リサージェンス理論に基づく量子非摂動解析
Resurgence Theory for Non-perturbative Quantum Analysis

Tatsuhiro Misumi
Research and Education Center for Natural Sciences, Keio University

カスタム検索第33回 サイエンス・カフェ

イベント・ニュース

月が落ちてきた 〜重力加速度 g の測定〜
講師： 古野泰二
日時： 2017年08月26日 ( 土 )へPPP2017@YITP 08/02/17
13:30〜15:30

はじめに：摂動論とリサージェンス理論

量子論における摂動論と非摂動解析
H = H0 + g 2 H

g21

摂動論：H0の固有状態に基づいて，量子揺らぎを計算

H = H0 + g 2 H

g21

非摂動解析：摂動パラメタの大きな領域ではHを対角化し厳密な固有値を得る必要あり経路積分と固定点
Z=Z

D exp( S[ ]) =自明な固定点(摂動的真空)
1X
Z0 =aq g 2q
q=0

摂動級数

X

Z

2saddles

非自明な固定点

Z / e

Ssol

⇠e

非摂動的寄与

S

=0

A
g2

摂動級数と非摂動的寄与の関係

X

aq g

q=0

摂動級数

2q

❌

exp



A
g2

非摂動的寄与「摂動的寄与と非摂動的寄与は関連付かない異なる寄与」というのが一般的な見方
本当にそうだろうか？

!

dxvergence encodes physical information about the saddles of ordinary integr
摂動計算とボレル和
ls of quantum mechanics and quantum field theory, as a consequence of Darb
3]. We recall a few relevant definitions and motivate (known) generalizati!
"
tions by using simple quantum
2 mechanics.
H0 + asymptoticg Hpertseries
ψ(x)
=
Eψ(x)
2 ) denote a perturbativethat satisfies the “Gevrey-1” cond2

P (g ) =

⇥

q=0

2q

aq g ,

摂動級数（漸近級数と仮定）はq

Gevrey1 : |aq | ⇥ CR q!

一般に階乗発散し収束半径0 aq / q!

' series tha
sitive constants C and#R [5, 7].$Known%examples&2 of perturbative
高次まで摂動計算を行っても意味のある情報は得られなさそうだが…
m dx condition [29]. We denote themechanics
and QFT satisfy the “Gevrey-1”
S
=
dt
−
V
(x)
f P ( ) by BP (t) and define it as2dt
ボレル変換：有限の収束半径を持つ級数に変換⇥

aq q
ボレル和：元の摂動級数を漸近級数として持つ解析関数
BP (t) :=
t .
q=0

q!

' at t = 0
Borel transform determines “a $germ of a holomorphic function”

dx!
call a few relevant definitions and motivate (known) generalizations

sing simple quantum mechanics.
encodesasymptotic
physical informationabout
the thesaddlesof ordinary
integravergence
perturbative
seriesthat
satisfies“Gevrey-1”
conditio
摂動計算とボレル和sform. The Borel transform
method is applicable to the following cls of quantum mechanics and quantum field theory, as a consequence of Darb
called Gevrey-1)⇥
3]. 2We recall a few
relevant
definitions
andmotivate(known)generalizatiqµ∂"Ptions(g )by= usinga!qsimple
g 2q , X
1
:
|a
|
⇥
CR
q!
(6.
1 Gevreyqqquantum
22 q mechanics.
1
2H
+
g
H
ψ(x)
=
Eψ(x)
P
(g
)
=
a
(g
)
,
|a
|
∑
Cq!
,
2 ) denoteq=0
q
q
0
pert
a perturbative asymptotic series that satisfies
the “Gevrey-1” condA
q=0
tants C and R [5,⇥7]. Known examples of perturbative series that ari
摂動級数（漸近級数と仮定）は
22qq
onstants.
The
BoreltransformBP(t)isdefinedas
P (gsatisfy) = theaq g“Gevrey-1”,
Gevrey1 : |aq[29].
| ⇥ CRs and QFT
conditionWeq!
denotethe Bora/q!
一般に階乗発散し収束半径0q
q=01
BP (t) and define it as
X
aq q
'
BP
(t)7].
=$Knownt examples, & of perturbative
sitive constants C and#R [5,series
tha
%
⇥
q!
2
aq=0
q q
m
dx
mechanics and QFT
satisfythe“Gevrey-1”
condition [29]. We denote theBP (t) :=
t .
(6.
ボレル変換
S
=
dt
−
V
(x)
q!
fummation
P ( ) by BP(t)defineq=0
it2 )andB(gis definedasas 2
dt
⇥
Z 1
aq q
nsform determines 2“a germofaholomorphic
function”
at
t
=
0,
wi
dt
2
°t/g
t (t).
.
B(g ) =BP (t) 2:=e
BP
ボレル和
q!
vergence. Next, one analyticallycontinues
theobtained germ BP (g
q=00
x t-plane, called the Borel plane.
We
also
assumethattheanalyt2$germ' Pat(gt2)=co0
Borel
transform
determines “aB(gof a holomorphic
function”
that theBorel resummation) reproduces
the original
sum

!

dx

The Borel transform method is applicable to the following class ofvrey-1) encodes physical information about the saddles of ordinary integr
vergence
摂動計算とボレル和µ∂ls of quantum
mechanics
andquantumfield
1
q theory, as a consequence of Darb
X
1
2
2 q
3].
recall aqfew
and, motivate (known) generalizati
P (gWe) =(g )relevant, |aq | definitions
∑ Cq!
(12)
A
!
"
q=0 simple quantum
tions by using
2 mechanics.
H0 + asymptoticg Hpertseries
ψ(x)
=
Eψ(x)
2 ) denote a perturbativethat satisfies the “Gevrey-1” cond
The Borel transform BP (t) is defined as⇥

12q 摂動級数（漸近級数と仮定）はq
X
P (g ) =aq g a, q q Gevrey 1 : |aq | ⇥ CR q!
a/q!
BP (t)=
t,(13)
一般に階乗発散し収束半径0
q
q=0
q!
2

q=0

' series tha
sitive constants C and#R [5, 7].$Known
examplesof
perturbative%&22
幾つかの例ではボレル和が厳密結果を与える！
nmechanics
B(g ) is definedas satisfy the “Gevrey-1”
m dx condition [29]. We denote theand QFT
S
=
dt
−
V
(x)
Z
1
f P ( ) by BP (t) and dt
define it2 as°t/g2(t). dt
B(g 2 ) =
e
BP
(14)
厳密結果
2
⇥
g
0
aq q
BP (t) :=t .
q!SYMoriginal
Watson-Nevanlinna-Sokal(80)cf.) 2x^4
量子力学，N=2
on S^4 sum
Borel resummation B(g) reproducestheP (g 2 ) correctly
q=0

e the integral and the sum. Otherwise, we need to define the sum in' at t = 0
Borel transform determines “a $germ of a holomorphic function”

!

dx

he divergence encodes physical information about the saddles of ordinary integral
tegrals of quantum
mechanicsinformationand quantumabout
field theory,
as a consequence
Darbovergence
encodes physicalthe saddlesof ordinaryof integr
摂動計算とボレル和mls [1,
3]. We recall
a few relevant
definitions
andmotivate
generalizationof quantum
mechanics
and quantum
field
theory,as a(known)consequenceof Darb
efinitions
by using
simplequantummechanics.
3].
We recalla few
relevant
definitions
and motivate (known) generalizati2 ) denote a perturbative asymptotic series that satisfies the “Gevrey-1” conditP(gtions by using simple quantum mechanics.

!

"

2

H +g Hψ(x) = Eψ(x)2 ) denote a perturbative
0 ⇥ asymptotic
pertseries that satisfies the “Gevrey-1” cond

P (g 2 ) =2

P (g ) =aq g 2q ,Gevrey 1 : |aq | ⇥ CRq q!
⇥
q=0
2q 摂動級数（漸近級数と仮定）はq

aq g ,

Gevrey1 : |aq | ⇥ CR q!

aq / q!
me positive constantsq=0
C and R [5,一般に階乗発散し収束半径07]. Known examples of perturbative
series thatntum mechanics and QFT satisfy the “Gevrey-1” condition [29]. We denote the B
sitiveC(t)and[5, 7].itKnown
examplesof
perturbative
seriestha
rm of constants
P ( ) by一般的にはボレル変換が正の実軸上に特異点を持つ
BP
andRdefine
as
2

#

$

%

&

'

m ⇥ dx condition [29]. We denote themechanics and QFT satisfy the “Gevrey-1”
S
=
dt
−
V
(x)
a
q
q
f P ( ) by BP (t) and define BP
it as(t) :=t .
2 q=0 dtq!
⇥

aq q
Z 1e±i✏BP (t)“a:=dtgermgt2 of ta .holomorphic
rmal Borel transform
determines
function” at t = 0,2 ⌥i✏
正実軸上の特異点のため
q!
B(g e ) =
e
BP
(t)
2 q=0
g
0
積分路に不定性
radius of convergence. Next,
one analytically continues
the obtained germ B
whole transform
complex t-plane,
called “athegerm
Borelofplane.
We also assume
thatatthe
ana0
Borel
determines
a holomorphic
function”
t=

$

'

!

dx

vergence encodes physical information about the saddles of ordinary integr
摂動計算とボレル和
ls of quantum mechanics and quantum field theory, as a consequence of Darb
3]. We recall a few relevant definitions and motivate (known) generalizati!
"
tions by using simple quantum
2 mechanics.
H0 + asymptoticg Hpertseries
ψ(x)
=
Eψ(x)
2 ) denote a perturbativethat satisfies the “Gevrey-1” cond2

P (g ) =⇥2q

aq g ,

q=0

摂動級数（漸近級数と仮定）はq

Gevrey1 : |aq | ⇥ CR q!

一般に階乗発散し収束半径0 aq / q!

' series tha
sitive constants C and#R [5, 7].$Known
examplesof
perturbative
%
&
2 2
2 ⌥i✏
2
B(gQFTe satisfy) = Re[B(giIm[B(g)] [29]. We denote them )] ±dx
mechanics andthe “Gevrey-1”
condition
S
=
dt
−
V
(x)
f P ( ) by BP (t) and define it asA
2

Im[B(g )] ⇡ e

g2

2

dt

積分路の不定性に付随して⇥ 符合の不定性を持つ虚部が出現

aq q
BP (t) :=t .
q!
この摂動ボレル和の不定虚部こそ非摂動寄与の情報を含む！
q=0' at t = 0
Borel transform determines “a $germ of a holomorphic function”

なぜ非摂動寄与の情報が摂動論に？
◆ 常微分方程式論のリサージェンス理論

Ecalle (81)



d'
dk '
F z, '(z),(z), ..., k (z) = 0dzdz• 常微分方程式には複数の形式的な漸近級数解が存在 z ⇡ g 2
冪級数型:

0 (z) =

X

aq z

q

指数+冪級数型: e

nA
z

n (z)

n (z) =

X

q
a(n)
z
q

q

q

• 常微分方程式の解は各漸近級数のボレル和の総和 = トランス級数
XnAn'± (z; ) = S± 0 (z) +e z S± n (z)σ：トランス級数パラメタn• 特定のarg[z]でトランス級数パラメタ σ が不連続 = ストークス現象'+ (z; ) $ ' (z; + s)
s：ストークスパラメタ• 解の連続性から各漸近級数が結びつく'+ (z; ) = ' (z;

+ s)

S+

0 (z)

S

0 (z) ⇡ se

A
z

S

1 (z)

!

寄与の級数部分を Φn (z) として，F (z) = eΦn (z) と表すことがある．こactionTrans-series
parameter，F (z, σ):物理量や分配関数，とSaddle-point
Alien derivative：，σ:
Stokesautomorphismはストークス線の上下
ここでは
one-parametertrans-seriesの場合に集中する．
である．特異点方向
θ についてのStokesautomorphism Sθ は以下で

なぜ非摂動寄与の情報が摂動論に？

Ecalle (81)phism と Alien derivative： Stokes automorphism はストークス線の上下◆リサージェンス理論とAlien
calculus
S
=
S
◦
S
,
(4)S は以下で
θ+θ−θ
ぐ群作用である．特異点方向
θ
についてのStokesautomorphismθ
"#!
・特異点方向上下のボレル和を繋ぐ群作用：Stokes
automorphismSθ = Id− Discθ = expe−ωθ z ∆ωθ .
(5)
h X !✓iωθSθ+ = Sθ− ◦ Sθ ,S✓ = Id Disc✓ = exp
e z !✓ (4)" derivative であり，以下のような#のある特異点 ωθ についての Alien !
・各特異点について外微分作用素：AlienSθ = Id − Discθ = expe−ωθ z ∆ωθ derivative.
(5)[∆ωθ , ∂z ] = −ωθ ∆ωθ .

ωθ [!✓ , @]=0(6)

˙ ω ≡ e−ω
異点方向θ のある特異点
ωθθ についてのAlien derivative であり，以下のようなAlien derivative∆∆ωθ を定義することで，以下の交換関係
θ・Alien calculusと通常の微分を関係付ける方程式：Bridge equation．
!✓e z !✓[∆'(z;, ∂)] /@ '(z;
)
=
−ω
∆
.
(6)
ωθ z
θ ωθ
˙ ω , ∂z ] = 0 ,[∆(7)θ˙ ω ≡ e−ωθ ∆ω を定義することで，以下の交換関係
・Bridgeに Dotted
Alien˙ eq.の左右辺比較により各漸近級数間の関係が判明！
derivative ∆θ[∆ωθ , ∂σ ] = 0 , θ
(8)
✓
◆
1
X n+l
A
lA
がわかる．
z
S✓ [∂nz=s1 e z n+l (9), ∂σexp[e] = 0.
A] n =nl=0
˙[∆ωθ , ∂z ] = 0 ,(7)e equation は Alien derivative(Alien calculus) と一般の演算を関係づなぜ非摂動寄与の情報が摂動論に？
S+

0 (z)

S

0 (z) ⇡ se

摂動ボレル和の虚部不定性AzS

Ecalle (81)1 (z)非摂動的寄与ある種のリサージェンス構造は量子論にも存在すると考えられ，
摂動ボレル和が非摂動的寄与の情報を含む．

・コーシーの積分定理による摂動-非摂動関係の抽出

摂動級数の高次項と非摂動寄与の低次項が関係摂動級数から非摂動寄与を原理的には求めることが可能！

摂動的ボレル和とBion配位との不定虚部相殺2 ) reproduces the original sum P (g 2 ) correctly
g B(g
0 the
at the theBorel
resummation
hange
integralandsum.
Otherwise, we need to define the sum in
ummation.
change the integral and thesum. Otherwise, we need to define
the sum in
2
2
he
Borel
resummation
B(g)
reproducestheoriginalsumP(g)
correctly
model, let us consider a factorially divergent series of the following one
esummation.
Lipatov(77)ange the integral and thesum.
Otherwise,
we
need
to
definethesumin
ボレル和の具体的な例
oy model, let us considera factoriallydivergentseriesofthefollowing
Bogomolny,
Fateyev(77) oneµ∂
1q
mmation.
2
X
ns
°g
2
∂.q
(g ) = Ca factoriallymodel, let usPconsider
divergentseries of the following (15)
one
1q! µ
2
X
例) 摂動級数が交代級数の場合°g
A
2
P (g ) = Cq=0 µq!
.
(15)
∂
1
Aq
2
X
q=0 °g without singularities on the positive realorm becomesan
2 analytic function
P (g ) = C
q!
.
(15)
A without singularities on the positive real
form becomes an analytic
function
µ
∂
q=0
1
q
X
°t
CA
∂without
BP
=C X
.
(16)
1 µ
q=
m becomesan(t)analytic
functionsingularitiesonthepositive
real
A°t
A+
t
CA
BP (t) = Cq=0µ ∂=. 特異点は負の実軸上
(16)
1
Aq
A+t
X
q=0°t
CA along the positive real axis
summation is well-defined
as an integral
BP (t) = C=.
(16)
Z 1 A as an A
+ t along the positive real axis
resummation is well-defined
integral
q=0 dt1 A/g2
2 有限かつ不定性
2°t/g 2 CAeEi( A/g )B(g ) = Z 1 2 e.
(17)
2g the positive realのない結果gA+t
ummation is well-definedasanintegral
along
axis
dt
CA
2
0
2
°t/g
B(g ) =e.
(17)
2
Z 10 gA+tially divergent2 series isdta typicalexampleof Borel summable divergent°t/g 2 CA
B(g ) =e.
(17)
2
gis a typicalA +exampletorially divergentof Borel summable divergent0
1 2 series
2 4, if perturbation
is not alternating, the factorially divergent seriesx 非調和振動子ex.) x + gseries2
ally
divergentseriesisatypicalexampleofBorelsummabledivergentorm
withsingularitiesonpositive
real
axis
and
the
Borel
resummmationd, if perturbation series is not alternating, the factorially divergent
series2
uities.
For instance,
supposethat thepert (g ) gives
sform withsingularities
on positive
realperturbation
axis and theseries
BorelPresummmation

ボレル和の具体的な例
Bender-Wu(73)Zinn-Justin(81)例）二重井戸型ポテンシャル
1E0,pert =
2

aq =

g

2

3q+1
q!
⇡

9 4
g
2

89 6
g
2

5013 8
g …
8

1 d2
1 2
(x) + x (12 dx2
2(for large q)

2 ⌥i✏

Im[Bpert (g e)] = Im1e0±i✏gx)2 (x) = E (x)

特異点が実軸上に出現

11BPpert (t) =⇡ t 1/3"Z

Lipatov(77)Bogomolny, Fateyev(77)

t
g2

dt e
g2 ⇡

#

1
3g 2

1
e
=⌥ 2
t 1/3
g

1
SI = 26g

摂動的ボレル和の不定虚部は非摂動寄与(2-Instanton)と関係？

including the relative phase parameter. Section VI is devoted to a summary a
Appendix A we give some details of four-instanton calculations.

ボレル和の具体的な例
II. QUANTUM MECHANICS WITH THE SINE-GORDON POT
Lipatov(77)Bogomolny, Fateyev(77)Bender-Wu(73)例）Sine-Gordon型ポテンシャル
Zinn-Justin(83)In this article, we focus
on the sine-Gordon quantum mechanics described bequation
1
1 d2
2
ψ(x)
+
sin
(2gx) ψ(x) = E ψ(x) ,Hψ(x) = −
2 dx2
8g2

where we follow the notation in Refs. [12, 33] except g is replaced by g2 here [5BPpert (t) =(for largeq) sine-Gordon quantum mechanics is given by [53]Lagrangian
for the
特異点が実軸上! "2
11 dx+ V (x),V (x)= 2 sin2 (2gx) .
L =
に出現
2
2 dt
8g

Int the1g2 → 0 limit, it reduces to the Schrödinger equation of the harmonic osci

"Z

#

t
1
1e
2
The energy eigenvalues of periodic
potentials split intog2bands of states. With
g
±i✏

dt e
2
2e
Im[Bpert (g e )] = Im=⌥ 2
2
are labeled by the0 Bloch angleby gg θ⇡∈ [0,t π] defined
1
2 ⌥i✏

!

ψ x+

π
2g

"

1
SI = 22g= eiθ ψ(x) .

摂動的ボレル和の不定虚部は非摂動寄与(2-Instanton)と関係？

In this article, we are interested in the lowest band, although excited bands can b

The energy eigenvalue E of the lowest band can be expressed in terms of the pa

s of S 2 as

10C.

1 instanton + 1 anti-instanton

I. インスタントン-反インスタントン配位
INTRODUCTION= Bion配位
The amplitude of one instanton and one anti-instanton amplitude is composed of two configura-V (x)tions [IaĪ] coordinate
and [ĪI], as shown in Fig. 2. In these cases, the interaction between the two constituentse in the action density, with x2 being
I. weINTRODUCTION
1 is attractive, and the quasiImoduli integral
1 is ill-defined. Therefore
introduce the Bogomolnyi–+Ī
ction density is depicted
as a(BZJ)
functionxI Ī ( in) two
= ways,arctaneprescription+arctan+ n−g2 >/(2g)Zinn-Justin[29, 30]: wefirst evaluateethe integral by taking
0, and thengg4
weanalyticallycontinuetheresultfrom−g2 > 0 back to g 2 > 0 in the complex g 2 plane. This1
2
al bion configurations for λ = 1/1000, λ = 1/1000
procedure provides the imaginary ambiguity depending on the path of the analytic continuation

x2 )

x1
0

−4

−4

2

0

−2

τ



2

0

−2



4

4

[ĪI] [I Ī] 

R



x1



Bion
2

V (x)






!!!!

0

!!!!



!!!!

2

!!!!

as −g 2 = e∓iπ g 2 .

xx

1 s(x1 , x2 )FIG. 2: A schematic figure of an example of one-instanton and one anti-instanton amplitude ([I Ī], [ĪI]).

figurations for λ1 = 1/1000, λ2 = 1/1000
Each horizontal line stands for the vacuum in the sine-Gordon potential.
4

ty is depicted in two実古典解ではないが，経路積分に含まれる配位
ways, as a function
The amplitude of one-instanton and one anti-instanton configuration [I Ī] corresponding to theleft of Fig. 2 is obtained astion density, with x2 being a coordinate

0
≤
φ
<
2π
V (R) =I Ī

τ

" 2 #ϵ#インスタントン間には
−g
2
|g
|≪1
2
Γ(ϵ)
[I Ī]ξ −2 =dR exp − 2 e−R − ϵR −→
−g
2
0 ✏R
exp[
R]
+
実質的な引力が働くため
"
#
" #
2

g

! ∞

−g 2 =e∓iπ g 2"−→− γ + log"#

2
= − γ + log 2
g

−

πx1

πx1

2

2

+O

1
ϵ

+ O(ϵ)

寄与の計算は困難….？
" #e∓iπ g 2∓ iπ + O1ϵ+ O(ϵ) ,(32)

0 ≤ φ < 2π

where we perform the integral in the first line by considering −g 2 > 0, and in the second lineτ

8⌅V (⌃ ) = (µB , µB ) 2 e ⇥⇤FIG. 1: A schematic figure of an example gof two insta

nstanton and one anti-instanton amplitude is composedandofµtwo
configura⇧j ⌃ r is the charge of the bion Bij .
B = i
The amplitude of one instanton
anti-instanton
amplitude
is composedof two configu
forandtheonevacuumin thesine-Gordon
potential.
This typeof integral,
as noted earlier,is addressed in bosonic
wn in Fig. 2. インスタントン-反インスタントン配位からの寄与
In these cases, the interaction between thetwo constituents
Bogomolny[19]. Both integrals are divergent at large separation, andtions [I Ī] and [ĪI], as shown in Fig. 2. In these cases, the interaction between the two constitueby ⌃ ⇧ 0 where molecular configurations are meaningless. The firsti moduli integral is ill-defined. Therefore we introducetothe
Bogomolnyi–of the uncorrelated[B]-[B] or [B]-[B]events, andis attractive, and the quasi moduli integral is double-counting
ill-defined. Therefore
we introduce
the Bogomoln

2 > 0, and thenption [29,・Bogomolny--Zinn-Justin(BZJ)処方30]: Zinn-Justin
we first evaluate
theintegralbytaking−g2(BZJ) prescription [29, 30]: we first evaluate the integral by taking −g2 > 0, gandt

Bogomolny(79) Zinn-Justin(81)The amplitudeof two instantons shown in Fig

he result from −g2 > 0 back to g2 > 0 in the complex2 g2 plane.

1.

を正の実数と見なす! 2∞

2 plane.
we analytically continue the result from −g −2iθ
> 0 −2back toThis
g > 0 in the complexC̃−R
g
+

"

#

2
T
[II]e
ξ
=
dR exp − 2 e − ϵRgaginary ambiguity
dependingpath of ambiguity
the analytic
continuation
procedure
provideson
thetheimaginarydependingon 0the path of the analytic
continua2.
モジュライ積分を実行

!!!!

C̃

!!!!


Z 1
[I Ī]⇠ 2 = exp

!!!!

2

R
e
✏R
g2
Figure 11. Defining left (right) bion-anti-bion amplitude [Bij B ij ] =0 , we0✓◆✓tion of left (right) Borel resummation
B0, =0 .◆

2
2

⇡
+ log
+ O(1/✏) +TheO(✏)
! domination+ log⌥bei⇡taken care of by m
⇥ 22 ) canshort-distance
of
I(g
2
g
g
2
2
by rotatingg ⇧ g , whereinteraĪI][I Ī]  constantwhere γ contour,is the[orEulerandtheξ bion-anti-bion
is an instant±±[ĪI] =[I Ī] + [ĪI] = 2⇠





!!!!

E

1,1



!!!!

!!!!

Sinegordon

and continuing the integral back to positive |g 2 |+i0± . The result, as
is two-fold ambiguous:

2

✓

2

◆

$

[Bij B ij ] =0± = Re [Bij B ij ] + i Im [Bij−S
B ij ] I=0± ⌅ e 4S2iξ≡e/ πg 0 ±=+ log 2 ± i⇡
N 1
gConsider a typical observable in CP theory with Nf ⇤ 1 fermio
observable will receive contributions to all orders in perturbation
perturbative contributions. Denote the lateral Borel summation
2 for
B0, =0± . Then write g 2 = |g 2 |ei , where ⇥ is the phase of the complex
to make sense,
ambiguities must cancel:
2 these two2iθ

!!!!

!!!!

as −g2 = e∓iπ g2 . と解析接続
3.

Here we have neglected terms of order O(g ) or

摂動的ボレル和の不定虚部を見事に相殺！

amplitudeby ξ ande anti-instanton. Preciselyamplitude
speaking,th
FIG. 2: A schematic figure of an example
of one-instantonand one
([I
Ī],
[Ī
Im B0, =0± + Im [BB] =0± = 0 , up to e 6S

horizontal lineandstandsthe vacuum inthe sine-Gordon
potential.
of an example Eachof one-instanton
one foranti-instanton
amplitude([I Ī], [ĪI]).

複素化理論の固定点としてのBion配位

これらの非摂動寄与がなぜ重要？

Behtash, Dunne, Schafer, Sulejmanpasic, Unsal(15)Fujimori, Kamata, TM, Nitta, Sakai (16)(17)・Bion配位は複素化された理論のHolomorphic EOMの古典解・複素解のOne-loop det + 準モジュライ積分 がBZJ処方と等価

Bion
トンネル現象xI ( ) = tanheSI

行き帰り遷移02

zI Ī (⌧ ) =



zT
!⌧0
!(⌧ + ⌧0 )
coth
tanh
2
2
2

e

2SI

(Re ± iIm)

tanh

!(⌧

⌧0 )
2

zT , ⌧ 0 2 C

2
2 ±i✏
g
!
g
e
結果の特異性(ストークス線)の避け方に依存して不定虚部が出現し，これが摂動ボレル和の不定虚部を相殺！

これらの非摂動寄与がなぜ重要？

Behtash, Dunne, Schafer, Sulejmanpasic, Unsal(15)Fujimori, Kamata, TM, Nitta, Sakai (16)(17)・Bion配位は複素化された理論のHolomorphic EOMの古典解・複素解のOne-loop det + 準モジュライ積分 がBZJ処方と等価

Bion
トンネル現象xI ( ) = tanh

e

SI

0

2



zT
!⌧0
!(⌧ + ⌧0 )!(⌧ ⌧0 )FIG. 2: Complex
zI Ī (⌧ ) FIG.
= 1: Realcoth
tanh
tanh
and complex solutions in the inverted tilted double well
2 The inverted2potential (on the real 2axis) is shown in black, ics2 with a double
potential.
the real bounce and associated critical and turning points are shownin red, and the pair of2S
complexand critical points
I bions and turningcr
are blue. The blue points correspond to z1 and zT , z⇤T in (6). Note
Tthat the motion takes place in the real and imaginary parts of thecomplex potential, as explained in the text.

real and imaginar
acteristic size of t
of0pg the two tunn

where we have used the Cauchy-Riemann equations ∂xVr =∂yVi , and ∂yVr = ∂xVi . An important aspect of (2) is that it
does not describe an ordinary two-dimensional classical mechanical system: the holomorphic classical mechanics is not“tilt”. The groun
ders in perturba
metry is sponta
is non-zero ande(Re ± iIm)z ,⌧ 2C

Double well po
2
2 ±i✏
g
!
g
e
結果の特異性(ストークス線)の避け方に依存して V (x) is an asym

不定虚部が出現し，これが摂動ボレル和の不定虚部を相殺！

どうして複素古典解が必要か？
0次元積分における最急降下法では積分径路を変形し複素固定点に繋がる径路(thimble)に分解
経路積分においても複素固定点を考えるのは自然· エアリー積分
Ai(g

2

)=
⇡

Z 1
r

1

Re[e

d exp

g
exp
4⇡

✓



i

✓

2
3g 2

3

3
◆

+

g2

◆

i( 3 /3+ /g 2 )1/g 2 = i/2
1/g 2 = 1]

どうして複素古典解が必要か？
最急降下法(Thimble分解)における複素固定点の寄与· エアリー積分Ai(g 2 ) =

Z 1

1

d exp



i

✓

3

3

複素平面上の2つの複素固定点

+

g2

◆

i
=±
g

i
g

i
g

arg[g 2 ] = 0+どうして複素古典解が必要か？
最急降下法(Thimble分解)における複素固定点の寄与· エアリー積分Ai(g 2 ) =

Z 1

d exp

1



i

✓

3

3

複素平面上の2つの複素固定点

+

g2

◆

i
=±
g

最急降下法：元の積分径路を，固定点を通り，虚部一定の最急降下径路に分解

C=

X

n J

i
g

i
g

最急降下径路分解= Thimble分解

arg[g 2 ] = 0+どうして複素古典解が必要か？
最急降下法(Thimble分解)における複素固定点の寄与· エアリー積分Ai(g 2 ) =

Z 1

d exp

1

・J



i

✓

3

3

+

g2

◆

Im[S] = Im[S0 ]Re[S]  Re[S0 ] 最急降下径路

・ n = hK , Ci
C=

最急上昇径路Kと

i
g

i
g

J

元の径路との交叉数Xn J

arg[g 2 ] = 0+どうして複素古典解が必要か？
最急降下法(Thimble分解)における複素固定点の寄与· エアリー積分J+

arg[g 2 ] = 0+n+ = hK+ , Ci = 0

K+

n = hK , Ci = 1
C=

X

n J

2⇡
arg[g ] =
3
2

C=J
J

までこの分解を維持K

arg[g 2 ] = 0+どうして複素古典解が必要か？
最急降下法(Thimble分解)における複素固定点の寄与· エアリー積分2⇡
arg[g ] =+32n+ = hK+ , Ci = 1

n = hK , Ci = 1
C=

X

n J

C = J+ + J

ストークス現象：特定のarg[g^2]で

K+

K

J+

J

thimble分解が不連続変化

2⇡
arg[g ] =+3
2

どうして複素古典解が必要か？
最急降下法(Thimble分解)における複素固定点の寄与
ストークス現象· エアリー積分arg[g 2 ] =

2⇡
!⇡
3

ストークス現象：特定のarg[g^2]でthimble分解が不連続変化
2⇡
2⇡
2
2
arg[g ] =arg[g ] =
+
3
3

C=J

C = J+ + J

摂動ボレル和の不定性はストークス線上
でのthimble分解の不定性に対応！
cf.) Real-time formalism, Finite-density lattice systemどうして複素古典解が必要か？
· 変形ベッセル積分 ⇡1Z( ) = p

Z 2

dz e

1
2

sin2 z

⇡
2

⇡e
= p

1
4

I0

✓

1
4

◆

z0 = 0
固定点：
z1 = ±⇡/2arg[ ] = 0

C = J0 (0 ) + J1 (0 )arg[λ]=0でのarg[ ] = 0+

C = J0 (0+ )ストークス現象J1 (0+ )・各Thimbleがトランス級数の各セクター（摂動+非摂動）に対応
・摂動寄与 J0 が虚部を持ち， arg[λ]=0± で不定になる

・適切にThimbleの寄与を加えることで不定性のない結果が得られる

Bionトランス級数としての厳密結果

(1)

=
t+
ive
ted
ϵ is
mes
ber
ert
=1
ubain
−V

厳密結果は複素解トランス級数で完全再現
Ex.) CPN量子力学の基底エネルギー⇣ ⌘E (2) = g 2mcoth gm2

sinh2 gm2

2
4

Ei

2m
g2

+ Ei
2

⇣

2m
g2

⌘

2

3

Fujimori, Kamata, TM,
Nitta, Sakai (16)(17)

1
X

2m 5
=
e
log 2
g
p=0

2pm
g2

Ep(2)

トランス級数厳密結果
can be Borel-resummed. Hence we rewrite thecoeﬃcientin the form of the Borel transform (See Appendix. A forthe details of calculations.) as
& ∞
e−t
(2)
2
E0 = g + 2m
dt
(7)
摂動的寄与
2m ,
t
−
2
0
g ±i0
'
(
"
#
✓
◆
Z &1∞
21)2
2 1)2
(p
+
(p
−
2m i⇡
(2)
E
2m dtedt et−t (p + 1) 2m+ (p+ 1) 2m+ 4mp2
Epp(2) ==2m+log±
非摂動的寄与
2m
2m
2
t
−
t
+
g
2
2
2
0
t
t
+
2
2
g
±i0
g
0
g ±i0
g
)
*
2m πi
2
複素解背景の摂動寄与+ 4mp γ + log 2 ±, (p ≥ 1). 複素解の半古典寄与(8)g2Note that the imaginary ambiguities
associated to the' ⇡ ei 0 cs[⌦(⌧ ⌧0 ), k]
(2)
Borel
resummationismanifest
in
the
first
term
of
E
複素p-bion解pi0
˜⇡ecs[⌦(⌧), k]with g 2 ±i0, which is 'compensatedby the⌧0imaginarypart(2)±iπ/2 in the last term of Ep+1 , reproducing the original
real E (2) precisely.
トランス級数内の各セクターの不定虚部は順次相殺され厳密結果に！
We
can now recognize the full resurgence structure to
all orders of nonperturbative exponential: imaginary am-

量子論のリサージェンス理論とトランス級数tUVrenormalons:t = !8!n/"0

1

1

CP(N!1) on R x S

Schiappa, Marino, Aniceto(08) Argyres, Dunne,Unsal(12) Our group(13)Instanton!!anti!instanton
singularities: t = 8! , 16 ! , ...

Neutral bions:t = 8 ! n /# (n=1,2,...)

・全ての複素古典解寄与の和では，不定虚部は相殺され厳密結果を得る
Upper
figure: The conjectured structure of the Borel plane for CPN 1 on R2 . Lower
emi-classical singularities associated with the neutral bion molecules in CPN 1 on small
0 = Im B[0,0]regime+ B[2,0][I Ī] + B[4,0][II Ī Ī] + ... 不定虚部の相殺
Nf = 0, the weak-coupling
has an extra
singularity closer toorigin than themalon pole on R2 . For Nf ⇥ 1, the location of the semi-classical and non-semi-classical
・複素古典解とトランス級数に基づく展開で非摂動的定式化が可能！？
ingularities
coincide. Although the theory moves from a weakly coupled description to
1
1
1
X
X
X
upled one, the structure
of
the
Borel
plane
singularities
eitherdo not change at all or2
2qnA/g 2
2q
F (g ) ⇡c(0,q) g +ec(n,k) g
mely mildly. We take this as evidence that the neutral bion molecules are トランス級数
the semi-classical
q=0
n=1
q=0
renormalons. This also gives us hope that even the theory on R2 may potentially be
bitrary N .

注.）異なるトポロジカル数毎にリサージェンス構造が存在that arise in the Borel summation of the perturbation theory cancel with the

量子論での各種の例• 1D Double-well, Sine-Gordon, CPN 模型
Betahsh,et,al(15) Fujimori, Kamata, TM, Nitta, Sakai(16)(17)• 2D CPN シグマ模型 with compactification
Argyres, Dunne, Unsal(13) TM, Nitta, Sakai(14)• 3D Chern-Simons理論 with exact results
Gukov, Marino, Putrov(16) Honda, Fujimori, TM,et.al.(17)• 4D N=2 超対称ゲージ理論 on S^4
Schiappa, Marino, Aniceto(13) Honda(16)• 行列模型 & 位相的弦理論

Marino(07) Marino, Schiappa, Weiss(09), Hatsuda, Dorigoni(15)ボレル平面上の特異点が複素古典解に対応．
そこからの寄与の和が厳密結果を与える!

具体例１：CPN sigma model[Fujimori, Kamata, TM, Nitta, Sakai(16)(17)]dimensional reduction· 1d limit : QM of a particle on sphere

CP1シグマ模型
potential with two minima (due to twisted b.c.)

instanton
・CP1模型 on R1· fractionalx S130

1 |@µ '|2
kink
L= 2
+
LF
2
2
g (1 + |'| ) (tunneling)
asymptotic-free theory

complex bion solution

・Twisted boundary conditions!

complex φ-plane!

'(y + L) = eimL '(y)FIG. 9: Fractional instanton configuration on S 2 in the reduced quantum mechanics is depicted. It corre-(m=π/L : Z2 t.b.c.)sponds to a single line from the north to the south pole which is rotated over the half of S 2 homogeneously.

The figure depicts the rotation of the line around the half sphere. The red arrows denote paths dependingon x2 with a constant x1 , while the blue arrows denote the x1 dependence of such paths.

→ BPS Fractional instantonscf.) m=π/L

Lee,Yi(97)Lee, Lu(97)Kraan, van Baal(97)Eto, et.al. (04)Bruckmann (05)

!

π1

+1/2

−1/2

+1/2

−1/2

π2

+1/2

−1/2

−1/2

+1/2

of freedom. If we
ϵ = 1, this model becomesa supersymmetricsystem whi
n this subsection, we use Lorentzian signature instead
ofsetEuclidean
signaturein orderfrom the 2d N = (2, 0) CP 1 sigma model by an analogous dimensional rediscussed insimply,
the previous subsection.
o Schrödinger equation later. To denote 1d quantities
we rewrite without subscrip

Small-L での有効理論

Since the fermion number ψ̄ψ commutes with the Hamiltonian, we can el(0)

the conserved fermion number and the associated induced potential. By projects the Lorentzian time t, ϕk=1(0) → ϕ, g1d → g, G11̄ → G etc. The Lagrangian of the CP
onto the subspace of the Hilbert space with a fixed fermion number, we obtain

Lagrangian (see Appendix A for details)
zian quantum mechanics withform
✏ =a 1fermion takes thebosonic

・CP1量子力学（：超対称✏ = n 2 N1 ：可解）∂ ϕ∂ ϕ̄
L =

!

t

t

− V (ϕϕ̄),
2

V (ϕϕ̄) ≡1 m2 ϕϕ̄
1−
− ϵm
2
2
g (1 + ϕϕ̄)1+"
1
we have chosen the fermion number so that the supersymmetric grou
L = 2 G ∂t ϕ∂t ϕ̄ − m2 ϕϕ̄ + iψ̄Dt ψ + whereϵm(1+ ϕ ∂ϕ log G)ψ̄ψ ,(II.2
is contained in the subspace of the Hilbert space. The potential V as a funcgg 2 (1 + ϕϕ̄)θ ≡ 2arctan|ϕ| is shown in Fig. 3.

G is theFubini-Study metric and Dt is the pullback of the covariant derivative・フェルミオン数射影よる基底状態実効理論

[H,

!
"
1 ¯| i = 0,
Dt ψ = ∂t + ∂t ϕ ∂ϕ log G ψ.
2
(1 + ϕϕ̄)22✏gmreduction
dimensionalm2¯] = 0
G=

V =

4

sin ✓

(II.2

✏mg cos ✓2✏g m on sphere· 1d limit : QM of a particle
!

#
"

!

✓ = 2 arctan |'|potential with two minima (due to twisted b.c.)・2つのlocal minimaを持つポテンシャル
Fig. 3: The potential V with the contribution of the fermion. The horizontal axis
北極と南極に対応

・Kink解m
SI = 2gθ ≡ 2arctan|ϕ|on CP 1 ∼= S2.
· fractional
instantonForϵ = 1, the ground state wave function Ψ0 , which preserves the supersy
kinkthezero energy solution of the Schrödinger equation(tunneling)HΨ0 = 0,bion
solutionwith theHamiltonian
H of the bosonic theory:
2つの極小点間のトンネル効果を表すcomplex
complex φ-plane
H = −g 2 (1 + ϕϕ̄)2∂ ∂+ V (ϕϕ̄).
∂ϕ ∂ ϕ̄

厳密結果

厳密な基底エネルギー in CP1[Fujimori, Kamata, TM, Nitta, Sakai(16)(17)]・基底状態エネルギー

E = E
E

E

(1)

(2)

=g

=g

2

2

(1)

✏=✏

n

(2)

2

✏+E

m
m coth 2 =g2coth gm2 Ei
4m
sinh2 gm2

m+g

3

✏ + O( ✏ )
1
X

2

2me

2pm
g2

p=0

⇣

2m
g2

⌘

+ Ei
2

⇣

2m
g2

⌘

3

2m 5
log 2
g

摂動寄与と非摂動寄与が含まれるはず厳密な基底エネルギー in CP1
E

(2)

=g

2

coth gm2 Ei
4m
sinh2 gm2

2

・摂動寄与
(2)
E0 ⇡ g 2

2m

1
X

(n

n=1

1)!

✓

2

g
2m

⇣

2m
g2

⌘

◆n

+ Ei
2

⇣

2m
g2

⌘

3

2m 5
log 2
g

ボレル和を実行 arg[g^2] = ±0・固定点寄与
(2)
Enp
0 ⇡

2m

1
X
p=1

e

2mp
g2

"

(p + 1)

2

1
X

n=1

(n

1)!

✓

ボレル和を実行 arg[g^2] = ±0

2

g
2m

◆n

+ (p

1
X

✓

2

◆n

g
1)
(n 1)!
2m
n=1
✓
◆#
2m
2
2p
+ log 2
g
2

1)

=
+
ve
d
is
es
er
rt
1
bn
V

Then, we can define the perturbative sum for the non-alternating series as the analytic ctinuation of B0 (g 2 ) in the g 2 complex plane from negative coupling, g 2 < 0, to the posit
real axis, g 2 > 0. This can be done in one of the two ways as shown in Fig. 2. Approach
the positive real axis clock-wise (from above) and counter-clock-wise (from below).

厳密な基底エネルギー in CP1
B0 (|g 2 | ± i ) = Re B0 (|g 2 |) ± i Im B0 (|g 2 |)where Im B0 (|g 2 |) ⇤ e 2SI ⇤ e 2A/g

2

(6

2
ambiguous part,of non-Borel-summability⇣ is the ⌘⇣ and is a manifestation⌘
3 [compare with (1.22mA definition of the Borel
2msum equivalent to what we described above through analy
m
Ei
+
Ei
coth g2
continuationin the complexgg2 -plane is the directional (sectorial)
Borel sum. Define
g2
2m
(2)
2
4
5
E
=
g
m
log
1
m
2
can be Borel-resummed.
werewritetheS P (gcoeﬃcient) ⇥ B (g ) =BP (t)edt,(62gsinh2Hence
2
g
g
2

2

2

⌅·ei

2

2

t/g 2

0

in the form of the Borel transform (See Appendix. A forthedetails of calculations.) as
・摂動寄与
t
& ∞
−t
e
C+
(2)
2
E0 = g + 2m
dt(7)ボレル平面上の2m ,t − g2 ±i00C
正実軸上の特異点
'
(
& ∞
2
2
(p
+
1)
(p
−
1)
Ep(2) = 2m
dt e−tFigureLateral, or right2m
and left, Borel sums. Dark circles are singularities (poles or bra
2m 9. +
t−t+
Whenever a singularity exists
2 ±i0
2 between the right and left Borel sums, the theory is non-Bo
0
gpoints).
gsummable. The singular direction in the t-plane corresponds to a Stokes line in the complex g -pla*
・固定点寄与 )see Fig.2. The difference of the sectorial sums in passing from ⇥ = 0 to ⇥ = 0 is the Stokes “jum2m across
πia Stokes ray.
2
# 1).
+ 4mp
(p2≥
(8)
✓
◆ w
Z 1 γ +"log 2 2± A special, case
of
this
is
the
lateral
Borel
sum.
The
functionB(g)
isassociated2(pjust above(p +g 1) contoursi⇡Borel resu
1) and just below the ray
2mright (left)(2)t
2 at angle ⇥, and is called
2

+

±

2

+ If there2m
+ 4mp
+ log
±
2m mation.
are no singular points in the ⇥ direction, then the2left and right Borel su
g2t+
the 0imaginaryt ambiguities
associatedandtheBorel thesummable in the ⇥-direction. A theory for wh
g 2 ±i0 are equal,g 2theory is sectorialto

Ep = 2mdte

Note that
there are no singularities on ⇥ = 0 is called Borel(2)summable in physics. In many cases, tha ray of first
singular pointsof the Borel
Borel resummation is manifest in is thetermof transformEp BP (t), as shown in Figure 9. Then,theory is non-Borel summable, but left and right Borel summable. The ambiguity describ
2with g ±i0, which is compensatedabove,
byassociatedthe imaginary
partwith whether we approach
the real positive axis from above or below
the complex g -plane, in the latter language, maps to the choice of the integration cont(2)in the Laplace-transform.theThe integral
is, of course, dependent on the choice of the conto±iπ/2 in the last term of Ep+1 , reproducing
original
2

1)

=
+
ve
d
is
es
er
rt
1
bn
V

Then, we can define the perturbative sum for the non-alternating series as the analytic ctinuation of B0 (g 2 ) in the g 2 complex plane from negative coupling, g 2 < 0, to the posit
real axis, g 2 > 0. This can be done in one of the two ways as shown in Fig. 2. Approach
the positive real axis clock-wise (from above) and counter-clock-wise (from below).

厳密な基底エネルギー in CP1
B0 (|g 2 | ± i ) = Re B0 (|g 2 |) ± i Im B0 (|g 2 |)where Im B0 (|g 2 |) ⇤ e 2SI ⇤ e 2A/g

2

(6

2
ambiguous part,of non-Borel-summability⇣ is the ⌘⇣ and is a manifestation⌘
3 [compare with (1.22mA definition of the Borel
2msum equivalent to what we described above through analy
m
Ei
+
Ei
coth g2
continuationin the complexgg2 -plane is the directional (sectorial)
Borel sum. Define
g2
2m
(2)
2
4
5
E
=
g
m
log
1
m
2
can be Borel-resummed.
werewritetheS P (gcoeﬃcient) ⇥ B (g ) =BP (t)edt,(62gsinh2Hence
2
g
g
2

2

2

⌅·ei

2

2

t/g 2

0

in the form of the Borel transform (See Appendix. A forthedetails of calculations.) as
・摂動寄与
t
& ∞
−t
e
C+
(2)
2
E0 = g + 2m
dt(7)ボレル平面上の2m ,t − g2 ±i00C
正実軸上の特異点
'
(
& ∞
2
2
0-bion背景での摂動的寄与
(p
+
1)
(p
−
1)
Ep(2) = 2m
dt e−tFigureLateral, or right2m
and left, Borel sums. Dark circles are singularities (poles or bra
2m 9. +
t−t+
Whenever a singularity exists
2 ±i0
2 between the right and left Borel sums, the theory is non-Bo
0
gpoints).
gsummable. The singular direction in the t-plane corresponds to a Stokes line in the complex g -pla*
・固定点寄与 )see Fig.2. The difference of the sectorial sums in passing from ⇥ = 0 to ⇥ = 0 is the Stokes “jum2m across
πia Stokes ray.
2
# 1).
+ 4mp
(p2≥
(8)
✓
◆ w
Z 1 γ +"log 2 2± A special, case
of
this
is
the
lateral
Borel
sum.
The
functionB(g)
isassociated2(pjust above(p +g 1) contoursi⇡Borel resu
1) and just below the ray
2mright (left)(2)t
2 at angle ⇥, and is called
2

+

±

2

+ If there2m
+ 4mp
+ log
±
2m mation.
are no singular points in the ⇥ direction, then the2left and right Borel su
g2t+
the 0imaginaryt ambiguities
associatedandtheBorel thesummable in the ⇥-direction. A theory for wh
g 2 ±i0 are equal,g 2theory is sectorialto

Ep = 2mdte

Note that
there are no singularities on ⇥ = 0 is called Borel(2)summable in physics. In many cases, tha ray of first
singular pointsof the Borel
Borel resummation is manifest in is thetermof transformEp BP (t), as shown in Figure 9. Then,theory is non-Borel summable, p-bionの半古典的寄与
but left and right Borel summable. The ambiguity describ
p-bion背景での摂動的寄与
2with g ±i0, which is compensatedabove,
byassociatedthe imaginary
partwith whether we approach
the real positive axis from above or below
the complex g -plane, in the latter language, maps to the choice of the integration cont(2)in the Laplace-transform.theThe integral
is, of course, dependent on the choice of the conto±iπ/2 in the last term of Ep+1 , reproducing
original
2

1)

=
+
ve
d
is
es
er
rt
1
bn
V

厳密な基底エネルギー in CP1
2

⇣

⌘

⇣

⌘

2

3

2m
2m
m
Ei
+
Ei
coth g2
g2
g2
2m 5
(2)
2
4
E =g
m
log 2
2Hencemcan be Borel-resummed.
werewritethecoeﬃcient2gsinh g2

in the form of the Borel transform (See Appendix. A forthedetails of calculations.) as
・摂動寄与
& ∞
−t
e
(2)
E0 = g 2 + 2m
dt
(7)
⌥2mi⇡
2m ,
t − g2 ±i0
0
'
(
& ∞
2
2
(p
+
1)
(p
−
1)
摂動的寄与の不定虚部は，1-bionの半古典的寄与の不定虚部と相殺
Ep(2) = 2m
dt e−t
2m +
2m
t
−
t
+
0
g 2 ±i0
g2
p=1
bion
)
*
・固定点寄与
2m πi
2
# 1).
+ 4mp
, (p2≥✓(8)◆
Z 1 γ +"log 2 2±2(p 1)(p +g 1)
2m i⇡
(2)
t
2
Ep = 2m

dte

+ 4mp
2m +
2m
t +associatedthe 0imaginaryt ambiguities
tog 2 ±i0g2+ log

Note thatthe(2)
Borel resummation is manifest in the first term of Ep
with g 2 ±i0, which is compensated by the imaginary part(2)
±iπ/2 in the last term of Ep+1 , reproducing the original

g2

±

2

1)

=
+
ve
d
is
es
er
rt
1
bn
V

厳密な基底エネルギー in CP1
2

⇣

⌘

⇣

⌘

2

3

2m
2m
m
Ei
+
Ei
coth g2
g2
g2
2m 5
(2)
2
4
E =g
m
log 2
2Hencemcan be Borel-resummed.
werewritethecoeﬃcient2gsinh g2

in the form of the Borel transform (See Appendix. A forthedetails of calculations.) as
・摂動寄与
& ∞
−t
e
(2)
E0 = g 2 + 2m
dt
(7)
2m ,
t − g2 ±i0
0
'
(
& ∞
2
2
(p
+
1)
(p
−
1)
Ep(2) = 2m
dt e−t
2m +
2m
t
−
t
+
2 ±i0
2
0
g
g
(p-1)-bion
相殺p-bion*・固定点寄与 )
2m πi
2
# 1).
+ 4mp
, (p2≥✓(8)
Z 1 γ +"log 2 2±2(p 1)(p +g 1)
2m
(2)
t
2

i⇡
Ep = 2mdte+ 4mp+ log 2 ±
2m +
2m
g
2
t +associatedNote that the 0imaginaryt ambiguities
to theg 2 ±i0
g2
(2)
Borel
resummationismanifest
in
the
first
term
of
E
p
(p-1)-bion背景での摂動的寄与の不定虚部は，p-bionの半古典的寄与2with g ±i0, which is compensated by the imaginary part(2)の不定虚部と相殺！
±iπ/2 in the last term of Ep+1 , reproducing the original

◆

1)

=
+
ve
d
is
es
er
rt
1
bn
V

For this case, the eﬀective potential between the BPS
components in bion configuration is modified as
Veﬀ ≈

2mi
4mi −mi τr
′
−
e
cos
φ
+
2ϵ
mi τ r ,r22gg

Schrödinger equation reduces to the recursion recalled Bender-Wu recursion relation,
%)
4 $
#
4
0 =
(k − i + 1)2 Bl−i,k−i+1i

厳密な基底エネルギー in CP1
(.16)

i=0

− (2k − 2i + 1)Bwith ϵ′ = 1 + 12 (ϵ − 1)N . By performing the Lefschetz
l−i,k−i + Bl−i,k−i−1
2
⇣
⌘
⇣
⌘
2 on this3
thimble integral (quasi moduli integral) based
2m
2m
l
m
#
Ei
+
Ei
eﬀective potential, we derive thecothcontributions
fromg 2N
2
2m 5
g2 4
+ g Ai (Bl−i,k + 2Bl−i−1,k−1(2)2
real and complex
E bions= gto themground state energylog*

2Hencei=1m2
can be Borel-resummed.
werewritethecoeﬃcient
2
g
sinh
2
g
$ 2 %2(ϵ −1)N−1+ Bl−i−2,k−2 ) − Bl,k−1 + ϵ(Bl,k − Bl−2,k−2′#gthe Borel
sin ϵtransform
π
in
the
form
of
(See
Appendix.
A
for
2
−
′
Ebion = −2mi
Γ (ϵ ) e2mπ
where Bl,k = 0 if l < 0, k < 0, k >(
2l.
i
the
detailsof
calculations.)
as
i=1
∞
・摂動寄与
& πiϵ
We now obtain Al in Epert = m l=0 Al η 2l . As
e& ∞ for θ = −0−t(.17)in Fig. Bender-Wu法による高次摂動係数
2 it has the asymptotic behavior
×
.
−πiϵ
e
e
for
θ
=
+0
(2)
E0 = g 2 + 2m
dt
(7)
1 Γ(l + 2(1 − ϵ))2m ,Al ∼ − l−1.
− g2 ±i0
Again, the imaginary ambiguity of the
0 bion tcontribution2
Γ(1 − ϵ)2
depends on a sign of the phase of complexified
coupling
'
(
&
′
∞
2
constant g 2 = |g 2 |eiθ . The contribution
vanishesatϵ=2 Now,
we
introduce
Borel transform and Borel r
(p
+
1)
(p
−
1)
摂動寄与と一致！
ϵ = 0, whichmation. The Borel resummation of Epert gives aEp(2)agrees= with2mthe supersymmetry.
dt e−t+2m
2m which has E
alyticfunctionpert as an asymptot
t
−
t
+
2
0
Contribution from Perturbative
vacuum :g 2We±i0 ries. Firstly,
g the Borel transform B[E ](t) of the1) To derive a perturba- * E (η2 ) = (∞ E η2l is defined as pert
・固定点寄与here
focus on the CP model.
pert
l=0 l
tive series of the ground stateenergy, we 2mredefine πi
the
2
# 1).
+the4mp
±
,
(p
≥
(8)
✓
◆
∞
Z 1 γas+ψ"logwave function andcoordinate= e−x Ψ(x),
|ϕ|
=
#
2
2
2
F
l l
2(p 1)(p +g 1)B[E
ηx, η ≡ √g (2). Then, the Hamiltoniant becomes
2 pert ](t) = 2m t , i⇡
′

2mi
g2

2

+ 4mp
+ logl=0 2l! ±
2m +
2m
g
2
t ambiguities
t +associated&0'g 2 ±i0
2
Note
that
the
imaginary
to
the
g
H̃
1
2 2 2
2
2 1
where t ∈ C is a Borel(2)parameter. Note, in the p= − (1 + η x ) ∂x + (1 − 4x ) ∂x + V (x),(.18)
Borel4 resummation is manifest
in the firstterm
Ep examples), the Borel tranmxcase (andlotsofof other
複素p-bion解の寄与と一致！
2B[Eon the real and positive apert ] has singularities
withthegpotential±i0, iswhich is compensated by theimaginary
part
where
the Borel plane of t. Now, the (lateral) Borel resu
(2)
tion is definedas±iπ/2 in2 the last
term
of
E
,
reproducing
theoriginal
2
2 2
x
1
−
η
x
p+1
V (x) = (1 − x )(1 + η 2 x2 )2 +
−ϵ
(.19)
.
±iδ
Emp = 2mdte

複素固定点(複素解)寄与
Euclidean e.o.m

SE
=
'

@V
2
@⌧ ' +@ '¯=0運動方程式の実解：Real bion解

'=

r

実解

!2

ei 0
i sinh !(⌧! 2 = m2 +⌧0 )モジュライパラメタは ⌧0 : position= ✏mg 20 : phase: kink profile

⌧+

1
4! 2
⌧ = log 2!
!
m2

理論(変数)の複素化· ' の実部と虚部を複素化' = 'R + i'I'¯ = 'R

変数の複素化

i'I ! '
˜

C
SU
(2)
SU
(2)
⇤
1
⇠
CP 1 ⇠
!
T
CP
=
=
C
U (1)
U (1)

CP1の複素化· Holomorphic actionに解析接続

S[', ']¯ ! S[', ']˜holomorphic

complex conjugate of ϕ, this is a solution of the complexified model and hence

ration “complex bion solution”.

運動方程式の複素解：Complex bion解ng that the shifted solution can also be rewritten into the kink-antikink form)−iφ+

+ e−ω(τ −τ− )−iφ−

"−1

r

!
"−1
−τ+ )+iφ+複素解ϕ̃ = eω(τ+ e−ω(τ −τ− )+iφ−, (III.23)

,

!2

ei 0
cosh !(⌧' =τ± :
position parameters
#
$
2
1
4ω
τ± = τ0 ±log 2+ πi ,
2
2ω
ω −m

'˜ =

⌧0 )

'⇤

π
φ ± = φ0 − ,2(III.24)d the fact that the shift τ0 → τ0 + ω1 πi
2 can be rewritten as the combination of

πiφ+ → ωτ+ ± iφ+ + πi
(mod
2πi)
and
ωτ
±
iφ
→
ωτ
±
iφ
+
−
−
−
−
2
2 (mod 2πi).

plex bion solution can also be viewed as a kink-antikink solution with complex

1
τr ≡ τ+ − τ− =
ω

#

log

4ω 2
ω 2 − m2$

+ πi . : “複素相対距離”(III.25)複素解のThimble積分実行
準モジュライパラメタ (Nearly massless modes)= kink間相対距離 ⌧r と相対位相rReal and complex bionsからの寄与

Z1
⇡
Z0

Z

d⌧r d r exp[ Ve↵ (⌧r ,r )]複素化されたモジュライ空間での実効ポテンシャル TM, Sakai, Nitta (14)Ve↵ (⌧r ,

r) =

4m
cos
2
g

m⌧r
e
+ 2✏m⌧rr

Lefschetz Thimble 積分法
CR =X

n = hCR , K i

n J
交叉数

J : upward flow → Thimble
K : down flow → Dual thimble(a) θ = −0
Thimbleは4次元空間上の面に対応

(⌧r ,

2
)
2
C
r

Fig. 9: Integration contour, Lefschetz thimbles and
上図は3次元射影 for ✓ = arg[g 2 ] < 0and (b) θ = +0 (seen from lower right). The orang

Lefschetz Thimble 積分法
CR =X

n = hCR , K i

n J
交叉数

J : upward flow → Thimble
K : down flow → Dual thimble
Thimbleは4次元空間上の面に対応

(⌧r ,

2
)
2
C
r

上図は3次元射影 for ✓ = arg[g 2 ] < 0(a) θ = −0(b) θ = +0

′ ∈ R, φ′ ∈ iR. We can cthimbleJ
corresponds
to
the
two
dimensionalplaneτ
σ contour, Lefschetz thimbles and dual thimbles for (a) θ = −0 (seen from upp: Integration

Lefschetz Thimble 積分法

ntial
b)
θ =satisfies+0 (seen from lower right). The orange lines are the original extended integration co! and green)"lines and surfaces indicate the thimbles and theifour colored (red, blue, yellow

2m X
ResinceV ≥the2ϵ integration
1 + log, andnLefschetz
ImZV = thimbles(σπ − θ)ϵ=direct
const.,
ctively. Note that
contourareproducts of
Zq.m.
=
2
ϵg

ion and lines in (φR , φI , τI ), their projected images are lines in the three-dimensional space.

′ ∈ R, φ′ ∈ iR. Integrating over the thimbles, we obtain· Thimble Jσ に沿った積分! 2 iθ "2ϵ
i
g e
′
′
−V
′
ur for θ = −0.ZσNote
the identical thimble
relatedby2 .the shift= thatdτK0 and
dφKe0 are =e−2πiϵσ
Γ (ϵ)
2m 2m
R
iR
#

#

2π. By taking into account how the original integration contour is decomposed in

efore, the ·bion
contribution
is
given
by
thimblenumbers
Kσ との交叉数
bles (see Fig.元の積分径路とdual
10), the sign of the intersection
can be determined as⎧⎧! 2 iθ "2ϵ⎨ eπiϵfor θ = −0
⎨ ( −1 , 1 , 0 )2 for
1 g e
θ
=
−0
[I Ī] (n=−1 , n0 , n1 ) = sin ϵπ Γ (ϵ) ×.
.
⎩ e−πiϵ for θ = +0m 2m ⎩( 0 , −1 , 1 )for θ = +0

resultthe
is consistent
with theone
bydepending
applying theBogomolny–Zinn-Justin
efore,
bion contribution
hastheobtained
ambiguity
on thesign of θ
ストークス現象
不定虚部を生み出す⎧
he divergent region τ → −∞,⎨|φ| ≤ π/2 [36]. In this calculation of the complex
Zσ=0 − Zσ=−1 for θ = −0
[I Ī] =.

phasebetween localturbative
BPScorrections
components
in the backgroundbion con∂ϕ ∂ ϕ̄oaround the multi-bion(2)
explicitly.
Moreover,if
weobservethatE/m quasiis an
figurations.
These
two
parameters
correspondt-s Ψ0 = ⟨ϕ|0⟩ = exp(−µ/g2 ). For ϵ ≈ even function of m/g2 , we can also understandto
the prescorrection
to the groundstate wave
moduliparameters,
which
are
nearlymassless
modes
ence
of
Borel-summablepart(second
term
of
the
first
aained by expanding the Schrödinger
line in Eq.(8)). Thus all the terms can now be repro複素固定点からの寄与
aroundcomplex bion solutions. We thenect to smallδϵ ≡ ϵ −the1 as real⟨ϕ|δΨ⟩. andduced through resurgence relation and the sign changee ground state energy E can also beof m/g 2 to
, if wecan ground-statecompute all the classicalp-bion conobtain the contribution
theenergyfromtributions.
thebion
solutions
基底エネルギーへの寄与(1)
2 (2)real and complex
Multi-bion
solutions : Nonperturbative contribuE
+
δϵ
E
+
·
·
·
.
(4)
7)
tions to*the ground state energy+ come from the saddle)2(ϵ−1)−S−βE
2 of thepointspath
integral
Z
=
DϕD
ϕ̃
e
∼
e
eﬃcients can be determined by the2mglarge β), where sin
ϵπcomplexified
2 −
(for
we
have
theg 2degrees ofSchrödinger perturbation
theory
as
E
=
−2m
Γ
(ϵ)
e
bion
C
C
C
C
freedombyregarding
ϕ
≡
ϕ
+
iϕ
and
ϕ̃
≡
ϕ
−
iϕ
0⟩, E (2) = −⟨δΨ|Hϵ=1|δΨ⟩/⟨0|0⟩, · · ·
R
I
R
I
2m
π
xas independent
holomorphic variables, and imposed theHϵ=1 . We find that these coeﬃcients+
periodic boundaryπiϵcondition ϕ(τ + β) = ϕ(τ ) and for
ambiguities and can be+ β = −0
etelyimaginary
e
for
θ
2
convergent power series with reϕ̃. The×Euclidean action SE = 0 dτ [∂τ ϕ∂τ ϕ̃/(g(1(.13)+.
−πiϵ2turbative exponential exp(−2m/g )e has two conservedfor θ Noether= +0charges asϕϕ̃)2 ) + V (ϕϕ̃)]Esociated with the complexification of the Euclidean time
✓
◆
(i)
2
translation
τ
→
τ
+
a
and
the
phase rotation (ϕ, ϕ̃) →Ep exp(−2pm/g ),
2m(5)
2m
2m
i⇡
ib
−ib
2
(e
ϕ,
e
ϕ̃)
(a,
b
∈
C).
Using
the
corresponding
con- 3g2=0= 2me g2 ✏ + servation
4m
+
log
±
e
✏
+
O(
✏
)
2 obtainlaws, we gcanthe following solution of2(i)the equation of motion with nontrivial contribution in aerm E0 corresponds to the perturβ → ∞ limit,
on the trivial vacuum (perturbative∞%cients of E (1) [21] areg2 ,Ep(1) = −2m, (p ≥ 1).
(2)

厳密結果の p=1
部分と完全に一致
iφ f (τ − τc )−iφ f (τ − τc )
ϕ
Z =e

(9)
✓ ,
◆
sin
α
sin
α
(p + 1)
(p 1)
2m i⇡
(2)
t
2
E
=
2m
dte
+
+
4mp
+
log
±
(6) p
2m
2m
2
g
2
where (τt + g2 parameters associated0 c , φc ) aret complex
g 2 ±i0 moduli
12E
are expanded in powers of g ,divergent asymptotic series, which

c

"

2

,

2

ϕ̃ = e #c22

with the symmetry and f (τ ) is the elliptic functionf (τ ) = cs(Ωτ, k) ≡ cn(Ωτ, k)/sn(Ωτ, k),(10)sociatedwiththeof theNoether
Euclideantime asϕϕ̃) ) +V (ϕϕ̃)]complexification
has two conserved
chargestranslation
τ → theτ +complexificationa and the phase ofrotation(ϕ, ϕ̃) →timesociatedwiththeEuclideanib(etranslation
ϕ, e−ib ϕ̃) (a,b ∈τ C).
thephasecorrespondingconτ →+ a Usingand therotation (ϕ,ϕ̃) →servationlaws,
we bcanthe following
solution of con(eib ϕ, e−ibϕ̃) (a,∈ obtain
C). Using
the corresponding
the equation of motion with nontrivial contribution in a
servation laws, we can obtain the following solution of
β → ∞ limit,[Fujimori, Kamata, TM, Nitta, Sakai(16)(17)]Multi-bion 解

the equation of motion with nontrivial contribution in af (τ − τc )Multi-bion解β →ϕ∞limit,
iφc f (τ − τc )
−iφ
c
=e
,
ϕ̃ = e
,
(9)
2
2
sin
α− τ )
sinf (τα − τ )f(τ• 整数(p,q)で分類される
c
c
iφc
−iφc
,moduliϕ̃ =parameterse,(9)where (τϕc ,=φce) are complex
associated
2
2
sin α
sin α
• pはbion数
with the symmetry and f (τ ) is the elliptic function
where (τc , φc ) are complex moduli parameters associated• qは解の形状を規定
f(τ
)=
cs(Ωτ,k)≡cn(Ωτ,k)/sn(Ωτ,k),(10)
with the symmetry and f (τ ) is the elliptic function
Fig. 1:k) Multi-bion
solution: k),kink profilef (τ ) = cs(Ωτ,≡ cn(Ωτ, k)/sn(Ωτ,(10) of Σ(τ ) = (1 −ϕϕ̃)/(1 + ϕϕ̃) for (p, q) = (3, 1), ϵ = 1, m = 1, g = 1/200,
β = 100 and τc = 0. Σ = ±1 (dashed lines) correspond to1
north and south poles (global and local S
minima)
of
CP
.
⇡ pS
+ 2⇡i✏l
bion

#
%
!

"

Fig
(2,
ram
θ=

2m
!+m
Sbion = 2 + 2✏ log! mthewhich satisfies the diﬀerential equation (∂ f )2g = Ω2 (f 2 +τ1)(f 2 + 1 − k 2 ). Solutions are characterized by two integers (p, q) for the period√β = (2pK + 4iqK ′ )/Ω, with(p, q)解は無限個のmulti-bion解のタワーを構成！
2K(k) and 4iK ′ (K ′ ≡ K( 1 − k 2 )) as the period of the
doubly periodic function cs. The parameters (α, Ω, k)
are given in terms of the period β, and their asymptotic

from
par
Th
red
spa

×

p
2 multi-bion backgroundeeturbativecorrections
around
the
4
2
(2)
g
g
explicitly.
Moreover,if
weobservethatE/m is an2s Ψ0 = ⟨ϕ|0⟩ = exp(−µ/g ). For ϵ ≈even function of m/g 2 , we can also$
understand the pres'('(pcorrection to the ground state wave
ence of Borel-summablepartiσ(second term of the first
iσ
ϵ− 2
Γ 1−
ained by expanding the Schrödinger Γline in Eq.(8)).
Thus all the 2terms can now be repro('(
× 'duced through.
(C21)
ect to small δϵ ≡ ϵ − 1 as ⟨ϕ|δΨ⟩.
iσ resurgence relationiσ and the sign change
1 m/g− ϵ2 ,+
Γ 1+e ground state energy E can also beΓ ofif we2 can compute all2 the classical p-bion contributions.
基底エネルギーへの寄与(2)Multi-bion solutions : Nonperturbative contribuE (1) + δϵ2 E
+
·
·
·
.
(4)
The leading order tions
termto Eq.
(17) isthe ground state energy+ come from the saddle
points of the path integral Z = DϕDϕ̃ e−SE ∼ e−βE
eﬃcients can be determined by the# (for large β), where we have%complexified$p of&2(1−ϵ)the degrees
Schrödinger perturbation theory as
C2m C
C
C
Zp|δΨ⟩/⟨0|0⟩,1· · · 2mβΓ
freedom(ϵ)
by regarding
ϕ
≡
ϕ
+
iϕ
and
ϕ̃
≡
ϕ
−
iϕ
0⟩, E (2) = −⟨δΨ|Hϵ=1
− 2m
∓πiϵ
R
I
R
I
2
g
≈
e holomorphic variables,. theas independentand imposed
Hϵ=1 . We find that these coeﬃcients2
Z0 and can
p!be Γ periodic(1 − ϵ)boundary condition ϕ(τg + β) = ϕ(τ ) and fort imaginary ambiguities+β
ely convergent power series with reϕ̃. The Euclidean action SE = 0 dτ [∂τ ϕ∂τ ϕ̃/(g 2 (1 +turbative exponential exp(−2m/g 2 )(C22)ϕϕ̃)2 ) + V (ϕϕ̃)] has two conserved Noether charges
as✓◆sociated with the complexification of the∞2m
2mEuclidean time
2m
i⇡
%
2
2 (ϕ, ϕ̃) → 3
2
2
(i)
2
translation
τ
→
τ
+
a
and
the
phase
rotation
g
g
=
2mpe
✏
+
4mp
+
log
±
e
✏
+
O(
✏
)
Ep exp(−2pm/g ),
(5)
ib
−ib
This is consistent withdilute(e ϕ, e theϕ̃) (a,b ∈g 2C).gasUsing
the corresponding con-In
2approximation.
=0
servation
laws,
we 1,
can Z
obtainthevanishes
following solution
ofto
thesupersymmetric
case
ϵ
=
/Z
due
p
0
(i)
the equation of motion with nontrivial contribution in aerm E0 corresponds to the perturthe factor 1/Γ(1−βϵ).
the near SUSY case, we obtain→ ∞In
limit,
on the trivial vacuum (perturbative
厳密結果との精確な一致∂ϕ ∂ ϕ̄

全ての複素固定点からの寄与

cients of E (1) [21] areg2 ,Ep(1) = −2m, (p ≥ 1).

iφc f (τ − τc )
" 2
ϕ
e
,
Z =
1
2

−iφc f (τ − τc )ϕ̃ = e #
2 ✓ ,
2pm 2

(9)

◆
sin
α
sin
α
∂
Z
(p + 1) −(p 21)
2m i⇡
p
(2)
t
2
E
=
2m
dte
+
+
4mp
+
log
±
g 2m,
(6)lim
p
2m
2
≈
2mβe
(C23)
g
2
where (τt + g parameters associated0 c , φc ) aret complex
g ±i0 moduli
ϵ→1with∂ϵZ0symmetry and f (τ ) is the elliptic functionthe2

E (2) are expanded in powers of g ,divergent asymptotic series, which22f (τ ) = cs(Ωτ, k) ≡ cn(Ωτ, k)/sn(Ωτ, k),(10)

完全なトランス級数展開

1)

=
+
ve
d
is
es
er
rt
1
bn
V

厳密な基底エネルギー in CP1
2

⇣

⌘

⇣

⌘

2

3

2m
2m
m
Ei
+
Ei
coth g2
g2
g2
2m 5
(2)
2
4
E =g
m
log 2
2Hencemcan be Borel-resummed.
werewritethecoeﬃcient2gsinh g2

in the form of the Borel transform (See Appendix. A forthedetails of calculations.) as
・摂動寄与
& ∞
−t
e
(2)
E0 = g 2 + 2m
dt
(7)
2m ,
t − g2 ±i0
0
'
(
& ∞
2
2
(p
+
1)
(p
−
1)
Bender-Wu摂動展開と完全に一致
Ep(2) = 2m
dt e−t
2m +
2m
t
−
t
+
0
g 2 ±i0g2*
・固定点寄与 )
2m πi
2
# 1).
+ 4mp
, (p2≥✓(8)
Z 1 γ +"log 2 2±2(p 1)(p +g 1)
(2)
t
2

2m i⇡
Ep = 2mdte+ 4mp+ log 2 ±
2m +
2m
g
2
t +associatedNote that the 0imaginaryt ambiguities
to theg 2 ±i0g2(2)
Borel resummation is manifest in the first term of Ep
2

Timble計算に基づくp-bion寄与と完全に一致with g ±i0,
which is compensated by the imaginary part(2)
±iπ/2 in the last term of Ep+1 , reproducing the original

◆

1)

=
+
ve
d
is
es
er
rt
1
bn
V

厳密な基底エネルギー in CP1
2

⇣

⌘

⇣

⌘

2

3

2m
2m
m
Ei
+
Ei
coth g2
g2
g2
2m 5
(2)
2
4
E =g
m
log 2
2Hencemcan be Borel-resummed.
werewritethecoeﬃcient2gsinh g2

in the form of the Borel transform (See Appendix. A forthedetails of calculations.) as
・摂動寄与
& ∞
−t
e
(2)
E0 = g 2 + 2m
dt
(7)
2m ,
t − g2 ±i0
0
'
(
& ∞
2
2
(p
+
1)
(p
−
1)
Bender-Wu摂動展開と完全に一致
Ep(2) = 2m
dt e−t
2m +
2m
t
−
t
+
0
g 2 ±i0g2*
・固定点寄与 )
2m πi
2
# 1).
+ 4mp
, (p2≥✓(8)
Z 1 γ +"log 2 2±2(p 1)(p +g 1)
(2)
t
2

2m i⇡
Ep = 2mdte+ 4mp+ log 2 ±
2m +
2m
g
2
t +associatedNote that the 0imaginaryt ambiguities
to theg 2 ±i0g2(2)
Borel resummation is manifest in the first term of Ep
2

with g ±i0,
which is compensated by the imaginary partp-bion周りでの摂動展開と完全に対応(2)±iπ/2 in the last term of Ep+1 , reproducing the original◆

全てのMulti-bion解の寄与に基づく
完全なリサージェンス構造を確認！

具体例２：Sine-Gordon model[Fujimori, Kamata, TM, Nitta, Sakai(16)(17)]h angle takes values between 0 ≤ α < 1 since α and α + Z are equivalent. T

1.condition
Expansion
the SUSY point
Sine-Gordon量子力学
minated from the boundary
byaroundthe redefinition
Ψ(θ) → Ψ̃(θ) ≡ e[Fujimori, Kamata, TM, Nitta, Sakai(17)]inger equation(III.2)fortheperiodicwave
functionΨ̃
becomes・Sine-Gordon量子力学
a. Smallδϵ expansion
"
!
2
mϵ
m
2
2
2
First, we consider
expansionaroundSUSY
point cos
(ϵ =θ 1,Ψ̃α =0).
The gen
=
E
Ψ̃.
sin
θ
−
H Ψ̃ =the−g(∂θ − iα)
+ the
4g2
2

given by the modified Bessel function of the first kind
! 2π state of the system described by this Schödowing, we discuss thegroundが整数もしくは半奇整数の場合に厳密に解ける
m⟨0|0⟩ =dθ exp (z cos θ) = 2πI0 (z),z ≡ 2.
g → ∞) ond state energy can be read
0 oﬀ from the low temperature limit (β・基底状態エネルギーe leadingRayleigh-Schrödinger
theorygives thepath
firstintegra
expanswhich
canorderbe definedfor(1)
each α byperturbation
thefollowing
Euclidean
(2) 2
3

E = E

eion
groundstateenergyθ(τ +β) =θ(τ ) asmod 2π✏+E✏ + O( ✏ )

"
#
"
⟨0|
cos θ|0⟩
m ∂
m
∂
(1)
"
E " −βH= −log I0 (z).
= −E =Z(α)∂ϵ= Tr[e] 2= ⟨0|0⟩Dθ exp(−SE2), ∂z
ϵ=1

is is the exact result which will be compared with the semiclassical multi-bi

Euclidean action is given by摂動寄与+非摂動寄与
cussed below. To extract
contribution, let us consider
the wea!the perturbative
"
#
$
%

0
0
has an imaginary ambiguity associated withtheπ branch
cut starting from t = 2z to infinity:
Multi-bion solutions! : Nonperturbative contribu−2z as∞ seriesz−tpressedastheconvergent
power
in
the
nonperturbative
exponentialeM.tions
We derive
semiclassicalcontributions
from themulti-bion
saddle points bye
1 energyito thetheground
state
come
from
saddle+dt +K0 (z),(III.15=−S
I0E(z) ± −βE!
"
π 0 Z = t(2zpoints of the path integral
DϕD∼ πe ∞− t)ϕ̃ eietz
thimblemethod.
We
will
see
that
their
expansioncoeﬃcientsaround
the
#
±
K
(z)
∂
m
ning
nonperturbative
part,
which
canceltheimaginaryambiguityoftheperturbative
0(1) where we have complexified
(1)
(1)
π
(for
large
β),
the
degrees
of
log
1
−
E
−
E
=
−
E
,
(III.17)par=where + is forz > 0.i Thus, weC obtainCthe
perturbative
p ambiguous
0 arg z < 0 and − is Cfor arg C
2 ∂zϕ those−2z inIiϕ±theK
freedombyas regarding≡powerϕR obtained+andϕ̃nonperturbative≡−p=1
iϕresults
nts
consistently
reproduce
from
theϕR
exact
discussed
0 (z)
0 (z)
be
expressedtheconvergent
seriesinexponential
e
as
I
I
π
(1)
of Eindependentasasholomorphic! variables, and imposed
the
・摂動寄与
"
# !
&
∞
ion.
i
,
#
±
K
(z)
∂
m
∞
periodic
condition
ϕ(τ
+
β)
=
ϕ(τ
)
and
for
z−t
0
(1)
−2z
(1) boundary(1)πi e(III.17)
m− ∂withlog
eof nonperturbative
1 1−
m ∂Ep exponential(1)
otes thep-th
power
E contribution
−
E
=
,
=
+
0
β ± i K (z) = − 2+log
log I0 (z) ± K0 (z) .
(III.16
E0 = −
dt
2
∂z
I
(z)
0
0
ϕ̃. The Euclidean
action S = 0t (2zdτπ[∂ϕ̃/(g2p=1∂z
(1 +
2 ∂z
π
−τt)ϕ∂%" τ! $ π E0

厳密な基底エネルギー in SGpiϕϕ̃)2 ) + V (ϕϕ̃)]has
twoconserved
Noether charges asK
(z)
±
∂
1
m
0
1.
Multi-bion
solutions
−2pz
(1)
π
−2z
∼
O(e
).
(III.18)E=
denotesthecontribution
with
p-th
power
of
nonperturbative
exponentialesociatedwiththecomplexificationoftheEuclideantimepi・固定点寄与2 ∂z p I0 (z) ± π K0 (z)%p "$ the phase rotationtranslation τ → τ +!a and(ϕ, ϕ̃) →i(z) corresponding± π K0the∂1mib−ib (1)−2pz con(e
ϕ,
e
ϕ̃)
(a,
b
∈
C).
Using
the multi-bion
solutionscorresponding
tothecomplexsaddle points of(III.18)
the
∼
O(e
).
E
=
2
p
i
rther expanded in 2powers
ofI0g(z) andtheleading order term is given by
±the
K
(z)
0
servation laws, we∂zcanp obtainfollowing
solution ofπ
the sine-Gordon QM (III.6). They can be easily obtained from those in thethe equation of motion with&2nontrivial
contribution
in a
2m 'p
摂動ボレル和の不定虚部はすべてのe further expanded in (1)powers of g and−the
2 leading order term is given bygβ→∞limit,+ · · · , of the Euclidean equation of(III.19)±ie The solutions
oring the azimuthalEangle
variable.
p = −mp-bion寄与を足し上げて初めて相殺
&
'
p
2m
−
f
(τ
−
τ
)
f+(τ· ·take−τcthe) form(1)cin
g 2 limitiφc−iφnontrivial
contributions
the
β
→
∞
c
·
,
(III.19)
E
=
−m
±ie
p 複素解
ϕ=e
,
ϕ̃
=
e
,
(9)
−2z for large z. This 2nonperturbative contribution with theused K0 (z)/I0 (z)sin
≈2πe
α
sin α
θ(τ
) −2z ffor(τ large− τc )z.
have
used
K
(z)/I
(z)
≈
πe
nonperturbative
with
the
where
(τ
,
φ
)
are
complexmoduliparameters
associated
0
0
c
c tan
=
, Thisiguity is expected
to correspondto thesemiclassicalpart ofcontribution
the p-bion(III.33)
contribution.
2 f (τ )sin
αthe elliptic function
withthesymmetry
andisambiguity is expected to correspond to the semiclassical part of the p-bion contribution.
re this exact result with the multi-bion contributions later.
f (τ) = resultcs(Ωτ,k) and≡ cn(Ωτ,k)/sn(Ωτ,k),function(10)
plexifiedposition
modulusf (τ ) istheelliptic
mpare this
exact
with
the
multi-bion
contributions
later.
expansion

all α expansion

Fig. 2: Multi bion solution: p = 3, q = 2, m = 1, ϵ = 1, g 2 = 1/20000, β = 100, τc =+enomenon. Thep originalpintegration
contourC
canbedeformed
anddecomposedasJ−2 J1 or
∓pπσ
p
∓πσ
p−1
0
ce the residues at σ ̸= 0 vanishes in the β → ∞ limit, the pole at σ = 0 gives the leadin

Tr(T± ) = 2 e!

2 sinh (±πσ) ± ipπesinh"(±πσ) δϵ + O(δϵ ) ,pending on arg g 2 . The ambiguous Borel resummation I0 (z)± πi K0 (z) corresponds to the integra

m for large β±

p-bion 寄与と厳密結果
2p

p
.
!
#
e δϵ = ϵ − 2j − 1 (∀j ∈ Z). The first term indicates that the trace Tr(T ) has a±"imβ − 2pm
−imβσ−mϵβ −mβg 2 Res eI+O(e, e at ).the QES p
e
X
=
−
i
(s1=
,··· 0,,s2pso) that the p-bion contribution (III.50)
at σ = 0 for δϵvanishes
σ=0pi=1 imaginary ambiguity of the per
he remaining nonperturbative part, which cancel the基底エネルギーへの寄与
#
$

1Z
us
the following
contribution
ofp thethe nonperturbative
p-bion solutions: exponentia(0)rt, wecanobtain
be expressedas thesemiclassicalconvergent
power
seriesin
Ep = lim lim −
= 0.
!
#
β→∞
δϵ→0
β Z0
$
'
%
&
!
"
2iσ−ϵ+1p∞Zp
1imβ − 2pm
4m
i
#
−imβσ
2
± π K0 (z)≈ − (1) e g(1)Res e m ∂Γ(−iσ)Γ(ϵ − iσ)(1) Z± ,2
− E0 σ=0=− the log
Ep expansion,=
Z0 in Ep (III.54)
2π 1 −g contribution
econd term
Eq.
gives
following
to
the
first
co
i
2 ∂z
I0 (z) ± π K0 (z)
p=1
round
energy:ere
Z± state
is givenby(1)−2
here Ep denotes the contribution
with
p-th
power
of
nonperturbative
exponential
e
&
#
$
%
*
)
2p
(
(
"
2m p
πi
Z
∂
1
Γ(2j
+
1)
− 2
p
(1)
gπiαs + .· · · .
%
"
!
$
(iσ
−
ϵ
)(1
−
s
s
)
+
···
EpZ±== lim lime− exp ± =
−m
±i
p
i
i
i−1
i
i
2j
2
β→∞ δϵ→0 m∂δϵ∂ i=1(4z)
1 β Z0 ± π K0 (z)s1 =±1s2p =±1
−2pz
(1)
∼
O(e
).
Ep =
i
2 ∂z p I0 (z) ± π K0 (z)the transfer
isusing
consistent
with matrix
the exact result for j = 0 (III.19), which indicates that the ima
⎛
⎞⎛
⎞
2
j=0
his
can bethefurther
expandedinで厳密結果との精確な一致
powersof g andthe leadingorderterm is given
by conπiα±πi(iσ−ϵ)πiα±πi(iσ)ty from
perturbation
seriesiscanceled
by
those
from
the
semiclassical
bion
e
e
e
e
⎠⎝
⎠,
T± = ⎝&
p
2m '±πi(iσ)±πi(iσ−ϵ)
−πiα
−πiα
−
e
e
e
e
(1)
+ ··· ,Ep = −m ±ie g2
Small α expansion around j ∈ Z, α = 0

can also be written as−2z respect to α. For ϵ = 2j + 1, the trace Tr(Txt, let us consider the expansion with

4次元Yang-MillsやQCDでは？

the Borel plane contains poles on the real axis, which generate ambiguities in the calculation,
depending on whether
the thepole correlation
is circumvented
from aboveofortwo
fromvector
below. currents
The class jof
Considerfunctionsµ =diagrams su↵ering from this problem
and the! are referred to as the renormalon diagrams"
corresponding non-Borel summability is 4the (in)famous
problem [12].
(−i) d x e−iqx ⟨0|Trenormalon(jµ (x)jν (0))|0⟩= qµ qν − q 2 gµ
Borel non-summability of the perturbation theory is not in itself surprising and was‘t Hooft(79)argued by Dyson long time ago3 [14]. This problem also appears in quantum
mechanics, but
22
with Q is=caused
−q by
. the
Wefactorial
now computeof the fer
there the divergence
proliferation theof thecontribution
number of the Feynman
diagrams. shown
In fact, one
suchdivergencecured by instanton–anti-instanton
eventsin finds
Fig.that
1 totheAdler isfunction
2[15, 16], and has a priori nothing to do with the renormalon problem.

QCDにおける赤外リノーマロン不定性◆アドラー関数と赤外リノーマロン

2
dΠ(Q
)
2
2
Then, we can define theperturbative sum for the non-alternating series as the
2
.
D(Q
)
=
4π
・Renormalized loop挿入によりfurther subtraction不要
2 coupling, g2 < 0, t
tinuation of B0 (g 2 ) in the g 2 complex plane fromdQ
negativeI.

INTRODUCTION
real axis, g 2 > 0. This can be done in one of the two ways as shown in Fig. 2.
22
the positive real axis clock-wise (from above)(from bel
↵
log(
k
/µ
) of selected diagramsk and counter-clock-wise
・各loop寄与は
0 s
2 it is not th
The setis gauge-invariant,but
I.

INTRODUCTION

B0 (|g 2 | ± i ) = Re B0 (|g 2 |) ± i Im B0 (|g 2 |)where Im B0 (|g 2 |) ⇤ e 2SI ⇤ e

that&contributes
・外線運動量Qだけに依存しUV
IR finite to renormalon divergence. It is selected hereis the ambiguous part, and is a manifestation of non-Borel-summability [compar
systematicinvestigationA definitionis postponed
to equivalent
Section
3. weRenormalonof the Borel sum
to what
described above thr"#
I.
INTRODUCTIONn! Z αs (µ)β0 t "continuation in the complex g -plane is the directional (sectorial) Borel sum. D
α2s #(µ)
n◆2 1974; Lautrup 1977; ’t H
1
inbubblediagrams(Gross
Neveu
1
2
2
X
X✓ &
BP (t) = α
(µ)
−
=
(1) 1
F
(
k̂
)
k̂
Q
↵
s
s 0
2
2
1 + still
α2s (µ)βD(Q ) =↵sd#k̂n 2⇡0 t/2↵s so prominentn!) ⇥discussions+B (gUVS P (g) = contr. BPedt,
0 ↵s log
diagrams
featureinof (t)renormalong2! " αns (µ)βµ
Figure1.
Left:thevacuumpolarization
withallcorrections.
Right:
Particular
contribution
to
the
2
k̂
αs (µ)0 0t
n=0n
BP (t) = αs (µ)−= polarization
vacuumoften
referred to aswith
the renormalon
diagram.
are even
identified
them. (1)2
1 + αs (µ)β0#
t/2nn! " αs (µ)β
The
Adler
requiresno additional
subtractionsts (µ)
tcan bebeyo
It was recently0suggested functionin [17,α18]that
IR renormalon
ambiguity cancellation
BP (t) = αs (µ)
−
=
(1)
2
understoodthe
in terms
of semi-classicalsolutionsappearingin the theory
Lagrangian.
Therefore
no regularizationi2renormalized
1 +QCDα instanton-monopole
(µ)β
t/2
C
t = − onnR3 ⇥ S1 , but which do not appears on R40. This idea was substantiated(2)by the +detailed
αs (µ)β0fermion loop insertions are 1renormalized. The renormalized
ferm
2

2

⌅·ei

2

2

t/g 2

0

2 analysis of two-dimensional models on R ⇥ S [19–22], which $have extra non-perturbative
C % int=−(2)saddles comparedto the theory on R2 (these are analogous
to the instanton-monopoles
22
正の実軸上の特異点− β0fmechanics
αs ln(−k/µ )L,+a C
αs (µ)β0 gauge theories). Since these theories reduce to quantum
for
small
resurgent2t = − 8π(2)1Capital 1
lettersareusedtodenotethe4-momenta,
and
small
letters
denotethespatial3-momenta.
α(µ)β
s ga2 (µ)
0withscheme-dependentCβ=N
T /(3π)the (pf
2
Figure/N
9.constant
Lateral, or rightandandleft, Borel
sums.
Dark
are singularities
20f
fcirclesβ2nS
I0 is caused by the appearance of logarithms in the one loop (3)suppression
vacuumpolarization
diagrams
B(g ) = ReB ±This
iπ
e
大まかには
2S
I
/N
c
作用
points). Whenever a singularity exists between the right and left Borel sums, the theo±Im[Be whichii ]one-loop
6 we revisit in this work on R3 ⇥ S 1 .
=0
β
(see
Section
2
and
[10,
11]
for
more
details.),scheme= −5/3.
theβ-function.
In0summable.
The thesingularMS
directionin the t-planeC
corresponds
to a Stokes line in the com
2
3−1/g
Although it is true that the perturbationseeseries
divergent,waspointedthat Dyson’s
Fig.2. isThedifference ofitthe
sectorial
sumsout
in passing
from ⇥ = 0argument
to ⇥ = 0 is the2e46
scaleに関連
across a Stokes
ray. (3) QCD
may not be entirely
valid [13].
±Unlessotherwise
stated,αs denotes
the strong
coupling renormalized
⇠
(⇤/Q)
2
2
g
2e−1/gA special case of this is the lateral Borel sum. The function B ± (g 2 ) is as(3)
2 ±contours just above and just below the ray at angle ⇥, and is called right (left)
2
−1/g
g
2e
8the ⇥ direction, then the left and righmation. If there are no singular points in
+

±

(4)

正則化されたQCD型理論？
QCDをweak coupling領域で扱うために変形が必要• ゲージ場作用をゲージ対称に変形する方向性
Double-trace deformation, et.al.

Unsal(09)• 随伴表現クォークなどを導入する方向性
QCD(adj.), Z3-QCD, et.al.

Unsal(07) Kouno, et.al. (12)Kanazawa, TM(14) Iritani,Itou,TM(15)Cherman(16), Sulejmanpasic(16)• 時空をコンパクト化する方向性
R3 x S1, T3 x S1, et.al.

Yamazaki,Yonekura (17)

QCD(adj.) on R × S
3

1◆細谷機構による非自明なPolyakov-loop holonomy in S^1
P = Tr (P exp[igcf.) SU(3)=0A4 dx4 ])C

q2 0

U(1)^(Nc-1) 中心対称相の出現0q1

cf.)SU(2)





SU(2)
中心対称
閉じ込め




⇥ = 0.326

SU(2)
非中心対称
非閉じ込め

q2 0

Schematic phase diagam

U(1)
中心対称
ヒッグス



0
Cossu, D’Elia (08)
Cossu, Hatanaka, Hosotani,q1Noaki(13)Kashiwa,
TM (13)Figure3. Contour plots of V(Nc = 3
TM, Kanazawa(14), }. Phase transitions
0.326 , 0.400

the split phase and then to the deconfi

This phase is Z3 -symmetric but br
At ⇥ ⌃ 0.248 there is a first-ord(0, ± , ⌅ ), (±2 /3, ⌅ /3, ⌅ /3)

PF =This is the so-called “split phase”

9

at x1 → +∞. The configurations ωL (ωL∗ ) and ωR (ωR∗ ) can be regarded as a domain wallU(1)相におけるソリトン解a path connecting the north pole m = (0, 0, +1) and the south pole m = (0, 0, −1) in theand anti-domain wall, respectively. A domain wall at each constant x2 slice corresponds totarget space, as illustrated in Fig. 2(a). A U(1) modulus is localized on these domain walls

characterizing
which point on the equator in the(Q=1/N)
target space a domain wall passes through
BPS解 : 1/N
分数インスタントン[56]. This U(1) modulus is twisted along the domain wall to satisfy the boundary condition

P = diag[1, e

] (P

= 1)

at x2 = 0 and x2 = 1. When one changes a constant xN
2 slice from x2 = 0 to x2 = 1, a path
2 i/N
4 i/N
2 (N 1)i/N, · · ·, e,ein the target space changes with sweeping a half of the sphere as the target space, as shownin Fig.2(b) and (c). Therefore, these configurations give maps from the space R × S 1 to a

・ZN twisted B.C.を課すことと等価
half of the target space. BPS configurations ωL and ωR carry a half of the unit instanton
・BPS解は1/N targetspaceへのマップに対応   ex.) SU(2) caseTarget spaceの半分 (SU(2)/U(1)) =   Monopoles    

Q=1    

()

Q=1/2

(a)
Instantons

Q=1
FIG. 2:

()

(b)

(c)

Q=1/3

(a) Domain wall, and (b) and (c) fractionalized instantons in the target space S 2 . (b)corresponds to the configurations ωR and ωL∗ while (c) corresponds to the configurations ωL and∗.
ωR

from: Bruckmann et al., hep-th/0309008

charge, Q = 1/2, while anti-BPS configurations ωL∗ and ωR∗ carry Q = −1/2. This fact alsoe Bogomol’nyi-Prasad-Sommerfield (BPS) monopole, there is also a KK monopole.
istence of KK-monopoles, which are perhaps the most crucial ingredient in our discu
QCD(adj), was discoveredin 1997,
independently by Lee and Yi using
D-branes in s
Magnetic
bionsによる閉じ込め
Unsal(07)eory [9] and by Kraan!and van Baalby using calorons configurations [10]. The mag#""
d topological charges
F, F F̃ ofthese monopoles
are normalized as= （磁荷,
インスタントン数）
BPS :(+1,+ 21 ),cf.)SU(2)BPS : (−1, − 12 )KK :(−1, + 12 ),KK : (+1, − 21 )here bar denotes antimonopoles. BPS
KKIn [8], we constructed the d (1,= 1/2)
3 dimensional(−1,long-distancetheory for QCD(adj) fo
1/2)
ted on R3 × S 1 by employing three tools: abelian duality, symmetries, and index theo
his strategy is, in essence, similar to the Seiberg-Witten Magnetic
constructionof prepotentibion凝縮= 2 SYM [4]. The unique lagrangian to order e−2S0 dictated by these considerations

閉じ込め

BPS

L

dQCD

KK

(2,0)

1
BPS
KK 0 1/2)= (∂σ)2 − b e−2S0 cos 2σ + iψ̄ I γµ ∂µ ψ(1,I 1/2)
+ c e−S(−1,cos σ(det ψ I ψ J + c.c.)I,J2(−1, −1/2)(1, −1/2)(2,0)(−2, 0)I
BPS
KK
here σ and ψ denote the dual photon and fermion. Dimensionless coordinates, meas

Figure 2: (Left)Magnetically(−1,andcharged monopoles carry−1/2) topologically(1, −1/2)
units of compactification
circumferenceL,
areused.induce
A detailed
microscopic
derivati
Fermionzero
mode交換に基づくMagnetic
bionの凝縮
modes.
Consequently,
theycannotabosonic
potentialforthe dual
Figure 2: (Left)Magnetically and topologically charged monopoles carry compulsory
fermiomodes.
Consequently,
theyforcannotgauge
induce a bosonicpotential for isthe dual
photon. (Right) inTo
is Lagrangian will be given
in section
2. Themass
gapbosonsmanifest
cally null,magneticallycharged
bions
have
no
external
fermioniclegs.
Hencally null, magnetically charged bions have no external fermionic legs. Hence, they induce theによる閉じ込め(massive
dual
photon)bosonic potential,
which implies mass for the dual photon and confinement. The figure is for
impliesmass
for thesize
dual
photonandconfinemen
grangian. The inverse bosonicof the potential,
mass gapwhichis thecharacteristicthe(whichchromoelectricwith n = 2. The combination of the BPSKK ofmonopoles
is not depicted) is an instan
caloron). It ofis present
confined phase,
but is not the source of the dual photon mass term.
nf =manifest
2. The combination
theinBPSKKbe, hence confinementwith
is alsoin dual formulation[8]. monopoles (which is not defe Bogomol’nyi-Prasad-Sommerfield (BPS) monopole, there is also a KK monopole.
istence of KK-monopoles, which are perhaps the most crucial ingredient in our discu
QCD(adj), was discoveredin 1997,
independently by Lee and Yi using
D-branes
Neutralbionsと赤外リノーマロン
Argyres-Unsal(12) in seory [9] and by Kraan!and van Baalby using calorons configurations [10]. The mag#""
d topological charges
F, F F̃ ofthese monopoles
are normalized as= （磁荷,
インスタントン数）
BPS :(+1,+ 21 ),cf.)SU(2)BPS : (−1, − 12 )KK :(−1, + 12 ),KK : (+1, − 21 )here bar denotes antimonopoles. BPS
KKIn [8], we constructed the d (1,= 1/2)
3 dimensional(−1,long-distancetheory for QCD(adj) fo
1/2)
Neutralandbion
ted on R3 × S 1 by employing three tools: abelian duality, symmetries,
index theo
his strategy is, in essence, similar to the Seiberg-Witten construction= of prepotenti= 2 SYM [4]. The unique lagrangian to order e−2S0 dictatedby these considerations
IR-renormalon
?
(2,0)

L

dQCD

1
BPS
KK
= (∂σ)2 − b e−2S0 cos 2σ + iψ̄ I γµ ∂µ ψI + c e−S0 cos σ(det ψ I ψ J + c.c.)I,J2(−1, −1/2)(1, −1/2)

here σ and ψ I denote the dual photon and fermion. Dimensionless coordinates, meas
Figure 2: (Left)Magnetically and topologically charged monopoles carry
Neutral bion(もしくは対応する複素解)の寄与は不定虚部
units of compactification
circumference
areused.induce
A detailed
modes.
Consequently, L,
theycannota bosonicmicroscopic
potential for derivati
the dualis Lagrangianを持ち，リノーマロン不定虚部を相殺すると予想出来る
will be given
in section
2. Themassbions
gaphavefor gaugebosonsis manifest
incally null,magneticallychargedno external
fermionic
legs. Henimpliesmass for thesize
dualofphoton
and confinemen
grangian. The inverse bosonicof the potential,
mass gapwhichis thecharacteristic
the chromoelectricnf =manifest
2. The combinationof the BPS- KK
be, hence confinementwith
is alsoin dual formulation[8]. monopoles (which is not deQCD(adj)での結果はQCDに繋がるか？





強結合
閉じ込め



非閉じ込め






弱結合
閉じ込め

?




(2つの相を分けるorderparameterは存在しない)


問題は弱結合-強結合閉じ込め相の連続性があるか否かQCD(adj)での結果はQCDに繋がるか？


強結合
閉じ込め

非閉じ込め





弱結合
閉じ込め

?



問題は弱結合-強結合閉じ込め相の連続性があるか否か(2つの相を分けるorder parameterは存在しない)

まとめ• 複素解の寄与まで含めたトランス級数によって，量子論の厳密結果を再現• 量子力学，2次元場の量子論，厳密に解ける理論などにおいては，リサージェンス構造を確認• QCDを含むリノーマロンを持つ理論においては，
defomed理論とのcontinuityが鍵
数学の分野も巻き込んだ今後の進展が期待される

昔からのインスタントン展開と何が違うの？
• トランス級数にリサージェンス理論による数学的裏付けが与えられた．
• 複素古典解とThimble分解という観点から経路積分を理解する(定義する)可能性が得られた．
• 分数インスタントンの複合状態(Bion)もしくはそれに対応する複素解がリノーマロンを説明する可能性が出てきた．

何が出来るのか？
• 格子理論と異なる見方に基づいた場の量子論・経路積分の非摂動的定式化の可能性

Kontsevich(14)• 複素古典解の寄与を少しでも取り入れることで，新
しい非摂動的近似法として使える• 非超対称場の量子論であっても，複素古典解を見つけることで，非摂動解析を近似的に実行出来る可能
Dolan, Burns ('80)
性．

•Resurgence and Transseries in Quantum, Gauge andString Theories — CERN 2014•

Resurgence and Localization in String Theory and
Quantum Field Theory — SIMONS CENTER 2015•

Resurgence, Physics and Numbers — PISA 2015•

Resurgence Gauge and String theory — IST 2016•

Stokes phenomenon, Resurgence and Physics at IRMA
— IRMA Strasbourg 2016•

Resurgence at Kavli IPMU — IPMU 2016 Dec.

•

RIMS-iTHEMS International Workshop on Resurgence
Theory — RIKEN Kobe 2017 Sep.

•

Resurgent Asymptotics in Physics and Mathematics —KITP-UCSB 2017