ) =

−G

F

m

2 H

8

√

2π

[2 +

m

2_H

s

_{− m}

2_H

−

m

2_H

s

ln(1 +

s

m

2_H

]

m

_H

� 1 TeV

λ(Q) =

λ(v)

1 −

3 4π2

λ(v) ln(

Q2 v2

)

V = µ

2

φ

∗

φ + λ

_|φ

∗

φ

_|

2

_{v =}

�

_−µ

2

_/λ

1

λ(v)

−

1

λ(Q)

=

3

4π

2

ln

Q

2

v

2

実験からの制約

量子補正には直接観測されない（＝ヒッグスボソン）粒子

の影響も含まれる

ヒッグス以外の種々の精密測定からヒッグスの質量を

予測可能

12 160 165 170 175 180 185 m_t [GeV] 80.20 80.30 80.40 80.50 80.60 80.70 M W [GeV] SM MSSM MH = 114 GeV MH = 400 GeV light SUSY heavy SUSY SM MSSM both models

Heinemeyer, Hollik, Stockinger, Weber, Weiglein ’08

experimental errors 90% CL: LEP2/Tevatron (today) Tevatron/LHC ILC/GigaZ !

"#$%&'($)&*+,&-&'()(-+./+)0(+1)&$%&-%+2.%(*

3//(4)5+,-(%64)6.$5+./++12+76&+-&%6&)67( 4.--(4)6.$5

!'

₎

4&$+8(+-(*&)(%9+:6)0+'

_;

9+).+)0(+

<6==5+'&55+

'

₎

65+-.#=0*>+?+)0(

7&4##'+(@,(4)&)6.$

7&*#(

./+)0(+<6==5+/6(*%

! ,-.86$=+)0(+

A;1B

'(40&$65'+C$(:+,0>5645DE

F-(4656.$+'(&5#-('($)+!

GCHE+/8

IJ

,-.K(4)6.$L+"'

)

MJNOCJE+P(QN

R0(+R.,+S#&-T+'&55

!

"#$%

&

%

!%

_*

#

₎( _'

!"##$%&'()&*&+,-'(.&/&#,00

不確定性原理の範囲内でvirtualな

中間状態が可能

量子補正

質量の予想

実験で生成・観測できるくらい軽そう

m

H

150GeV 以外 のヒッグスが発見されたら

標準理論を低エネルギー極限とするような新しい物理の

存在を示唆

13 Higgs Physics 15 M H [GeV/ c 2 ] 600 400 500 100 200 300 0 3 5 7 9 11 13 15 17 19 log₁₀ ! [GeV] Triviality

EW vacuum is absolute minimum

EW Precision

Fig. 4. Bounds on the Higgs-boson mass that follow from requirements that the electroweak theory be consistent up to the energy scale Λ. The upper bound follows from triviality conditions; the lower bound follows from the requirement that V (v) < V (0). Also shown is the range of masses permitted at the 95% confidence level by precision measurements.

where v = (G_F√2)−1/2 _{≈ 246 GeV is the vacuum expectation value of the}

Higgs field times √2, we find that Λ ≤ MHexp ! 4π2v2 3M_H2 " . (3.13)

For any given Higgs-boson mass, there is a maximum energy scale Λ! at which the theory ceases to make sense. The description of the Higgs boson as an elementary scalar is at best an effective theory, valid over a finite range of energies.

This perturbative analysis breaks down when the Higgs-boson mass ap-proaches 1 TeV/c2 and the interactions become strong. Lattice analyses [25] indicate that, for the theory to describe physics to an accuracy of a few percent up to a few TeV, the mass of the Higgs boson can be no more than about 710 ± 60 GeV/c2. Another way of putting this result is that, if the elementary Higgs boson takes on the largest mass allowed by perturbative

標準理論が正しいと  

仮定した場合に許される 

 

ヒッグスの質量

ヒッグスの生成過程

生成過程によって終状態の粒子、トポロジーが違う

探索のしやすさは、断面積だけでは決まらない

（次に説明するように）ヒッグスの崩壊比も探索には

大きく影響を与える

14 102 103 104 105 100 200 300 400 500 qq ! Wh qq ! Zh gg ! h bb ! h qb ! qth gg,qq ! tth qq ! qqh m_h [GeV] ! [fb] SM Higgs production LHC

TeV4LHC Higgs working group

3.1.2 Higgs production at hadron machines

In the Standard Model, the main production mechanisms for Higgs particles at hadron colliders make use of the fact that the Higgs boson couples preferentially to the heavy particles, that is the massive W and Z vector bosons, the top quark and, to a lesser extent, the bottom quark. The four main production processes, the Feynman diagrams of which are displayed in Fig. 3.1, are thus: the associated production with W/Z bosons [241, 242], the weak vector boson fusion processes [112, 243–246], the gluon–gluon fusion mechanism [185] and the associated Higgs production with heavy top [247, 248] or bottom [249, 250] quarks:

associated production with W/Z : q ¯_{q −→ V + H} (3.1) vector boson fusion : _{qq −→ V} ∗V∗ _{−→ qq + H} (3.2) gluon − gluon fusion : gg −→ H (3.3) associated production with heavy quarks : gg, q ¯_{q −→ Q ¯}Q + H (3.4)

q ¯ q V∗ • H V • q q V ∗ V∗ H q q • g g H Q • g g H Q ¯ Q

Figure 3.1: The dominant SM Higgs boson production mechanisms in hadronic collisions. There are also several mechanisms for the pair production of the Higgs particles

Higgs pair production : pp −→ HH + X (3.5)

and the relevant sub–processes are the gg → HH mechanism, which proceeds through heavy top and bottom quark loops [251,252], the associated double production with massive gauge bosons [253, 254], q ¯_{q → HHV , and the vector boson fusion mechanisms qq → V} ∗V∗ _→ HHqq [255, 256]; see also Ref. [254]. However, because of the suppression by the additional electroweak couplings, they have much smaller production cross sections than the single Higgs production mechanisms listed above.

117

Qは重いクォーク(t,b)

V = W or Z

ヒッグスの崩壊

結合はわかっているので、位相空間で積分すれば崩壊幅を

計算できる

ゲージボソン

フェルミオン

運動学的に許される範囲で最も重い粒子に崩壊しやすい

15

2.2

Decays into electroweak gauge bosons

2.2.1 Two body decays

Above the W W and ZZ kinematical thresholds, the Higgs boson will decay mainly into pairs of massive gauge bosons; Fig. 2.9a. The decay widths are directly proportional to the HV V couplings given in eq. (2.2) which, as discussed in the beginning of this chapter, correspond to the JPC = 0++ assignment of the SM Higgs boson spin and parity quantum numbers. These are S–wave couplings, ∼ !"1 · !"2 in the laboratory frame, and linear in sin θ, with θ

being the angle between the Higgs and one of the vector bosons.

a)

•

H V V

•

b) H V f ¯ f

•

c) H f₃ ¯ f₄ f₁ ¯ f₂

Figure 2.9: Diagrams for the Higgs boson decays into real and/or virtual gauge bosons. The partial width for a Higgs boson decaying into two real gauge bosons, H → V V with V = W or Z, are given by [32, 145] Γ(H → V V ) = GµM 3 H 16√2π δV √ 1 − 4x (1 − 4x + 12x2) , x = M 2 V M_H2 (2.27)

with δ_W = 2 and δ_Z = 1. For large enough Higgs boson masses, when the phase space factors can be ignored, the decay width into W W bosons is two times larger than the decay width into ZZ bosons and the branching ratios for the decays would be, respectively, 2/3 and 1/3 if no other decay channel is kinematically open.

For large Higgs masses, the vector bosons are longitudinally polarized [159] Γ_L Γ_L + Γ_T = 1 − 4x + 4x2 1 − 4x + 12x2 MH!MV −→ 1 (2.28)

while the L, T polarization states are democratically populated near the threshold, at x = 1/4. Since the longitudinal wave functions are linear in the energy, the width grows as the third power of the Higgs mass, Γ(H → V V ) ∝ MH3 . As discussed in §1.4.1, a heavy Higgs

boson would be obese since its total decay width becomes comparable to its mass

Γ(H → W W + ZZ) ∼ 0.5 TeV [MH/1 TeV]3 (2.29)

and behaves hardly as a resonance.

82

2.1

Decays to quarks and leptons

2.1.1 The Born approximation

In the Born approximation, the partial width of the Higgs boson decay into fermion pairs, Fig. 2.1, is given by [111, 145]

Γ_{Born(H → f ¯}f ) = GµNc

4√2π MH m

2

f βf3 (2.6)

with β = (1 − 4m2f/MH2 )1/2 being the velocity of the fermions in the final state and Nc the

color factor N_c = 3 (1) for quarks (leptons). In the lepton case, only decays into τ+τ− pairs and, to a much lesser extent, decays into muon pairs are relevant.

•

H f

¯ f

Figure 2.1: The Feynman diagram for the Higgs boson decays into fermions.

The partial decay widths exhibit a strong suppression near threshold, Γ(H → f ¯f ) ∼ β_f3 _{→ 0 for MH % 2mf}. This is typical for the decay of a Higgs particle with a scalar coupling eq. (2.3). If the Higgs boson were a pseudoscalar A boson with couplings given in eq. (2.5), the partial decay width would have been suppressed only by a factor β_f [146]

Γ_{Born(A → f ¯}f ) = GµNc

4√2π MH m

2

f βf (2.7)

More generally, and to anticipate the discussions that we will have on the Higgs CP– properties, for a Φ boson with mixed CP–even and CP–odd couplings g_{Φ ¯f f ∝ a + ibγ5}, the differential rate for the fermionic decay Φ(p_{+) → f(p, s) ¯}f(¯p, ¯s) where s and ¯s denote the polarization vectors of the fermions and the four–momenta are such that p_{± = p ± ¯}p, is given by [see Ref. [147] for instance]

dΓ dΩ(s, ¯s) = β_f 64π2_M Φ ! (|a|2 + |b|2)"1 2M 2 Φ − m2f + m2fs·¯s # +(|a|2 − |b|2)"p_{+·s p+·¯s −} 1 2M 2 Φs·¯s + m2fs·¯s − m2f # −Re(ab∗)%_µνρσpµ₊pν₋sρs¯σ _{− 2Im(ab}∗)m_fp_{+·(s + ¯s)}$ (2.8) The terms proportional to Re(ab∗) and Im(ab∗) represent the CP–violating part of the cou-plings. Averaging over the polarizations of the two fermions, these two terms disappear and we are left with the two contributions ∝ 1₂|a|2(M_Φ2 _−2m2_f−2m2_{f) and ∝} 1_2|b|2(M_Φ2 _−2m2_f+2m2_f) which reproduce the β_f3 and β_f threshold behaviors of the pure CP–even (b = 0) and CP–odd (a = 0) states noted above.

74 � ₁ 2m_H � spin,color |M|2dLIP S dLIP S = (2π)4δ4(q − p1 − p2) d3p₁ (2π)32E 1 d3p₂ (2π)32E 2

ヒッグスの崩壊

結合はわかっているので、位相空間で積分すれば崩壊幅を

計算できる

ゲージボソン

フェルミオン

運動学的に許される範囲で最も重い粒子に崩壊しやすい

15

2.2

Decays into electroweak gauge bosons

2.2.1 Two body decays

Above the W W and ZZ kinematical thresholds, the Higgs boson will decay mainly into pairs of massive gauge bosons; Fig. 2.9a. The decay widths are directly proportional to the HV V couplings given in eq. (2.2) which, as discussed in the beginning of this chapter, correspond to the JPC = 0++ assignment of the SM Higgs boson spin and parity quantum numbers. These are S–wave couplings, ∼ !"1 · !"2 in the laboratory frame, and linear in sin θ, with θ

being the angle between the Higgs and one of the vector bosons.

a)

•

H V V

•

b) H V f ¯ f

•

c) H f₃ ¯ f₄ f₁ ¯ f₂

Figure 2.9: Diagrams for the Higgs boson decays into real and/or virtual gauge bosons. The partial width for a Higgs boson decaying into two real gauge bosons, H → V V with V = W or Z, are given by [32, 145] Γ(H → V V ) = GµM 3 H 16√2π δV √ 1 − 4x (1 − 4x + 12x2) , x = M 2 V M_H2 (2.27)

with δ_W = 2 and δ_Z = 1. For large enough Higgs boson masses, when the phase space factors can be ignored, the decay width into W W bosons is two times larger than the decay width into ZZ bosons and the branching ratios for the decays would be, respectively, 2/3 and 1/3 if no other decay channel is kinematically open.

For large Higgs masses, the vector bosons are longitudinally polarized [159] Γ_L Γ_L + Γ_T = 1 − 4x + 4x2 1 − 4x + 12x2 MH!MV −→ 1 (2.28)

while the L, T polarization states are democratically populated near the threshold, at x = 1/4. Since the longitudinal wave functions are linear in the energy, the width grows as the third power of the Higgs mass, Γ(H → V V ) ∝ MH3 . As discussed in §1.4.1, a heavy Higgs

boson would be obese since its total decay width becomes comparable to its mass

Γ(H → W W + ZZ) ∼ 0.5 TeV [MH/1 TeV]3 (2.29)

and behaves hardly as a resonance.

82

2.1

Decays to quarks and leptons

2.1.1 The Born approximation

In the Born approximation, the partial width of the Higgs boson decay into fermion pairs, Fig. 2.1, is given by [111, 145]

Γ_{Born(H → f ¯}f ) = GµNc

4√2π MH m

2

f βf3 (2.6)

with β = (1 − 4m2f/MH2 )1/2 being the velocity of the fermions in the final state and Nc the

color factor N_c = 3 (1) for quarks (leptons). In the lepton case, only decays into τ+τ− pairs and, to a much lesser extent, decays into muon pairs are relevant.

•

H f

¯ f

Figure 2.1: The Feynman diagram for the Higgs boson decays into fermions.

The partial decay widths exhibit a strong suppression near threshold, Γ(H → f ¯f ) ∼ β_f3 _{→ 0 for MH % 2mf}. This is typical for the decay of a Higgs particle with a scalar coupling eq. (2.3). If the Higgs boson were a pseudoscalar A boson with couplings given in eq. (2.5), the partial decay width would have been suppressed only by a factor β_f [146]

Γ_{Born(A → f ¯}f ) = GµNc

4√2π MH m

2

f βf (2.7)

More generally, and to anticipate the discussions that we will have on the Higgs CP– properties, for a Φ boson with mixed CP–even and CP–odd couplings g_{Φ ¯f f ∝ a + ibγ5}, the differential rate for the fermionic decay Φ(p_{+) → f(p, s) ¯}f(¯p, ¯s) where s and ¯s denote the polarization vectors of the fermions and the four–momenta are such that p_{± = p ± ¯}p, is given by [see Ref. [147] for instance]

dΓ dΩ(s, ¯s) = β_f 64π2_M Φ ! (|a|2 + |b|2)"1 2M 2 Φ − m2f + m2fs·¯s # +(|a|2 − |b|2)"p_{+·s p+·¯s −} 1 2M 2 Φs·¯s + m2fs·¯s − m2f # −Re(ab∗)%_µνρσpµ₊pν₋sρs¯σ _{− 2Im(ab}∗)m_fp_{+·(s + ¯s)}$ (2.8) The terms proportional to Re(ab∗) and Im(ab∗) represent the CP–violating part of the cou-plings. Averaging over the polarizations of the two fermions, these two terms disappear and we are left with the two contributions ∝ 1₂|a|2(M_Φ2 _−2m2_f−2m2_{f) and ∝} 1_2|b|2(M_Φ2 _−2m2_f+2m2_f) which reproduce the β_f3 and β_f threshold behaviors of the pure CP–even (b = 0) and CP–odd (a = 0) states noted above.

74 � ₁ 2m_H � spin,color |M|2dLIP S dLIP S = (2π)4δ4(q − p1 − p2) d3p₁ (2π)32E 1 d3p₂ (2π)32E 2

崩壊幅がヒッグス自身の質量に依存していることに注意

プラスその依存性

崩壊比

フェルミオンへの崩壊幅はm

H

に比例する一方、ゲージ

ボソンの場合m

H

の３乗に比例する

H→WW崩壊が可能なヒッグス質量になるとH→WWが

優勢

16

Figure 2.25: The SM Higgs boson decay branching ratios as a function of MH.

Figure 2.26: The SM Higgs boson total decay width as a function of MH.

112 18 C. Quigg 1 10 100 1000 200 400 600 800 1000 Partial Width [GeV] M_Higgs [GeV/c2] W+W! Z0Z0 _ t t

Fig. 6. Partial widths for the prominent decay modes of a heavy Higgs boson.

Higgs Mass [GeV/c2]

Higgs Width [GeV]

500 400 300 200 100 0 100 10 1 0.1 0.01 0.001

Fig. 7. Higgs-boson total width as a function of mass.

∝ m

3H

!" !"

ATLAS

LHC Summary

1994年にLHC計画承認

4つの衝突地点（＝検出器）

ATLAS, CMS 他2つ

7 TeV + 7 TeV の陽子・陽子衝突型加速器

7 TeV の陽子の速さ = 光速ー10km/h

これまでの世界最高

0.98TeV + 0.98TeV (光速ー450km/h)

1000台以上の超伝導磁石(8.3T)

2010年

3.5+3.5TeVでの衝突・データ収集

21

ヒッグス探索のこれまで

直接探索より m

H

> 114 (GeV) @95% CL

2001年まで行われたCERNでのLEP実験（電子・

陽電子衝突型実験）

LHCはLEP実験のトンネルを使っている

直接探索より m

H

160GeV 付近は除外

現在も行われている米国フェルミ国立加速器研究所の

Tevatron実験（陽子・反陽子衝突型実験）

種々の精密測定と放射補正より（10ページ参照）

      m

H

< 182 (GeV) @95% CL

22

LHCでの探索

H→WWがないとき（軽いとき）

gg→H→γγ,qqH(→γγ）が

最有力

崩壊比は小さいが背景事象

（BG）が比較的少ない

ttH(→bb), qqH(→bb)が有望

gg→H→bbは背景事象

（BG）が多すぎてダメ

qqH(→ττ）も有望

H→WWが可能なとき（重いとき）

gg→H→WW(ZZ)が有望

23 Figure 2.25: The SM Higgs boson decay branching ratios as a function of MH.

Figure 2.26: The SM Higgs boson total decay width as a function of MH.

112 102 103 104 105 100 200 300 400 500 qq ! Wh qq ! Zh gg ! h bb ! h qb ! qth gg,qq ! tth qq ! qqh m_h [GeV] ! [fb] SM Higgs production LHC

LHCでのヒッグス探索能力

図はLHC実験の1つATLAS実験単独での結果予想

実験グループはもう1つ（CMS）ある

非常に軽くなければ数年で発見可能

24

!"#

標準理論ヒッグスの質量

必要な統計量 ルミノシテ

ィ(fb

-1

)

S/

√

B

ルミノシティ 1,2年目 a few fb-1 _{/ yr} 2,3,4年目 a few x 10 fb-1 _/yr

予想は非常に難しい

ヒッグス発見後にやること

ゲージ対称性により、ゲージボソンの質量は本来ゼロ

カイラル対称性により、フェルミオンの質量は本来ゼロ

標準理論では、素性の違う2種類の粒子に質量を与える

メカニズムが一つ

省エネだが若干（かなり）怪しい

ヒッグスセクターには原理がない

25

結合定数 質量

の関係を確認することが最重要

結果次第では発見された

ヒッグスは標準理論の枠外

今回のまとめ

ヒッグスを探してその性質を探ることが現在の素粒子物理学

の最重要課題

GWS模型が真に正しいのか？

ヒッグスセクターの物理法則を支配する原理の欠如

エネルギーフロンティア最新実験LHC

ヒッグスが 素粒子として 存在すればLHCで発見可能

26