磁場を有する余剰次元
理論の次元脱構築
龍田佳幸(DESY)

with 富谷昭夫(華中師範大)based on arXiv:1703.05263

標準理論と世代数
クォーク・レプトンは世代構造がある
表現は同一．Higgs粒子との(湯川)結合定数の大きさが違う

これとか

これとか

これとか

これも

新物理と標準理論の物理標準理論を超えた物理

標準理論の背後の物理

超対称性

SU(3) SU(2) U(1)

余剰次元

3世代構造

暗黒物質

主なトピッククォークのフレーバー構造

暗黒エネルギーレプトンのフレーバー構造

インフレーション

Higgs機構超弦理論

imitation of [Svetitsky’s talk @ Lattice 2017](個人的な意見では)標準理論自体を理解しつつ，標準理論を超えた物理をやりたい

目次
イントロ
余剰次元模型の基礎知識
トーラスコンパクト化と磁場
次元脱構築「トーラス＋磁場」の脱構築
まとめと展望目次
イントロ
余剰次元模型の基礎知識
トーラスコンパクト化と磁場
次元脱構築「トーラス＋磁場」の脱構築
まとめと展望how some of the open questions mentioned in lecture 4 can be addressed in
March 18, 2008 9:47
World Scientific Review Volume - 9in x 6inthe warped setting, resulting in a “complete” model.

Kaluza-Klein展開と波動関数

agashe1.2. Lecture 1
March 18, 2008 9:47
World Scientific Review Volume - 9in x 6in
高次元場を，余剰次元方向の運動量の完全系で展開
4
Kaustubh
Agashe1.2.1.
Basics of Kaluza-Klein
Decomposition

余剰次元空間がコンパクト空間なら運動量は離散化される
Considerfollowing
5D action
for aon(real)
scalarhenceimpose theperiodic
boundary
conditions
the fields
as fieldwell, (herei.e., weandrequireµ
forth,
thecoordinatesxwill
denotethe usual
4Dthe
and
the
coordinateyΦ(y = 2πR) = Φ(y).
Then,
we
can
(Fourier)
expand5Dscalarfield
as1)の場合：
c.f., [Dawson, Mohapatra ’08]
例えば，円弧(Swillfollows:denote the4extra dimension):
Kaustubh Agashe
!
!
"n=+∞
%
#
$
2
S5D = Φ d=4 x√ 1 dy !
∂M φΦ(n)(∂
Φ)
−
M
ΦΦ
(1.1)
iny/R
M
(x)e
(1.2)
impose periodic boundary conditions on the fields as well, i.e., we r
2πR n=−∞
Φ(y = 2πR) = Φ(y). Then, we 2can (Fourier) expand
the 5D scalar fi
3
Since gravitational law falls oﬀ as 1/r and not 1/r at long disfollows: front has
周期境界条件：
whereitthecoeﬃcientbeenchosen forpropernormalization.
= 2⇡R)= (y)Suptances,
is4D＋1D
clear thatin we
mustcompactify
the(y
extra
dimension.
Substituting this expansion into S5D and using
the orthonormality
of
n=+∞
1
!
pose we compactify the extra dimension
onacircle
(S
),
i.e.,
with
y
un1
iny/R
(n)
iny/R
profiles of the Fourier modes in theΦextrae) to inteφ (x)e= √dimension (i.e.,
restricted (−∞ < y < ∞), but with y 2πRidentified with y + 2πRa . We
n=−∞ 4D action:
grate over the extra dimension, we obtain the following
a Equivalently, we can restrict the range of y: 0 ≤ y ≤ 2πR, imposing the condition that

&
'
"
%
#
$
%
$
(
2
!
where
the
coeﬃcient
in
front
has
been
chosenfor
propernormalizati
n
2
4
(n)
µ (n)
(n) (n)− M + 2 φ φ
S4D = d x∂µ φ
∂ φ
Substituting this expansion into S5DR and using the orthonormaniny/Rprofiles of the Fourier modes in the extra dimension (i.e.,
e
) to
(1.3)
grate over the extra dimension, we obtain the following 4D action:
質量＋運動量を持った4Dの場だけで記述できている
This implies that from "the 4D pointof view%the
5D scalar
field appears
&
'
%
#
$
$
(
2
!

and deconstructing points of view.
h are to decompose extra dimensional parts ofensional fields into a complete set spanned byde wavefunctions. A novel proposal by Bachas
quantized magnetic fluxes in toroidal compactII. MULTIPLE ZERO-MODES WIT
where the magnetic fluxes provide a degenerlowest KK-expanded wavefunctions. The deトーラス＝2D平面＋周期境界条件
the lowest
KK modes should be regarded as
Here, we briefly review an original theo
eplication of matters after dimensional reducRef. [2]. We consider a six dimensional (y5et, yal.
6 pointed
トーラスの座標：
the Bachas’s
proposal, Ibanez
ory that has magnetic fluxes (or magne2e degenerated mode functions can be analytiin
toroidalcompactification
T. It cont
R
z
⌘
(y
+
iy
)/(2⇡R)!
複素座標：
56
sed by an elliptic function and discussed theiring flux background 2b⇡ = T 2 F of the fiel
properties [2].
(ib/2)dz ∧ dz̄, which is provided by a vect(b)such higher
dimensional
theoriescontain
dihF
i
=
hF
6 0
ゲージ場の強さのうちの余剰次元成分の期待値：
56 i =
coupling constants and are non-renormalizable.
b(b)
A (z) = Im (z̄dz),
theories possess less predictability in general.
5次元ではこのような成分はない
2
rkani-Hamed et al. proposed a splendid ul-

トーラスと磁場

where two Cartesian coordinates of the fif
string theoryで言うﬂux compactiﬁcation

rections y5 and y6 are expressed by z ≡ (yds
the Dirac charge quantization, The single-valuednesswhere
αakane.waseda.jp Diracʼs charge quantization：[Bachas ’95; Cremades, Ibanez,
Marcheano’04]
of the 6D acti
@mail.ccnu.edu.cn
tractible loops, e.g., zZ→ z + 1complex→ z+1+qb= m ∈ Z.
2π>0possesse
磁場の”総量"：(2) hF i = b
we decoT2
U(1) 電荷： q(ψ+,0 , ψcomplex v
qb
possesses= m ∈ Z.
(2)
we decom
2π(ψ+,0 , ψ−,We perform the KK decomposition of the 6D Weyl
spinora normali[Cremades,
Ibanez, Marchesano’04]Ψ and scalar Φ by a complete
set asthe lack o
KK展開したmasslessmode(s)が満たす式
izable in t!
(5) are loΨ(xµ , z) =χn (xµ ) ⊗ ψn (z),(3)
their sche
n
n
=
0,
1,
2,
...
!
point is thΦ(xµ , z) =ϕn (xµ ) ⊗ φn (z).
(4)
to
the
fam
n
though m※コンパクトスケール＝1
of the
spi
for
n
=
0,
1,
2,
·
·
·
,
where
χ
denotes4D
Weylspinors.
n2⇡R=1の単位を採用：
ワイルスピノル場
tra. Indee
Here, the KK-decomposed wavefunctions in the extra dispectra ofmensional parts ψn and φn are chosen to be eigenstateswith D = 2@/@ z̄ + ⇡mz
iD
/
=
m
n
n
n
of the covariant derivative D = 2∂/∂ z̄ + πmz and theLaplace operator ∆ = {D† , D}/2 for extra dimensionsasフラックスからの寄与/ n = mn ψn and ∆φn = m2n φn , respectively, as definediDψ
余剰次元座標での微分(The
KK
inスカラー場も同様
Ref. [2]. In the Letter, we adopt a unit 2πR = 1 for
4πm(n +
the compactification radius R and set a U (1) chargeas†Letter,
th
with
=
{D
,
D}/2
q = 1. On(1) for m > 0, the lowest
n the flux nbackgroundnand φn fo
KK-decomposed modes ψ0 are m-multiply degenerated asin Ref. [6]

ゼロモード方程式=mnspectrathe spinor
mode offunctionsof themensional parts ψn and φn are chosen to be eigenstates thoughn = 0, 1, 2,derivative· · · , where Dχn =denotesof theforcovariant
2∂/∂4Dz̄ +Weylπmzspinors.
and the of the spinor, an exception re
Here,
the KK-decomposed
wavefunctionsin theextra di- as tra. Indeed, it is straightforwa
Laplace
operator∆ = {D† , D}/2
for extra
dimensions
spectra of the spinor are calcu
mensional
parts
ψ
and
φ
are
chosentobeeigenstates2nn/ n = mn ψn and ∆φn = mn φn , respectively, as definediDψ
of the covariant derivative D = 2∂/∂ z̄ + πmz and the(The KK mass spect2
in Ref.
[2].
In
theLetter,
we
adopt
a
unit
2πR
=
1
for
m
=4
†
[Cremades, Ibanez, Marchesanon’04]
Laplace operator ∆ = {D , D}/2 for extra dimensions as
4πm(n + 1/2).) Alth2
the compactification
radius
R
and
set
a
U
(1)
chargeas/ n = mn ψn and ∆φn = mn φn , respectively, as definedi解は楕円関数で与えられる：Dψ
Letter,thespectra
concrete
w
(The
KK
mass
of
th
q = 1.in On
theflux
background
(1)
for
m
>
0,
the
lowestRef. [2]. In the Letter, we adopt a unit 2πR = 1 for+,0,0
where,
and+ φ1/2).)n ≥ 1 are4πm(nthn for Although
the compactification
radius
R
and
set
a
U
(1)
chargeas
KK-decomposed
modes
ψ
are
m-multiply
degeneratedasT
0,0 ：規格化可能な解なし=(,)
the concrete+,0in Ref.
[6]. wavefunctq = 1. On the flux background (1) for m 0> 0, thelowest ,0 Letter,and φnIII.
for n ≥
1 are analytic"# degenerated as(DE)CONST
KK-decomposed modes ψ0 are m-multiply
j/m
in Ref. [6].
jπimzIm z# (mz, mi),ψ+,0 (z) = N eϑ "(5) III. (DE)CONSTRUCTID
j/m
0j
πimzIm z
DIMEN
ψ+,0 (z) = N eϑ(mz, mi),(5)0
In order to (de)coj=0, 1, 2, …, m-1
with j = 0, 1, · · · , m − 1 and the Jacobi theta function
from thInsions,order weto start(de)constructwith jヤコビのテータ関数：= 0, 1, · · · , m − 1 and the Jacobi theta function
sions,
we start from
the1Wilso
diagram
in
Fig.
or n
" #
diagraminparts
Fig. 1associatedor namely d
!
The
α " #
2
πi(α+ℓ)!
parts associated with thα τ) =
ϑ(ν,e πi(α+ℓ)2ττ+2πi(α+ℓ)(ν+β)(6) The satisfy+2πi(α+ℓ)(ν+β) ,the Dirichlet b
ϑ
(ν,
τ
)
=
e
,
(6)
satisfy the Dirichlet boundary
β β
ℓ∈Z
of4D
theWeyl
4D spinor
Weyl inspinorℓ∈Zof thethree

ゼロモード解とその縮退

↵,

：実変数

⌫, ⌧ ：複素変数⎡⎤⎡⎤
4Dで同じ表現の場が複数個に縮退
!
!
!
&&
'
1
!
!
†
⎣
L =1
η̄ȳ i ⎣ΓM QM (ȳ)ηȳ+M̂ − QM (ȳ −M̂ )ηȳ−M̂ ⎦ − M'0 ⎦†：m個の各モード
L→= 世代数
ΓM QM (ȳ)ηȳ+M̂ − QM (ȳ − M̂ )ηȳ−M̂
2 ȳ η̄ȳ i
M=5,6,72 ȳ
pic from [Abe et al. ’14]
M =5,6,7⎡

波動関数の局在確率密度：

j+,0 (z)2[Cremades, Ibanez, Marchesano ’04]

をプロットしてみる

波動関数のガウシアン的局在：j(y6 j)2+,0 ⇠ e

0.8

0
0

0.6
0.4

0.2
0.4

0.2

0.6
0.8

j=0

1

1

0.8

0
0

0.6
0.4

0.2
0.4

0.2

0.6
0.8

10

10

向

1

2

2

1

1

0.8

0
0

0.6
0.4

0.2

方

1

y6

2

j=1

0.4

0.2

0.6
0.8

10

j=2

y6方向への局在性は，低エネルギー有効理論で，結合定数などの物理量に影響この機構を使った現象論
3世代模型[Abe, Choi, Kobayashi, Ohki ’08; Abe, Fujimoto, Kobayashi, Nishiwaki, Miura, Sakamoto,YT ’15; …]離散フレーバー対称性 [Abe, Choi, Kobayashi, Ohki ’08; Abe, Kobayashi, Ohki, Sumita,YT ’14; …]
MSSM-like models[Abe, Kobayashi, Oikawa, Sumita ’14; Abe, Kobayashi, Sumita,YT ’14; …]

動的超対称性の破れ[Abe, Kobayashi, Sumita ’16; Higaki,YT ’16]

Type I seesaw[Buchmuller, Schweizer ’17; Ishida, Nishiwaki,YT ’17 & in preparation]SUSY SO(10) + split SUSYSUSY SO(10) + ﬂavor mixings
モジュライ固定[Buchmuller, Dierigl, Ruehle, Schweizer ’15]
[Buchmuller, Schweizer ’17]

[Braun, Hebecker,Trapletti ’07; Buchmuller, Dierigl, Ruehle, Schweizer ’16; …]

目次
イントロ
余剰次元模型の基礎知識
トーラスコンパクト化と磁場
次元脱構築「トーラス＋磁場」の脱構築
まとめと展望UV completion
余剰次元模型は不幸にも「non-renormalizable」
何がしかの理論の有効理論として考えるべき(私の知る限りでは)

UV completionとしては2つの可能性が濃厚

超弦理論へ埋め込む

See a good textbook, [Ibanez, Uranga ’12]

利点：openstring sectorとして単純に埋め込める．e.g., magnetized D-branes
難点：超弦理論がやはり難しい

次元脱構築により4D理論の寄せ集めとして理解する
利点：4Dの知識だけで理解可能．格子理論の知識も使える．一般の系にも応用可能
難点：場合によっては計算コストはそこそこ必要

今回はこちらaccording to Wikipedia

次元脱構築

形而上学の仕組みを解体し，その可能性の要素を抽出して再構築を試みる
脱構築＝
哲学的思考の方法．(=deconstruction)

余剰次元空間を格子化する：[Arkani-Hamed, Cohen, Georgi ’01]

：4D＋S1連続理論

our 4D方向

5D方向
y=0

our 4D方向

y=2πR：4D連続理論 1D格子理論
5D方向

y=0

y=2πR

複数個の繰り込み可能な4D理論だけで有効的に5D理論を実現
5D S1：Arkani-Hamed, Cohen, Georgi ’01
5D QED：Hill, Leibovich ’02

5D RS：Abe, Kobayashi, Maru,Yoshioka ’03
6D T2 with flux：our work

次元脱構築 (続き)複数個のweak gauge groups G，strong gauge groups Gs
を繋げる：

Gsがカイラル凝縮して，複数個の非線形シグマ模型に
3/ fs

ゲージ結合定数と，凝縮期待値を調整する： a = 1/(gfs ), R = N a
ゲージ場の質量行列を対角化すると， mk ' 2⇡|k|/R

KKスペクトラム(近似的に)再現できている

目次
イントロ
余剰次元模型の基礎知識
トーラスコンパクト化と磁場
次元脱構築「トーラス＋磁場」の脱構築
まとめと展望ドメインウォール(DW)・フェルミオン
定式化の前に，Nielsen-Ninomiyaの定理の復習：[Nielsen, Ninomiya ’81]

1) 格子上の並進対称性
2) カイラル対称性の全てが満たされるとき，

3) エルミート性

フェルミオン・ダブラーが生じる

4) スピナー場の双一次形式
5) 相互作用の局所性

d次元空間のとき，2d-1個の不要モード

余剰次元空間への磁場はtopological indexを稼ぐ

指数定理に適用可

ダブラーを避ける為，DWフェルミオンを用いる [Kaplan ’09]
→ DW用の格子数が大きいときOVフェルミオンへ

DWフェルミオン用の1次元空間を加え，境界条件とその方向のバルク質量によりダブラーを重くする

III.

This isdufrom the scratch,anddard domain-wall
Dirac operator.
(DE)CONSTRUCTION
OF MAGNETIZED

j/m y6 )) in
nctionsQ
toroidaldirectionsacquireM
πimzImz (y5T,(Hthetransfer
matrix= (1the− HMM mi),
thatDIMENSIONS
the
Möbius domain-wall
Diracoperatoz) = N eϑ(mz,(5))/(1 + HM ).
laptopanddesktopckerneloperator(theHermitianMöbius-Dirac
opera0lowing expectation values:
more appropriate chiral symmetry [9] only thr
HM is given as
shown,
itis possible
In order to(de)constructtoroidalTo
dimenmass
spectra andmagnetizedmode functions.
realize
Gaussian- t!
extensionoffermion
theseventh
in compari· · , m − 1 and the Jacobi theta function
sions, we startfromtheWilsonon
the 3Ddirection“Moose”
localized
zero-modes,
itisnecessarytoselectappropriate
2DW
KK
wavefunctions
asdiagram
in
Fig.
1
or
namelydiscretizedthreedimensions.
standard
one,
e.g.,
Ref.
[10].
Also,
it
is
expec
HM = Γ 7 1
,
(11)
bases
out
of
eigenvectors
in
Eq.
(14)
such
that
all
of
kinetic
(y
=
̸
1),
5
2
The parts associatedwiththeseventhdimensionoffieldsQ5!
(y5πi(α+ℓ), y6 )2 τ +2πi(α+ℓ)(ν+β)= 2 + DW
(12)
terms
are
canonicallynormalized.
This
isbecause,
on
the
KK
mass
spectramMöbius domain-wall fermion with N7 latticnν, τ ) =e,(6)satisfytheDirichlet
boundary
condition.
The
Lagrangian
exp[−iby](y=1),
there is no reason to be eigenstates of a covariant6
re Γ7 is a chirality operator, namely the Paulimatrix 5 lattice,comes
asymptoticallyequivalent
tothethatstandremarkable
that
negat
ℓ∈Z
of the4D Weylspinorin
three
Cartesiancoordinates
reads:
translational
operatorỸ
defined
in
Ref.
[15].
Note
and DW is the Wilson–Dirac operator with the
mass
5
6
2N
points [11].
The
Hermitiandoma
7 lattice
the zero-mode
wavefunctions
in Ref.
[2] arealleigenstatesthelowesteigenvalues,
0<
2 isexp[ibyrequired to5 ],realize a correct
Q6 (y
)=
(13)
0 . Here,
5 ,My06<ȳ =(y5the, yoperator.
, y7 ) of the
Dirac
This
is dueto [12]the fact
6
7 dardoperatorỸ .DW
operatorH
withandthe thuscorresponding
structure, andwe set domain-wall
M0 = 1. We
adopt
same
ory,
the
(de)
ce of parametersthat
as ⎡those
reported
in
Ref.
[13].
Note
the
Möbiusdomain-wall
Diracoperator
can realize⎤practicalinfrared
regulator
filed
istakegiven
in Refs.
[9, 11In
ourcalculation,wethenumberoflatyifMwe start∈ {1/N
,
2/N
,
·
·
·
,
1}
(M
=
5,
6),
and
b
=
M
M
the overlap-Diracwhich ischirality
projection
via
!more
!
!
& operator,' in the1fromappropriatechiralsymmetry[9]
only
throughaslight†ticepointsseventhdirectionissixteen,
i.e.,
N
=
16,
7
⎣
= limit η̄ofΓand(ȳ)ηan
− QM
(ȳ
− M̂
)ηȳ−M̂ ⎦ particle− M0η̄ȳ ηȳin LtheitMhasexact
signȳ iN7 → ∞
M Q
ȳ+M̂
szedrequiredfromtheconsistencyofsingle12 ȳ extension of the seventh direction
andin
setcomparison
N5 = N6ȳ = 30,with
20 (monopoles).
and
10 for comparison.
Also, thaNotetheM =5,6,7
tion, the theory
becomesnonlocal and phenomenologH
=
sgn
(H
DW
M
rat3. Our ),
we
fix
the
numberofmagnetic
fluxes
as
m
=
code
⎡
⎤
[14].
2
ynctionunnatural.
standard
one,
e.g.,
Ref.
[10].
Also,
it
is
expected
that
the
els
is
controlled
by
N
!
!
is
implementedbyPython
3.4
and
Numpy
with
Cython&'1 flux background,†
o introduce magnetic
assume+that⎦ points−η̄ȳ ⎣domain-wall
QMwe
(ȳ)η
Q
−from
M̂ where)ηȳ−−
2η̄ȳ ηȳmatrix, the calculation(8) sgn in(HMöbiusfermionN
latticebeM (ȳwith
ȳ+M̂
M̂
the
scratch,andisperformed
our
7
the
KK
spectra
are
the
sign
function
b
2
M )d
rat
functions QM (y5 , y6 )ȳin theMtoroidal
directionsacquire=5,6,7
andstandard
desktop
computers.
setup aspreviously
comes
asymptoticallyequivalentlaptop
torationalthe
one viation
withByapproximationis thedefinedfollowing expectation values:
betweenthecoshown,
itispossibletorealize
localization
profilesofthe
2N!7 lattice points [11]. ZERO-MODE
The Hermitian
domain-wall
Dirac
V. (DE)CONSTRUCTED
KK
wavefunctions
as
well
as
severallower
modesof theN7
ries
can
be
recognized1(y5 ̸= 1),
same
as
those
in
[Al-Hashimi,
Wiese
’08]
1
−
(T
(H
))is
2
MIt
HDW [12] with (12)the corresponding
Q5 (y5 , y6 ) operator=WAVEFUNCTIONSKK mass spectra Pauli–Villars
mnsgn= 4πmn
(7)
as
shown
in
Fig.
2.
(H
=Ref.
,
exp[−ibycoordinate(y5 =y5 , y6 , y7 ) is a discretizedfor1), of the Dirac operator as the same as rat
theand
work
[7].
M )in
6]
N
.
Next,
we
de
N
61 +
remarkablethatchirality modes
neverappear(T (Hinfrared
regulator
is givenin11]negativeasM ))in 7
se diagram which
is definedas yM filed∈Here,
QM (ȳ)isRefs.
a link[9,functionfor the M -th dimension,
the lowest eigenvalues, as recognized in the continuum (D),itheQ (y , y ) = exp[iby ],(13)· · · , 1}6(M5 =6 5, 6, 7), M̂5 is a unit vector
which ensures the gauge invariance [8]. To
be precise, afwavefunctions
ψ
(
0
ory,
and
thus
the
(de)construction
mechanism
also
lead
to
1=
with
the
transfer
matrixT (HirectionandΓM ,is2/N
the gamma
matrices 5,
forwavefunctions,ter integrating
out
only
the
seventh directionof thelattiM ) = (1 − HMrderobtainzero-mode
the
main
re
yM ∈to
{1/N
6),
and
b
M
M , · · · , 1} (M = H
chirality
projection
vialink
thefunction
presence
nonvanishingfluxes= fermion
sgnrat(HMwith),(9) Qof7 (ȳ)
easilyfindsatisfactory
DW cized
mensions.
As
is
the
case
with
the
latticefieldsatrivial=1Thekerneloperator(theHermitianMöbius1each ofHKK
m tois requiredfromthe consistency problem,of single particle
2
M lev(monopoles).
Note
that
the
degeneracy
of
solve
an
eigenvalue
η
=
η
(x)
and
η̄
=
η̄
(x)
are
regardedasandaPauli–Villars
infrared
regulator
field,
chiralfermions
T
(H
)
=
FIG.
1.
A schemaȳȳȳM wavefunctionsmodefefunction[14]. ȳ
tor)
H
is
given
as
els
is
controlled
by
N
,
while
the
mass
squaredratiosofM
7
1+Hrmion fields eachwhereother in thethe 4D
Minkowski
arerealized in
the residual
2D,
similarly
tothe
a derivation
of
M
Moose
diagram(ormatrixsign
function
sgn
(H
)
by
use
of
M
rat
the
KK
spectraaredetermined
by
N
and
N
.
The
de(D)
(D)
5
6
uum
theory(5).
Itisd M0 is a realH
positiveψparameteroforder
the (y),overlap-Dirac operator (14)in thecontext of the 4D latwith
(y)
=
λ
ψ
DW
n nticeother
2Ddirectionn approximation
viationthe continuum
and(de)constructed
theo- ass
rational
is defined
as between
W
areIV.
interested
in
the
low
energyspectrumofgauge
theory.
Forpractical
reasons,
we
employthe(DE)CONSTRUCTED ZERO-MODE
Hthan
Γis
,of tonents
arediscretelyr
M =7stanries can berecognized,
dependingonnumbersN5utilnecessaryator, thus we WAVEFUNCTIONS
extract the low lying modesMöbius domain-wall
Diracoperator rather
thethe2 + Dzero-mode
W
N
7
and
N
.
Next,
we
depictthe(de)constructed6
1 − (T (HM ))in (D),iy ≡ (y5 , y6 ). Eigenvalues and eigenvectors
thewe conclude
with ȳ that= (y5 ,they6 , y7sgnrat (HM ) =(10)
wavefunctions
ψ,is= 0, 1,2) operator,
in Fig.in
3, thewhere
one can
0 a (iwhereΓchirality
namelytheN
Wilson-Diracop.
latticegaug
77order toobtain zero-mode
wavefunctions,main1KK-decomposed(de)constructed.
alue
problemcorrespond
tothethe+ easily(T (Hfind))M satisfactory agreements between each of zero：共変微分の部分から
σ3 , and DW is the Wilson–Dirac
operator w
is to solve an eigenvalue problem,

定式化

トーラス用の2次元の格子(y , y )に加え，DW用の1次元格子(y )を考える：

背景磁場に対応するリンク関数を仮定解きたいエネルギー固有値方程式：/

the KK spectra are determ
viation between the contin
ries can be recognized, deand N6 . Next, we depict(D),iwavefunctions ψ0(i =easily find satisfactory ag
mode wavefunctions for E
uum theory (5). It is also
nents are discretely
××××××realiz
we conclude thatthemagn
××××××
(de)constructed.

KKスペクトラムの脱構築(DE)CONSTRUCTED ZERO-MODE
WAVEFUNCTIONS

r to obtain
zero-mode wavefunctions, the main
KKスペクトラム＝固有値
solve an eigenvalue problem,
HDW ψn(D) (y) = λn ψn(D) (y),m2n /m21(14)

5

これand eigenvectors in the≡ (y5 , y6 ). Eigenvalues
4
problemcorrespondtothe
KK-decomposed
HDWは数値的に求める
3

パラメータなど
7D方向(DW用)の格子数：N7=16T2の格子数：N5, N6=10, 20, 30
m=3：ゼロモード 3＝3世代××××××


××××××


2





××××××

1






×××

0









n
1

2

3

4

5

エネルギー準位

KKスペクトラムのうち，軽いモードは誤差も少なく再現

the KK spectr
viation between
IV. (DE)CONSTRUCTED ZERO-MODE
ries can be rec
WAVEFUNCTIONS
and N6 . Next,
wavefunctions
In ゼロモード波動関数＝ゼロ固有値の固有ベクトルorder to obtain zero-mode wavefunctions, the main
easily find sati
task is to solve an eigenvalue problem,
mode wavefunc
uum theory (5HDW ψn(D) (y) = λn ψn(D) (y),=0for (14)0nents are discr
これこれeigenvectors in thewhere y ≡ (y5 , y6 ). Eigenvalues
andwe conclude th(de)constructedeigenvalue problem
correspond to the KK-decomposed
HDWは数値的に求めるゼロモードの脱構築

グレーの局面：連続理論のゼロモード

ゼロモード波動関数の概形も再現

目次
イントロ
余剰次元模型の基礎知識
トーラスコンパクト化と磁場
次元脱構築「トーラス＋磁場」の脱構築
まとめと展望まとめ「トーラス＋フラックス」という枠組みは，カイラルフェルミ
オンと，その世代数という重要な構造を作れる
現象論＆宇宙論模型も次々と構築されている

次元脱構築手法を「トーラス＋フラックス」の系に適用
ドメインウォールフェルミオンを用いた，4D連続理論＋3D格子で脱構築可能

$ j
%† k
i
d z ψ+,0 (z) ψ+,0 (z) φ0 (z)
2

=
T2

今後の展望
× χi0 (xµ )χ̄j0 (xµ )ϕk0 (xµ ).

(15)4dimensions is expressed as" are calculated byThus,
the eﬀective Yukawa couplings
××××××2zero-modeµµµ
4D有効理論における3点結合定数＝湯川結合定数：
dimensions
is expressedofasthreeoverlap
integrations
kinds
of
the
waveL
∝
d
z
Ψ(x
,
z)
Ψ̄(x
,
z)Φ(x
, z)
Yukawa
××××××T2"
functions[2]:#"&

2
µ
µ
µ
$ j
%† k
：6D連続理論
L
∝
d
z
Ψ(x
,
z)
Ψ̄(x
,
z)Φ(x
,
z)
2
i
"
 Yukawa=
d z ψ+,0 (z) ψ+,0 (z) φ0 (z)
 2
$
%
T
j
2 † k&
2i
#T"
yijk=
d z ψ+,0 (z)ψ+,0%(z) φ0 (z),(16)

$
† k
j
i
22
T
=
d z ψ+,0 (z) ψ+,0 (z) φ0 (z) × χi0 (xµ )χ̄j0 (xµ )ϕk0 (xµ ).
(15


T2

up to an overall factor.
Since
the
zero-mode
wavefuncj
i theµ eﬀectiveµ
kYukawaµThus,
couplings
are calculated b
× χ0 (x )χ̄0 (x )ϕ0 (x ).
(15)tions succeeded
to beoverlap(de)constructed,
corresponding
次元縮減後，4D有効理論の湯川結合定数に
integrationstheof threekinds of the zero-mode wavnYukawacoupling
constants
shouldbe arecalculated
as follows:
Thus,
the
eﬀective
Yukawacouplings
calculated
byfunctions[2]:
4
5
overlap integrationsof three $kinds of thezero-mode wave'
"
%
†
(D)
(D),i
(D),j
(D),k
$ (17)%† ：脱構築版
y
=
ψ
(y)
ψ
(y)
φ
(y).
j
functions[2]:2i
ijk symbols)uum (black crossy =d z ψ (z) ψ (z) φk (z),(16y"heories, where the square

ijk

T2

+,0

+,0

0

$ j
%† k
i
those of N5 =N6== 30 d z ψ+,0 (z) ψ+,0 (z) φ0 (z),
yijk
(16)
up
to
an
overallfactor.
Since
the zero-mode wavefun
Detailed analyses
T 2of the (de)constructed Yukawa couplings
2

連続理論と脱構築版のズレを見積もって，
tions succeeded to be (de)constructed, the correspondin

are left for our future work. In particular, corrections fromup to an overall factor.Yukawa
Since coupling
the zero-mode
wavefuncconstants
should be
calculated as follow
カットオフ(＝格子スケール)への制限を求める
small valuesof N5 , N6 and N7 to Yukawa couplingsaretions succeeded to be (de)constructed,
the corresponding'$ (D),jbe de%† (D),kcharacteristic features of our(D)setup. Those
might
(D),i
Yukawa coupling constants shouldbecalculated
as follows:
D DISCUSSION
y
=
ψ
(y)
ψ
(y) φ
(y).
(17
ijk
tected in the future experiments [17].