modes. Thi s means that it effectively ac ts as an IR cut off
separating t hose soft modes from the integration p rocess.
By successively lowering the scale k, the effective average
action Γ
k
includes increasingly more f luctuations and, in
the limit k → 0, all quantum fluctuations are integrated
out.
In order to compute Γ
k
, one has to truncate the system of
vertex functions, since the right-hand side of Eq. (12)
involves the two-point function Γ
ð2Þ
k
, which, itself, couples
through the flow to higher n-point functions. A typical
truncation scheme for these vertex functions is given by a
derivative expansion, which we will focus on in the
following; cf. also the introductory Refs. [17–27].
Along the lines of our previous study [4], we choose the
following (Euclidean) truncation for the linearly realized
Oð4Þ QMM based on Eq. (4) in Sec. II A and introduce the
k-dependent higher-derivative couplings C
2;k
and Z
2;k
,as
well as C
i;k
, i ¼ 3; …; 8:
Γ
k
¼
Z
x
Z
k
2
ð∂
μ
φÞ · ∂
μ
φ þ U
k
− h
ESB
σ þ C
2;k
ðφ · ∂
μ
φÞ
2
þ Z
2;k
φ
2
ð∂
μ
φÞ · ∂
μ
φ
− C
3;k
½ð∂
μ
φÞ · ∂
μ
φ
2
− C
4;k
½ð∂
μ
φÞ · ∂
ν
φ
2
− C
5;k
φ · ð∂
μ
∂
μ
φÞð∂
ν
φÞ · ∂
ν
φ
− C
6;k
φ
2
ð∂
μ
∂
ν
φÞ · ∂
μ
∂
ν
φ − C
7;k
ðφ · ∂
μ
∂
μ
φÞ
2
− C
8;k
φ
2
ð∂
μ
∂
μ
φÞ
2
þ
¯
ψðZ
ψ
k
γ
μ
∂
μ
þ y
k
Φ
5
Þψ
: ð14Þ
This truncation constitutes a derivative expansion up to
order Oð∂
4
Þ. It is well beyond the LPA and its minimal
extension known as the LPA
0
. The first would only
consider a scale-dependent effective potential U
k
, which
is a function of the Oð4Þ invariant ρ,
ρ ≡ φ · φ ¼ π
2
þ σ
2
; ð15Þ
while the latter would also take into account the scaling of
the field variables by means of the flow of the wave-
function renormalization factors for bosons and fermions,
Z
k
and Z
ψ
k
. In both of these two approximations, the higher-
derivative couplings would be absent; i.e., they are set to
zero. In light of Ref. [4], the momentum-independent
coupling C
1;k
ðφ · φÞ
2
is omitted in the above equation. It
is to be identified with the quartic interaction term of the
effective potential.
The parameter h
ESB
≠ 0 explicitly breaks the Oð4Þ
symmetry, as mentioned in the last section, and remains
k independent. For the scale-dependent factors Z
k
, Z
ψ
k
, and
the Yukawa interaction y
k
we suppress a general field
dependence. The couplings fC
2;k
;Z
2;k
g as well as fC
i;k
g,
i ¼ 3; …; 8, beyond the LPA
0
form a complete set of terms
of order Oðφ
4
; ∂
2
Þ and Oðφ
4
; ∂
4
Þ, respectively. As an
extension of Ref. [4] [cf. Eq. (20) therein], the low-energy
couplings in Eq. (14) now include momentum-dependent
σπ and σ self-interactions.
On the level of the two-point functions Γ
ð2Þ
k
, we define
different effective wave-function renormalization factors
for the σ and π fields,
Z
σ
k
¼ Z
k
þ 2σ
2
ðZ
2;k
þ C
2;k
Þ
− 2σ
2
p
2
ðC
6;k
þ C
7;k
þ C
8;k
Þ; ð16Þ
Z
π
k
¼ Z
k
þ 2σ
2
Z
2;k
− 2σ
2
p
2
ðC
6;k
þ C
8;k
Þ; ð17Þ
where p is the external momentum from the functional
derivatives with respect to the fields. The corrections to Z
k
in these definitions obviously arise from the presence of
the higher-derivative couplings. It should be noted that the
distinction between Z
σ
k
and Z
π
k
is not in contradiction to the
Oð4Þ symmetry of the model. As soon as the σ field
acquires a nonvanishing expectation value, σ ≠ 0, the
wave-function renormalizations will naturally split.
For later reference, we define the following renormalized
quantities based on the wave-function renormalization
factors Z
π
k
and Z
ψ
k
:
˜σ ¼
ffiffiffiffiffiffi
Z
π
k
p
σ;
˜
π ¼
ffiffiffiffiffiffi
Z
π
k
p
π;
˜
ψ ¼
ffiffiffiffiffiffi
Z
ψ
k
q
ψ;
˜
¯
ψ ¼
ffiffiffiffiffiffi
Z
ψ
k
q
¯
ψ;
˜
h
ESB
¼
h
ESB
ffiffiffiffiffiffi
Z
π
k
p
;
˜
y
k
¼
y
k
Z
ψ
k
ffiffiffiffiffiffi
Z
π
k
p
;
˜
C
i;k
¼
C
i;k
ðZ
π
k
Þ
2
;i¼ 1; …; 8;
˜
Z
2;k
¼
Z
2;k
ðZ
π
k
Þ
2
; ð18Þ
where we will, for simplicity, evaluate the above wave-
function renormalization factors at vanishing external
momentum, p ¼ 0; cf. the discussion in Appendix B.
By choosing Z
π
k
for both bosonic fields in the definitions
(18) one directly obtains the correct renormalization factors
in the nonlinear model, as presented in the next section.
The flow equations for all scale-dependent quantities in
truncation (14) are obtained by projecting the functional
derivatives of Eq. (12) (the flows of vertex functions) onto
the respective coupling. The resulting expressions for these
DIVOTGEY, ESER, and MITTER PHYS. REV. D 99, 054023 (2019)
054023-4
剩余15页未读，继续阅读
weixin_38687648
- 粉丝: 2
- 资源: 937
我的内容管理
展开
最新资源
- 解决Eclipse配置与导入Java工程常见问题
- 真空发生器：工作原理与抽吸性能分析
- 爱立信RBS6201开站流程详解
- 电脑开机声音解析：故障诊断指南
- JAVA实现贪吃蛇游戏
- 模糊神经网络实现与自学习能力探索
- PID型模糊神经网络控制器设计与学习算法
- 模糊神经网络在自适应PID控制器中的应用
- C++实现的学生成绩管理系统设计
- 802.1D STP 实现与优化：二层交换机中的生成树协议
- 解决Windows无法完成SD卡格式化的九种方法
- 软件测试方法：Beta与Alpha测试详解
- 软件测试周期详解：从需求分析到维护测试
- CMMI模型详解：软件企业能力提升的关键
- 移动Web开发框架选择：jQueryMobile、jQTouch、SenchaTouch对比
- Java程序设计试题与复习指南
资源上传下载、课程学习等过程中有任何疑问或建议，欢迎提出宝贵意见哦~我们会及时处理！
点击此处反馈
