Happy 60th Birthdays! Ishikawa-san & Kawamoto-san

Y. Kikukawa

Glashow-Weinberg-Salam model

on the lattice

A construction with exact gauge invariance

Institute of Physics, University of Tokyo

based on :

D. Kadoh and Y.K., JHEP 0805:095 (2008), 0802:063 (2008)

D.~Kadoh, Y.~Nakayama and Y.K., JHEP 0412, 006 (2004)

Y. Nakayama and Y.K., Nucl. Phys. B597, 519 (2001)

Happy 60th Birthdays !

Ishikawa-san

&

We learned Mechanics and Quantum Mechanics

from him !

1984年10月 _{゛0 夕北 海 道 大 学} 理学部 物理学科

Ishikawa-san’s nickname was

first,

“Glue” (from Glueball)

D E P A R I A A E N T O F P H Y s : C b H O K K A : D O u N : V E R S : T Y 繁ゆ° ON

I z o:

2 つの ″χν 衡 る.

4a)

■_

= 0

とら5ことこず

ヌと

: q Ii Ta \.€

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_ヽぃノ

_{ヽ 「}

と ら3と寺 _とィょ

_{旦上ユ _}

加t t t L 仙血 也 壼 L E 里 名 1 ≦Lヒ 罐ヨ る べL 翌1 もして来 わよ ' ハ t

I

22ヨ 會

メ

2 B

l),,11"- E !.,O あ 3

_っイ

L摺

_1[11::[III:::=___lI」[_Lil_____」:」 `2__二``ili___:【`:1_11::__12[」 `:::L」 2___li:―」12L_______―_―――― 0 ￨ ハ 占

え ′

■

｀1カ￨

_{l? u ' t - -}

_L

」ヽ)

´′

Bっざ

｀日物t lhゐ3と

卜可ゝtSLヾ

ぅ

´ D L P A R ! M L N 1 0 卜 P H 、 b i C 5 H O K K A : D C ) u N : V E R S : T Y ヽ 2 1 ) 、

( : b L

2 塙

潔 弁

T I 五 十 y B

_{斗 い t こ￨}

_{十 ミ L} ば

1lc街

ll九

411

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Ut z )

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ナSことこだ

アセ.

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￨ ￨ ´ ブ

] てi g

1 . ,' ) tt tt) k 7 V13 ∫2フ ちι_てt￢

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" π

弱 3 o S と 肩唆

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= Xr'l

と後 π は ヒ ■ 了 _でユい_な

_これJ｀

2 蹴 憬一 十にI † 一 恢 ヽ /

一

ぜ

十づ

ギ

2+ たЧ・

Υ = C 2

じ とP A κ l M L ` ` i O I P H Y S i t S H O K K A : D O u N : V E R S : T Y

SAPPOottN

上 ￢LXノ YI

‐ Eθ L

曖.、

bll

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ヒr蔵どtぅ

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Ｌ

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庄 標

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て

｀おヽF l 年 と

1 夫 ■ 曰月9・_力｀￨こtF L 可

セ 求

ごめ ∫。 対

/志可

`屋

土`人■ て午湧 Sと

=夜 71島Lく。

Univ. Festival 1985

Thank you very much, Ishikawa-sensei

Problem of lattice fermions : Species doubling

Dirac equation on the discrete space

∂

k

ψ(x, t) =

!

ψ(x + ˆ

_{ka, t) − ψ(x, t)}

"

/a

H =

3

!

k=1

α

k

1

i

∂

∂x

k

+

_{βm =⇒ H}

_lat

=

3

!

k=1

α

k

1

2

_i

"

∂

k

−

∂

_k†

#

+

_{β m}

E

= ±

!

"

"

#

3

$

k=1

1

a

2

sin

2

(p

k

a

) + m

2

,

p

1

, p

2

, p

3

∈

%

−

π

a

,

π

a

&

H

_lat

eigenvalues

species doublers

γ

5

= (−i)α

1

α

2

α

3

⇒

(−1)

n

×

(−i)α

₁

α

₂

α

₃

α

k

sin(p

k

a

) ! (−α

k

)q

k

p

k

=

π/a + q

k

,

|q

k

| ! π/a

In[41]:= x@p_D :=è!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!Sin@p * PiD * Sin@p * PiD + H0.1L^2 ;!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! In[40]:= values =8m Ø 0.005<;

In[44]:= Plot@8Evaluate@x@pD êê. valuesD<, 8p, -1.1, 1.1<,

PlotRange Ø8-0.2, 2.0<, AxesLabel Ø 8 p a ê p, W<D; -1 -0.5 0.5 1 a p ÅÅÅÅÅÅÅÅ_p 0.5 1 1.5 2 W damped-oscillations !"#$.nb 1

chirality flip !

Wilson-Dirac fermion

S

w

= a

4

!

x

¯

ψ(x)

"

γ

µ

1

2

#∇

µ

− ∇

† µ

$ +

a

2

#∇

µ

∇

† µ

$ + m

0

%

ψ

(x)

doubler mass :

m

₀

+

!

µ

a

2

" 2

a

sin

k

µ

a

2

#

2

! m

0

+

2n

a

n

= numbers of π

No-Go Theorem (Nielsen-Ninomiya)

S

= a

4

!

x

¯

ψ(x) D ψ(x) =

"

π/a −π/a

d

4

k

(2π)

4

ψ

¯

(−k) ˜

D(k) ψ(k)

˜

D

_(k)

is a periodic and analytic function of momentum

k

_µ

1.

2.

3.

4.

˜

D(k) ∝ iγ

µ

k

µ

for |k

µ

| " π/a

˜

D

_(k)

is invertible for all except

k

_µ

k

_µ

= 0

γ

5

D

˜

(k) + ˜

D

(k)γ

5

= 0

Explicit Breaking of

chiral symmetry

∂

l

∂k

l

D

˜

(k) =

!

x

e

ikx

_(ix)

l

D

_{(x) < ∞}

_=⇒

_{#D(x)# < Ce}

−γ|x|

Matsuyama-san once told me,

He was interested in the Species Doubling Problem of lattice

fermions, when he was a graduate student.

But Ishikawa-san did not like it and not allow him to study

such a topic. He did not like the topics such as “axiomatic

approach to QFT”, too. (I heard from Yabuki-san)

So he studied

not in his office (at the desk next to Ishikawa-san’s)

but in the library, hiding (^^;

Maybe, Ishikawa-san’s message was that

one should be more “physics-oriented”, not “mathematical

formalism-oriented”.

Also, I think, he is very practical.

But, ....

the standard model

= SU(3)xSU(2)xU(1) “chiral” gauge theory

Chiral symmetry

17 対称性の自然破綻 , 弱 い 相 互 作 用 の 基 本 的 性 質 は Ｖ マ イ ナ ス Ａ 理 論 と い う も の で 表 わ さ れ る 。 こ れ に よ れ ば 弱 い 相 互 作 用 に は 一_般 に 粒 子 の 左 巻 き 成 分 と 、 反 粒 子 の 右 巻 き 成 分 の み が 関 与 す る 。 ま た 粒 子 が 反 粒 子 に 変 わ る こ と も な い か ら 、 全 カ イ ラ リ テ ィ 、 す な わ ち   討 型 畔 蒔 中 ０ 踏 ︱ 卦 型 磐 蒔 中 ０ 洋   は 相 互 作 用 の 結 果 変 わ る こ と は な い 。 し か し ニ ュ ー ト リ ノ 以 外 の 粒 子 は 質 量 を も つ か ら 、 そ れ ら が 運 動 す る と き に 左 と 右 と が 混 じ っ て し ま い 、 結 局 カ イ ラ リ テ ィ の 保 存 が 破 れ る の で あ る 。 以 上 で 超 伝 導 休 と の ア ナ ロ ジ ー が だ い た い お わ か り で あ ろ う 。 レ プ ト ン や ク ォ ー ク は 元 来 は 質 量 を も た な い が 、 実 際 の 世 界 で は そ れ が 自 発 的 に 破 れ 、 質 量 の あ る 粒 子 と し て 現 わ れ る と 考 え る の で あ る 。 し か し こ れ だ け で は あ く ま で 一 つ の 可 能 性 に す ぎ な い 。 ほ か に も う 少 し 理 論 的 な 支 え が ほ し い 。 幸 い に し て Ｎ Ｇ 波 の 存 在 が そ れ を 果 し て く れ る 。 Ｂ Ｃ Ｓ 理 論 と の 対 比 を 具 体 的 に し て 説 明 す れ ば 、 こ の 世 界 に は 粒 子 と 反 粒 子 ︵ 例 え ば ｑ と 可 ︶ が カ イ ラ リ テ ィ ゼ ロ の ク ー パ ー 対 と し て 沈 澱 し て い る 。 一 つ の 対 を ぶ ち こ わ せ ば 質 量 の あ る タ ォ ー タ と 反 ク ォ ー タ が 発 生 す る 。 ま た 対 の 分 布 を 少 し か き 乱 せ ば Ｎ Ｇ 波 が 生 ず る が 、 こ れ は ス ピ ン が ゼ ロ 、 パ リ テ ィ が マ イ ナ ス 、 す な わ ち パ イ 中 ︱︱︱︱ 子 な ど と 同 じ 性 質 を も つ 。 パ イ 中 問 子 の 質 量 が 他 の ハ ド ロ ン に 比 べ て 特 に 小 さ い の は 、 こ れ が Ｎ Ｇ 量 子 的 な 粒 子 だ か ら で は な い か 。 実 際 に は パ イ 中 間 子 は 三 種 類 あ る し 、 も っ と 重 い Ｋ 中 間 子 や エ ー タ 中 間 子 η も 同 じ ス ビ ン 、 パ リ テ ィ の 粒 子 で あ る 。 こ れ ら を 一 様 に 取 り あ つ か う に は 、 カ イ ラ ル 対 称 性 に タ ォ ー ク の 香 り の 変 化 を 含 め 、 ま た こ れ ら の 中 間 子 の 質 量 が 完 全 に ゼ ロ で な い の は 、 自 発 的 な 破 れ だ け で は な い 要 素

,夕吉才套

左巻 きス ピンを,追 い越 しなが ら見ると右巻 きに見える カ イ ラ ル ︵ ｏ_〓 ａ じ と は ﹁ 手 に 関 す る ﹂ 、 つ ま り 右 利 き と 左 利 き の 区 別 を 意 味 す る 。 鰤 ペ ー ジ で ニ ュ ー ト リ ノ に ス ピ ン が 左 巻 き と 右 巻 き ︵ 運 動 方 向 に 対 し て ︶ の 二 種 類 が あ る こ と を 述 べ た が 、 こ の 二 つ は そ れ ぞ れ カ イ ラ リ テ ィ 量_を も つ ・_と 呼 ば れ る 。 実 際 の 世 界 で は ニ ュ ー ト リ ノ ν は 左 巻 き 、 反 ニ ュ ー ト リ ノ フ は 右 巻 き の も の し か 存 在 し な い た め に パ リ テ ィ の 破 れ が 起 こ る こ と は す で に 説 明 し た と お り で あ る 。 け れ ど も こ れ は 質 量 の な い ニ ュ ー ト リ ノ に 特 有 の 事 情 で 、 も し 質 量 が あ れ ば 右 、 左 両 方 の 成 分 が 存 在 せ ね ば な ら な い 。 な ぜ な ら ば 、 質 量 を も つ 粒 子 は 光 の 速 度 以 下 で し か 走 れ な い 。 あ る 粒 子 の ス ピ ン が 運 動 の 方 向 に 左 巻 き で も 、 そ れ を 追 い 越 す よ う な 運 動 座 標 系 か ら 見 れ ば 運 動 方 向 が 逆 転 し て 、 右 巻 き に 変 わ っ て し ま う か ら で あ る 。 「クォーク」  南部陽一郎

It was from Ishikawa-san that I first heard the name

“Nambu”. It was when his collected papers was

!

ν

₋

(x)

e

₋

(x)

"

Y =₋1₂

,

ν

+

(x)

Y =0

e

+

(x)

Y =₋₁

!

u

₋i

(x)

d

₋i

(x)

"

Y =1₆

,

u

+ i

(x)

Y =+ 2 3

d

+ i

(x)

_{Y =} − 1 3

Z0 W! 1 TeV 1 GeV 1 MeV -1 0 +1 e ! " t c u b s d Z W 1 keV 1 eV Weak Scale QCD Scale W Atomic Scale # Charge Mass top 174 GeV charm 1.25 GeV up 1.5 MeV bottom 4.20 GeV strange 95 MeV down 3 MeV " 1.78 GeV ! 105 MeV e 0.511 MeV Q=+2/3 e Q=-1/3 e Q=-e quark lepton neutrino # " ! e < 2 eV Q=0

The scalar Higgs field suffers from the

hier-archy problem: the Higgs mass and VEV are

very sensitive to Ultraviolet (UV) scale physics

through quantum corrections.

t H H W, Z, γ H H H H H

δm

2_H

_∼

1

16π

2 !

λ

2_t

, g

2

, λ

_H"

Λ

2_UV

Naturalness requires these quadratically

diver-gent contributions to be cut off by new physics

at ∼ 1 TeV.

!

W,Z,

higgs

top

Figure 1:

The most significant quadratically divergent contributions to the

Higgs mass in the Standard Model.

give

top loop

₋

_8π32

λ

2t

Λ

2

∼ −(2 TeV)

2

SU (2) gauge boson loops

_64π9 2

g

2

Λ

2

∼ (700 GeV)

2

Higgs loop

_16π1 2

λ

2

Λ

2

∼ (500 GeV)

2

_.

The total Higgs mass-squared includes the sum of these loop contributions and

a tree-level mass-squared parameter.

To obtain a weak-scale expectation value for the Higgs without worse than

10% fine tuning, the top, gauge, and Higgs loops must be cut off at scales

satisfying

Λ

top ∼<

2 TeV

Λ

gauge ∼<

5 TeV

Λ

Higgs ∼<

10 TeV.

(1)

We see that the Standard Model with a cut-off near the maximum attainable

energy at the Tevatron (∼ 1 TeV) is natural, and we should not be surprised

that we have not observed any new physics. However, the Standard Model with

a cut-off of order the LHC energy would be fine tuned, and so we should expect

to see new physics at the LHC.

More specifically, we expect new physics that cuts off the divergent top

loop at or below 2 TeV. In a weakly coupled theory this implies that there are

new particles with masses at or below 2 TeV. These particles must couple to the

Higgs, giving rise to a new loop diagram that cancels the quadratically divergent

contribution from the top loop. For this cancellation to be natural, the new

particles must be related to the top quark by some symmetry, implying that the

new particles have similar quantum numbers to top quarks. Thus naturalness

arguments predict a new multiplet of colored particles with mass below 2 TeV,

particles that would be easily produced at the LHC. In supersymmetry these

new particles are of course the top squarks.

Similarly, the contributions from SU (2) gauge loops must be canceled by

new particles related to the Standard Model SU (2) gauge bosons by symmetry,

and the masses of these particles must be at or below 5 TeV for the cancellation

to be natural. Finally, the Higgs loop requires new particles related to the Higgs

itself at or below 10 TeV. Given the LHC’s 14 TeV center-of-mass energy, these

predictions are very exciting, and encourage us to explore different possibilities

for what the new particles could be.

4

3 higgs tree (200 GeV) 2 ~ 2 h m gauge top loops

Figure 2. The fine tuning required to obtain an acceptable Higgs mass in the Standard Model with cut-off 10 TeV.

found no PBSM? Setting Λ = 1 TeV in the above formulas we find that the most dangerous con-tribution from the top loop is only about (200 GeV)2

. Thus no fine tuning is required, the SM with no new physics up to 1 TeV is perfectly nat-ural, and we should not be surprised that we have not yet seen deviations from it at colliders.

In the following, we will turn the argument around and use the hierarchy problem to predict what forms of new physics exist at what scale in order to solve the hierarchy problem. Consider for example the correction to the Higgs mass form the top loop. Limiting this contribution to be no larger than about 10 times the Higgs mass (limit-ing fine-tun(limit-ing to less than 1 part in 10) we find a maximum cut-off for Λ = 2 TeV. In other words, we predict the existence of new particles with masses less than or equal to 2 TeV which can-cel the quadratically divergent Higgs mass con-tribution from the top quark. In order for this cancellation to occur naturally, the new particles must be related to the top quark by a symmetry. In practice this means that the new particles have to have similar quantum numbers to top quarks. Thus the hierarchy problem predicts a new mul-tiplet of particles with mass below 2 TeV which carry color and are easily produced at the LHC. In supersymmetric extensions these new particles

are of course the top squarks.

The contributions from gauge loops also need to be canceled by new particles which are related to the SM SU (2) × U(1) gauge bosons by a sym-metry. The masses of these states are predicted to be at or below 5 TeV for the cancellation to be natural. Similarly, the Higgs loop requires new states related to the Higgs at 10 TeV. We see that the hierarchy problem can be used to obtain specific predictions.

SM loop maximum mass of new particles

top 2 TeV

weak bosons 5 TeV

Higgs 10 TeV

2.1. Supersymmetry and the hierarchy One successful approach to solving the hierar-chy problem is based on supersymmetry (SUSY). Loosely speaking, in SUSY every quadratically divergent loop diagram in Figure 1. has a super-partner, a diagram with superpartners running in the loop (Figure 3.). The diagrams with super-partners exactly cancel the quadratic divergences of the SM diagrams. Generically, this happens because superpartner coupling constants are re-lated to SM coupling constants by supersymme-try, but superpartner loops have the opposite sign from their SM partner because of opposite spin-statistics.

In the limit of unbroken supersymmetry the di-agrams cancel completely. If weak scale SUSY occurs in nature superpartner masses softly break the supersymmetry. Then the cancellation only takes place above the scale of superpartner masses MSUSY. Below MSUSY only the SM particles

ex-ist, thus there the SM loop diagrams (Figure 1) are not canceled but the cut-off Λ is replaced by MSUSY.

Experimental bounds on superpartner masses are somewhere in the few hundred GeV range, much lower than the upper bound of 2 TeV from fine tuning constraints top loops. Neverthe-less, the MSSM is already somewhat fine tuned. The problem arises from the experimental lower bound on the Higgs mass. As is well known, the tree level Higgs mass in the MSSM is bounded

3

higgs

tree

(200 GeV) 2

~

2

h

m

gauge

top

loops

Figure 2. The fine tuning required to obtain an acceptable Higgs mass in the Standard Model with cut-off 10 TeV.

found no PBSM? Setting Λ = 1 TeV in the above formulas we find that the most dangerous con-tribution from the top loop is only about (200 GeV)2

. Thus no fine tuning is required, the SM with no new physics up to 1 TeV is perfectly nat-ural, and we should not be surprised that we have not yet seen deviations from it at colliders.

In the following, we will turn the argument around and use the hierarchy problem to predict what forms of new physics exist at what scale in order to solve the hierarchy problem. Consider for example the correction to the Higgs mass form the top loop. Limiting this contribution to be no larger than about 10 times the Higgs mass (limit-ing fine-tun(limit-ing to less than 1 part in 10) we find a maximum cut-off for Λ = 2 TeV. In other words, we predict the existence of new particles with masses less than or equal to 2 TeV which can-cel the quadratically divergent Higgs mass con-tribution from the top quark. In order for this cancellation to occur naturally, the new particles must be related to the top quark by a symmetry. In practice this means that the new particles have to have similar quantum numbers to top quarks. Thus the hierarchy problem predicts a new mul-tiplet of particles with mass below 2 TeV which carry color and are easily produced at the LHC. In supersymmetric extensions these new particles

are of course the top squarks.

The contributions from gauge loops also need to be canceled by new particles which are related to the SM SU (2) × U(1) gauge bosons by a sym-metry. The masses of these states are predicted to be at or below 5 TeV for the cancellation to be natural. Similarly, the Higgs loop requires new states related to the Higgs at 10 TeV. We see that the hierarchy problem can be used to obtain specific predictions.

SM loop maximum mass of new particles

top 2 TeV

weak bosons 5 TeV

Higgs 10 TeV

2.1. Supersymmetry and the hierarchy One successful approach to solving the hierar-chy problem is based on supersymmetry (SUSY). Loosely speaking, in SUSY every quadratically divergent loop diagram in Figure 1. has a super-partner, a diagram with superpartners running in the loop (Figure 3.). The diagrams with super-partners exactly cancel the quadratic divergences of the SM diagrams. Generically, this happens because superpartner coupling constants are re-lated to SM coupling constants by supersymme-try, but superpartner loops have the opposite sign from their SM partner because of opposite spin-statistics.

In the limit of unbroken supersymmetry the di-agrams cancel completely. If weak scale SUSY occurs in nature superpartner masses softly break the supersymmetry. Then the cancellation only takes place above the scale of superpartner masses MSUSY . Below MSUSY only the SM particles

ex-ist, thus there the SM loop diagrams (Figure 1) are not canceled but the cut-off Λ is replaced by MSUSY .

Experimental bounds on superpartner masses are somewhere in the few hundred GeV range, much lower than the upper bound of 2 TeV from fine tuning constraints top loops. Neverthe-less, the MSSM is already somewhat fine tuned. The problem arises from the experimental lower bound on the Higgs mass. As is well known, the tree level Higgs mass in the MSSM is bounded

Fine-tuning required for a cutoff of 10 TeV For less than 10% fin tuning,

M. Schmaltz, hep-ph/0210415

(Naturalness)

LEP paradox

Waht is behind the Higgs sector ?

•

Flavor problems（yukawa coupling, mixing & CP

violations [Kobayashi-Maskawa (^^)], #generations, ..）

•

GUT

•

Cosmological problems

（

_{Baryon number asymmetry, dark matter, dark energy）}

New particles ?

New interactions ?

(dynamical mechanisms)

New symmetries ?

Dynamics of chiral gauge theories

may play important roles !!

•

chiral fermions : fund. unit of matter

•

realization of gauge & chiral symmetries

Need some

non-perturbative tools !

両手の鳴る音は知る。

片手の鳴る音はいかに？

     ー

_{禅の公案 ー}

What is the sound of one hand clapping ?

     

ー a koan from Zen Buddhism ー

SU(3)xSU(2)xU(1), SO(10), etc. ???

How to construct Glashow-Weinberg-Salam model

on the lattice ?

the last part of this talk

1.

chiral lattice gauge theories based on overlap D / the G-W rel.

2.

gauge anomalies in the lattice

SU(2)

L

x U(1)

Y

chiral gauge theory

3.

topology of the space of SU(2)xU(1) lattice gauge fields

4.

our approach & results

•

explicit construction of the smooth measure term

•

proof of the global integrability conditions

[reconstruction theorem]

5.

discussion

•

an extention to the standard model (the inclusion of SU(3) )

overlap Dirac op. / the GW rel.

Neuberger(1997,98)

D =

1

2

_a

!

1 +

_γ

5

H

w

"H

2 w

#

chiral operator

Luscher ; Hasenfratz, Niedermayer(1998)

γ

5

D + Dγ

5

= 2

aDγ

5

D

chiral fermion

ˆ

γ

5

ψ

±

(x) = ± ψ

±

(x)

¯

ψ

±

(x)γ

5

= ∓ ¯

ψ

±

(x)

{vi(x) | ˆγ5vi(x) = −vi(x) (i = 1, · · · , N₋)} {¯vi(x) | ¯vi(x)γ5 = +¯vi(x) (i = 1, · · · , ¯N₋)}

ψ

₋

(x) =

!

i

v

i

(x)c

i

¯

ψ

₋

(x) =

!

i

¯

c

i

v

¯

i

(x)

Path Integral Quantization

Z

=

!

D[ψ

−

]D[ ¯

ψ

−

] e

−a4 P x ψ¯−Dψ−(x)

=

! "

i

dc

_i

"

j

d¯

c

_j

e

− P ij c¯jMjici

= det M

ji Mji = a4 ! x ¯ v_jDv_i(x)

“overlap formula”

ˆ

γ

5

≡ γ

5

(1 − 2aD) = −

H

w

!H

2 w

Path Integral Measure depends on gauge fields !

˜ v_i(x) = vj(x)! ˜Q−1 " ji ˜ c_{i = ˜}Q_ijc_j

det ˜

_Q

complex

phase !

Narayanan-Neuberger(1993)

variation of effective action & gauge anomaly

δ

_η

Γ

eff

= Tr

!

(δ

η

D

) ˆ

P

−

D

−1

P

+

"

+

#

i

(v

i

, δ

η

v

i

)

= iTrωγ

5

(1 − D) − i

!

i

(v

i

, δ

ω

v

i

)

η

µ

(x) = −i∇

µ

ω

(x)

δ

_η

U

(x, µ) = iη

µ

(x)U (x, µ)

Γ

eff

= ln det(¯

v

_k

Dv

_j

)

* different situation from Dirac fermions in Vector-like theories like QCD

1.

2.

3.

gauge invariance ?

integrability ?

locality ?

[ topology of the space of gauge fields

non-trivial due to Admissibility cond. ]

Luscher(99)

[ admissibility cond. cf. Hernandez, Jansen, Luscher(98) ]

the gauge-field dependence must be fixed ...

[ gauge anomaly cancellations ]

1.

explicit construction of the smooth measure term, which fulfills

requirements of locality, gauge invariance & local integrability

2.

proof of the reconstruction theorem (global integrability conditions)

L

_η

_{= i}

!

i

(v

i

, δ

η

v

i

) =

!

x

η

µ

(x)j

µ

(x)

•

SU(2)xU(1) gauge anomaly

•

topology of space of SU(2)xU(1) gauge fields

key issues ...

applying this formulation to quarks and leptons ...

our results on the lattice GWS model :

η

µ

(x) = η

(2)

µ

(x) ⊕ η

(1) µ

(x)

v

_j(b)

(x) =

!γ

5

C

−1

⊗

iσ

2

" [v

j

(x)]

∗

v

(a)

j

(x) = v

j

(x)

a pair of doublets (a,b)

measure defined globally !

cf. Nuberger(98) Bar-Campos (00)

[SU(2

m

Q

[U(1)]

!

_{topological sectors ( ⇐ the admissibility cond. )}

m

µν

=

_2πi

1

!

s,t

ln U

_µν(1)

(

x + sˆµ + t ˆν)

Q =

!

x∈Γ4

tr{γ

5

(

1 − D)(x, x)}|

_U(2) [SU(2 m Q [U(1)] SU(2) U(1) ~ T^n

!

Global SU(2) anomaly (single SU(2) doublet)

O. Bär and I. Campos, Nucl. Phys. B581, 499 (2000)

non-contractible loops

Uµ(x) = eiA

T

µ(x)g(x)g(x + ˆµ)−1_U

[w](x, µ)V[m](x, µ)

U(1) degrees of freedom

Tn

[U (1)] × M[SU(2)]

measure term smooth on

pure SU(2) theory

our approach

•

covers all SU(2) topological sectors with vanishing U(1)

magnetic fluxes

•

global integrability is proved rigorously

even number of SU(2) doublets, U(1) Wilson line parts

•

explicit with two simplifications cf. U(1), Luscher (99)

★

direct proof of gauge anomaly cancellation in

★

separate treatment of the Wilson line

•

some non-perturbative applications ?

the Glashow-Weinberg-Salam model on the lattice

in finite volume

L

4

Y.~Nakayama and Y.K., Nucl. Phys. B597, 519 (2001)

D.~Kadoh, Y.~Nakayama and Y.K., JHEP 0412, 006 (2004)

D.~Kadoh and Y.K., JHEP 0805:095 (2008), 0802:063 (2008)

•

a perturbative computation of the EW contributions to

muon g-2 (one-loop check, beyond one-loop)

•

a computation of the effect of quarks & leptons to the

sphaleron rate at finite temp. (at one-loop, top quark?)

•

a lattice construction of models of dynamical EW symmetry

breaking

• “SU(2) walking” technicolor model can be put on the

lattice (!)

Appelquist et al.,

Dietrich, Sannino, Tuominen (05) [minimal]

• cf. recent activitiy to study QCD-like theories in/close to

the conformal window

•

T. Appelquist, G.T. Flemming and E.T. Neil (Yale Univ.)

Phys.Rev.Lett. 100:171607,2008 (

arXiv:0712.0609)

•

...

•

a study of Electroweak phase transition(1st order possible?)

•

some other non-perturbative applications ?

related results

•

N=(2,2) 2-dim. SuperQCD with exact chiral and Q

symmetries Sugino-Kikukawa(2008)

arXiv:0811.0916

•

reflection positivity in Lattice QCD with overlap Dirac

operator Usui-Kikukawa(2009) in preparation

•

....

•

Strongly coupled fourth family and a first-order

Electroweak phase transiton

Kohda-Yasuda-Kikukawa(2009)

appear very soon

work in continuum (^^;

Happy 60th Birthdays !

Ishikawa-san

&