红外固定点与爬行电弱对称破缺的新机制

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"Crawling technicolor" 在物理学领域，尤其是粒子物理和量子场论中，"Crawling technicolor" 是一个关于电弱对称性动态破缺的新机制。这个概念源于对Nambu-Goldstone (NG) 模式的分析，当在非平凡的红外（IR）不动点αIR实现比例不变性时，Callan-Symanzik方程揭示出了一种不寻常的行为。传统的Wigner-Weyl（WW）模式下的缩放定律在这种情况下不再适用。在这个新的机制中，运行耦合α在红外极限内并不直接趋向于αIR，而是"爬行"到接近但不过这个点。这种现象的关键在于，αIR处的NG机制导致了一个无质量的膨胀子σ的存在。膨胀子在红外膨胀（IR expansion）中，即ε ≡ αIR - α 范围内变得显著，并且可以被识别为希格斯玻色子。与通常与Wigner-Weyl模式固定点或Coleman-Weinberg势相关的膨胀不同，这里的NG模式膨胀是真实的，因此更为自然。膨胀子（或称为狄拉克玻色子）的质量平方与ε、β函数在αIR处的斜率β′、狄拉克衰减常数Fσ以及真空技术凝聚物的平方平均值 ⟨G^2⟩vac 有关。这种关系揭示了耦合强度如何随能量尺度变化，以及电弱对称性的破缺如何在不同的能量域内表现。理论的有效性依赖于Zumino对于Dilaton拉格朗日量的一致性条件。通过对希格斯势的封闭形式的研究，作者发现与标准模型相比，存在与β'相关的偏差。如果耦合α在αIR附近的爬行区域覆盖了超过TeV尺度的足够宽的能量范围，这将对标准模型的预测产生显著影响，可能抑制某些效应。这篇开放获取的文章由O. Cat`a、R.J. Crewther 和 Lewis C. Tunstall等人共同撰写，他们在不同的研究机构工作，包括德国锡根大学、澳大利亚阿德莱德大学和瑞士伯尔尼大学。文章在2019年7月19日接收，同年11月8日发表，为理解和探索电弱对称性破缺提供了新的视角和理论工具。

for θ

→ 0, where the dimensionless constant c

does not

depend on μ or M.

The result (32) must not be confused with hyperscaling

relations in mass-deformed theories [70–72], where the

conformal invariance of a gauge theory in the WW mode is

explicitly broken by a mass term −m

ψψ in the Lagrangian.

Hyperscaling is a property of the scaling WW mode. All

VEVs hOi

vac

and “hadron” masses M

inside

inside the

conformal window

scale with fractional powers of the

Lagrangian parameter m:

hOi

vac

∼ m

can

þγ

ðα

Þg=f1þγ

ðα

Þg

;

inside

∼ m

1=f1þγ

ðα

Þg

: ð33Þ

Not surprisingly, both hOi

vac

and M

inside

in Eq. (33)

vanish in the limit m → 0 of scale invariance. That is not

the case for Eq. (32) because, unlike m , M is not a variable

current fermion mass in a Lagrangian. Rather, M is a fixed

non-Lagrangian constant associated with a condensate,

such as a decay constant or a non-NG technihadron mass

arising from nonzero constituent masses of m ¼ 0 fermions

(Appendix A). The key property of a scaling NG-mode

fixed point is that amplitudes depend on the nonzero scales

M in the scale-invariant limit θ

→ 0 (Appendix D), so

unlike Eq. (33), the right-hand side of Eq. (32) does not

vanish in that limit. Furthermore, if we add −m

ψψ to the

Lagrangian, nonzero results such as Eq. (32) are corrected

by terms linear in m, as expected in chiral perturbation

theory or its chiral-scale extension [13–15]; fractional

powers of m are never seen. There is no such thing as

hyperscaling in crawling TC.

Now let us check what happens when amplitudes are

treated as functions of α; μ; fpg as α tends to α

at fixed μ.

Comparison with the β∂=∂α term in the CS equation (8)

shows that the right-hand side of Eq. (15) arises from the

singular α dependence of the condensate as α approaches

for fixed μ:

∂

∂α

hvacjOð0Þjvaci

∼

− α

hσjOð0Þjvaci

: ð34Þ

This singularity is to be expected.

The operator ∂=∂α

inserts O at zero-momentum transfer, so a pole due to a

zero-mass particle (the dilaton) coupled to O will produce a

singular result. Note that it is only at the fixed point that this

is allowed. A singularity or lack of smoothness in the α

dependence of any amplitude within the interval 0 < α <

would be a disaster: it would indicate a lack of

analyticity, such as a Landau pole, at a finite space-like

momentum.

Similarly, the fixed-μ limit α → α

applied to Eq. (31) is

singular:

M ∼ μðα

− αÞ

−1=β

fconstantg; fixed μ: ð35Þ

Note that this implies

∂

∂α

∼

ðα

− αÞβ

ð36Þ

and hence, from Eq. (32),

∂

∂α

hvacjOð0Þjvaci

∼

ðα

− αÞβ

hvacjOð0Þjvaci

;

ð37Þ

which shows that Eq. (34) is consistent with the soft-dilaton

theorem (17).

Equation (35) implies that, for M to remain finite in the

scaling limit θ

→ 0, μ tends to 0 according to the rule

μ ∝ ðα

− αÞ

1=β

: ð38Þ

Singularities are removed when the μ, α dependence of

amplitudes is eliminated in terms of physical quantities.

A simple example of μ dependence being related to a

soft-dilaton amplitude is when M in Eq. (31) is the mass

of a non-NG particle P. Then the scalar analogue

[51,74] of the Goldberger-Treiman relation (Fig. 2) applies:

∂

∂μ

¼ M

¼ F

σPP

: ð39Þ

We close this section with a discussion of the analogue of

the low-energy theorem (15) for QCD. There the relevant

FIG. 2. Generation of the mass M

of a non-NG particle P via

the dominant σ pole in hPjθ

jPi, where −g

σPP

PP defines the

σPP coupling. In the scale-invariant limit θ

→ 0, M

remains

nonzero.

In walking TC, there must be a phase transition at the sill of

the conformal window [73] that causes fermions to condense and

hence create NG and non-NG technihadrons outside the con-

formal window, but the distinction between spectra inside and

outside the window is usually left unclear. We reserve the term

“condensate” for VEVs which are nonzero in a symmetry limit

such as m → 0.

Perhaps this could be exploited in searches for NG-mode

fixed points on the lattice (Sec. VII).

This is simil ar to what happens in the large-N

limit of QCD,

where the singularity f

∼

ﬃﬃﬃﬃﬃﬃ

is eliminated by writing every-

thing in terms of the pion decay constant f

CAT

A, CREWTHER, and TUNSTALL PHYS. REV. D 100, 095007 (2019)

095007-8

equations have extra terms because quarks q have mass

≠ 0. When the trace anomaly [56–59] (with the vacuum

expectation value subtracted)



βðα

4α

μν

Aμν

þð1 þγ

ðα

ÞÞ



− fVEVg

ð40Þ

and the CS equation



∂

∂μ

þβðα

∂

∂α

− γ

ðα

∂

∂m

þγ

ðα



× hvacjOð0Þjvaci¼0 ð41Þ

are compared, we find



∂

∂μ

þ γ

ðα



hvacjOð0Þjvaci¼i lim

q→0

iq·x



vac





qqðxÞ − θ

ðxÞ



Oð0Þ



vac



subtr

: ð42Þ

If heavy quarks have been decoupled, and the limit m

→ 0 is taken for the light quarks q ¼ u, d, s as the IR fixed point is

approached, dilaton pole terms from both θ

and

qq may survive the limit [13–15]:



∂

∂μ

þ γ

ðα



hvacjOð0Þjvaci → f

hσjOð0Þjvaci



1 − ð 3 − γ

ðα

sIR

ÞÞðf







: ð43Þ

III. CRAWLING TC: HIDDEN

ELECTROWEAK-SCALE SYMMETRY

TC is based on the idea [76–78] that electroweak

symmetry “breaking” is the dynamical effect of a gauge

theory which resembles QCD but whose coupling becomes

strong at scales of a few TeV. The trigger for this effect

is a techniquark condensate h

ψψi

vac

≠ 0. The resulting

technipions become the longitudinal components of the



and Z

bosons, while the masses and couplings of the

other technihadrons are estimated by scaling up QCD

quantities, where the electroweak scale v ≃ 246 GeV plays

the role of the pion decay constant f

≃ 93 MeV.

An attractive feature of TC is that the hierarchy problem

is avoided: the mechanism for mass generation does not

rely on elementary Higgs-like scalars. Instead, masses are

generated dynamically through dimensional transmutation

[67], as in QCD.

When TC was invented, the Particle Data Group (PDG)

tables did not include QCD scalar J

¼ 0

þþ

resonances

below ≈1 GeV,

so for many years, it was thought, by

analogy with QCD, that TC scalar particles would not be

seen below the TeV scale.

There is now strong evidence for a light, broad 0

þþ

resonance f

ð500Þ in the QCD meson spectrum with mass

≃ 441 MeV [79–81] (evidence which seems to have

been mostly overlooked in the TC literature), and also for a

narrow Higgs boson h at m

≃ 125 GeV [82,83]. Given

these facts, can h be the TC version of the f

ð500Þ? At first

sight, the answer to this question is negative. An applica-

tion of the scaling rules mentioned above requires the TC

analogue f

of f

to have a large mass [84]

≈ ðv=f

Þm

¼ OðTeVÞ; ð44Þ

also, they seem to imply an OðTeVÞ width except for the

fact that the f

ð500Þ has plenty of phase space for its decay

into two pions, whereas there are no technipions for f

decay into and (for a mass of 125 GeV) no phase space for

it to decay into W

−

or Z

. But it is evident that this

estimate for the mass is much too large.

A convincing explanation for why the observed mass

≃ 125 MeV is so small relative to TeV scales is hard to

find. That is a key problem shared by all theories of

dynamical Higgs mass generation, including TC and its

extensions. The most promising strategy is to suppose that

the Higgs is a pseudo-NG (pNG) boson of a hidden

symmetry. Then the mass acquired by the pNG boson

due to explicit symmetry breaking is protected by the

underlying symmetry [85,86]. A light Higgs mass can arise

if explicit symmetry breaking is due to physics at the

electroweak scale and hence small relative to the scale of

dynamical symmetry breaking.

In composite Higgs models [87–91], where the hidden

symmetry is internal, this mechanism is well understood:

the Higgs boson and all would-be NG bosons are placed in

the same multiplet of an extended group such as SOð5Þ

[92–94]. For a recent review of these models, see chapter III

of Ref. [95].

The ϵð700Þ was excluded from the PDG tables in 1974. Its

successor f

ð500Þ was first mentioned in 1996, but became a

well-defined resonance only in the 2008 tables.

For consistency, the γ

terms in Eqs. (40) and (41) must

have opposite signs (unlike Ref. [14] where conventions

were changed during review). Here we choose the definition

¼ −μ∂ ln m

=∂μ [75]. Then

qq has dynamical dimension

3 − γ

ðα

sIR

Þ at a QCD fixed point α

sIR

, and similarly for

ψψ in

crawling TC, where the notation becomes γ

ðαÞ and α

CRAWLING TECHNICOLOR PHYS. REV. D 100, 095007 (2019)

095007-9

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红外固定点与爬行电弱对称破缺的新机制

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