超対称性と繰り込みー3次元非線形シグマ模型の立場

The renormalizability of 3-dimensional
non-linear sigma models
１．Introduction
２．Bosonic O(N) model
３．Supersymmetric models
４．Q(N-2) model
5.Summary
With K. Higashijima, K.Hitotsumatsu, M.Tsuzuki
１Introduction
The three dimensional non-linear sigma models
are nonrenormalizable within the perturbative
method.
Nonperturbative method: WRG approach
Progress of Theoretical Physics 110 (2003) 563
K.Higashijima and E.I
large-N expansion
2.Bosonic O(N) model
Lagrangian:
Phys. Rev. D43 (1991) 3428
V.G.Koures and K.T.Mahanthappa
The constraint may be implemented by
introducing a Lagrange multiplier field
.
The Euclidean path integral:
Integrating over the dynamical fields
From the Legendre transformation, we see the
effective action.
Because of the O(N) symmetry, the v.e.v of
the dynamical fields may be written as
The effective potential:
The stationary conditions of this potential:
The critical point:
There are two phases.
１．
The O(N)-symmetry is broken.
There are (N-1) massless NG bosons
２．
The O(N)-symmetry is unbroken.
There are N massive (
) bosons.
Gap eq.
In leading order :
The βfunction (leading order)
Renormalizablity
The effective action:
Feynman rules in O(N)-symmetric phase
(
)
Counterterm Lagrangian:
Gap eq.
We consider
leading order.
propagator in the next-to-
The problems in higher order
～order
(linear divergence)
These diagrams need the
additional counterterms.
～order
（log divergence）
3.Supersymmetric models
The 3-dimensional
O(N) model:
supersymmetric
is a Lagrange multiplier superfield.
Infinite parts from these diagrams cancel.
The fermionic parts are O(N) Gross-Neveu
model.
Using
,
We see the auxiliary field
fermion bound states.
corresponds to
The next-to –leading order β function
Gap eq.
We consider
leading order.
propagator in the next-to-
The wave function renormalization:
SUSY
The cases of
N-1
CP model
Inami, Saito and Yamamoto Prog. Theor. Phys. 103 （2000）1283
U(1) gauge auxiliary field:
This model also has two phases:
:SU(N)-symmetric, massive phase
:SU(N) broken, massless phase
We can assign U(1) charge for chiral and antichiral superfields as follow.
U(1) symmetric phase
U(1) broken phase
Next-to-leading order corrections to the
propagator.
The β function of this model has no next-toleading corrections.
The wave function renormalizations:
←gauge fixing
Is
supersymmetry responsible for the
vanishing of the next-to-leading order
corrections to βfunction of the model?
The results of WRG approach
The βfunction of target space metric :
When the target space is an Einstein-Kaehler manifold,
the βfunction of the coupling constant are obtained.
Einstein-Kaehler condition:
The value of h for hermitian symmetric spaces.
G/H
Dimensions(complex)
h
SU(N)/[SU(N-1)×U(1)]
N-1
N
SU(N)/SU(N-M)×U(M)
M(N-M)
N
SO(N)/SO(N-2)×U(1)
N-2
N-2
Sp(N)/U(N)
SO(2N)/U(N)
E6/[SO(10) ×U(1)]
E7/[E6×U(1)]
N(N+1)/2
N(N+1)/2
16
27
N+1
N-1
12
18
The examples of the Einstein-Kaehler cases
model
´t Hooft coupling：
model
1/N next-to-leading?
4.Q(N-2) model
model
O(N) condition:
There are two multiplier superfields.
There are three phases.
①
SO(N)-symmetric ,massive theory
②
New phase
SO(N) broken, massless theory
The differences between ① and ② phases:
① In effective action, the gauge fields have
Chern-Simons interaction.
The Parity is broken.
②
The gauged U(1) symmetry is broken.
We calculate the βfunction to next-to-leading order and
obtain the next-to-leading order correction.
Next-to-leading order corrections to the
propagator.
Some
3-dimesional supersymmetric sigma
models have next-to-leading order correction of the
βfunction.
Wilson的繰り込み群による利点と問題点
SUSYは、カットオフ無限大の極限をとると回復す
る方程式。
CP^NやQ^Nなどを系統的に考えることができる。
cosetの理論を考えた時、globalな対称性が破れ
た相しか見ていないことになる。臨界点が存在す
ることはわかったが、どのような相転移がおこって
いるかわからない。
繰り込み可能性に対する、近似の正当性の判断
がつきにくい。
１/N展開による利点と問題点
相構造、相転移、質量の様子などの理解が
できた。（フェルミオンに注目して考え直すと
束縛状態についての理解も得られる。）
繰り込み可能性。
個々の模型に対して、それぞれ考えなくては
いけない。
CP^Nでは、ゲージ場の補助場が入るため、
繰り込みをすると、ゲージ固定からSUSYは
破れる。（ボゾンとフェルミオンの異常次元が
一致しない）
図の待機場所

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