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首页夸克介子波动揭示低能耦合生成机制
本文主要探讨的是夸克介子模型在低能物理中的一个关键进展,即夸克-介子涨落动态生成低能耦合。在这个研究中,作者们针对线性O(4)夸克介子模型进行了扩展,该模型通常用于描述量子色动力学(QCD)下的强相互作用行为。低能限的计算是研究的重点,因为这有助于理解在较低能量尺度下,夸克和介子(特别是π介子,代表质子和中子的内部结构)的相互作用行为。 分析的核心内容是将线性模型中的有效作用转换成一个非线性实现的介子作用,这里的“非线性”指的是Goldstone boson非线性规范理论,其中π介子作为Goldstone bosons。这种转换涉及将模型中的高阶导数相互作用项,也就是那些由四点顶点函数决定的项,通过低能耦合进行参数化。低能耦合反映了夸克和介子的波动动力学,它们不是硬编码的常数,而是随着能量降低而动态生成的。 四点顶点函数在完全O(4)对称的近似下被计算,这个对称性考虑了动量相关的σπ交互作用和σ自交互作用。这意味着这些高阶导数耦合不仅仅是模型假设的结果,而是夸克和介子量子涨落的自然产物,它们的强度和行为受到量子力学波动性的深刻影响。 尽管研究者在低能量范围内限制了自由度的数量,但波动动力学展现出的定性特征使得他们能够深入讨论有效范围和纯仿生有效模型的重整化尺度。有效的重整化尺度对于确保理论在不同能量尺度上的一致性至关重要,它反映了量子修正的累积效应。 总结来说,这篇论文通过对夸克-介子涨落的细致分析,揭示了在量子场论背景下,如何从基本原理出发,通过动力学过程生成低能物理的有效参数,这对于理解和预测强相互作用在低能条件下的行为具有重要意义。这不仅深化了我们对夸克-胶子复合系统的理解,也为未来的实验验证提供了理论基础。
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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
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