跷跷板机制与Bogoliubov变换：自然性的新洞察

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"跷跷板机制的自然性和Bogoliubov变换" 本文是一篇发表在《Physics Letters B》上的开放获取文章，作者是Kazuo Fujikawa和Anca Tureanu，探讨了中微子质量跷跷板机制的新视角。在此观点下，中微子的小质量被视为两个大质量之间的差异，这一概念与Bogoliubov变换有关，而Bogoliubov变换在描述BCS理论时起着关键作用。作者特别关注了在单一风味模型中，当右旋中微子的质量项严重违反C对称性且具有良好的CP性质时，如何通过Bogoliubov变换解释Majorana中微子的自然性。中微子质量跷跷板机制是一种解释中微子质量非零但极小的理论框架。在标准模型中，中微子是无质量的，但在引入额外的重粒子（如右旋中微子）后，通过与轻左旋中微子的相互作用，可以产生微小的质量。跷跷板机制的核心在于，虽然重粒子有极大的质量，但通过相互作用，轻粒子（即观测到的中微子）的质量可以是这两个质量的差值，因此极其微小。 Bogoliubov变换在量子场论中广泛用于处理超导现象和凝聚态物理，它允许我们用无能隙的准粒子来描述原本有能隙的系统。在本文中，作者将这一变换应用到中微子质量跷跷板机制的Majorana中微子上，揭示了Majorana中微子自然出现的原因。Majorana中微子是一种自身的反粒子，它们的存在需要违反电荷共轭(C)和宇称(P)对称性，但这里讨论的是在违反C对称性下，同时保持CP对称性的情况。通过对具有右旋中微子质量mR在10^4至10^15 GeV范围内的典型模型进行分析，研究发现，要使跷跷板机制在自然设定下运作，即无量纲耦合常数保持小，这是非常重要的。自然性在这里指的是理论中的参数不需要精细调整就能得到观测到的现象。如果耦合常数过大，那么就需要人为地设定一些参数，使得中微子质量保持在极小的水平，这违反了自然法则。文章详细讨论了如何通过Bogoliubov变换理解和计算中微子的质量，并可能对中微子的CP违反性质、宇宙学以及粒子物理学的其他领域产生影响。通过这种新的视角，作者提供了理解中微子质量起源的一种新途径，这对于未来的中微子实验和理论研究有着重要的意义。

Physics Letters B 767 (2017) 199–204

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

Naturalness in see-saw mechanism and Bogoliubov transformation

Kazuo Fujikawa

a,b

, Anca Tureanu

a,∗

Department of Physics, University of Helsinki, P.O. Box 64, FIN-00014 Helsinki, Finland

Quantum Hadron Physics Laboratory, RIKEN Nishina Center, Wako 351-0198, Japan

a r t i c l e i n f o a b s t r a c t

Article history:

Received

4 January 2017

Accepted

29 January 2017

Available

online 1 February 2017

Editor: J.

Hisano

We present an alternative perspective on the see-saw mechanism for the neutrino mass, according to

which the small neutrino mass is given as a difference of two large masses. This view emerges when an

analogue of the Bogoliubov transformation is used to describe Majorana neutrinos in the Lagrangian of

the see-saw mechanism, which is analogous to the BCS theory. The Bogoliubov transformation clariﬁes

the natural appearance of Majorana fermions when C is strongly violated by the right-handed neutrino

mass term with good CP in the single ﬂavor model. Analyzing typical models with m

=10

to 10

GeV,

it is shown that a hitherto unrecognized ﬁne tuning of the order m

=10

−15

to 10

−26

is required to

make the commonly perceived see-saw mechanism work in a natural setting, namely, when none of the

dimensionless coupling constants are very small.

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

When one discusses the natural appearance of the observed

very small neutrino masses [1], one often refers to the see-saw

mechanism [2–4] the precise form of which depends on speciﬁc

models [5]. Those models are characterized by a very large mass

scale and thus the natural appearance of the tiny neutrino mass is

rather surprising. Naturalness is an esthetic notion and thus sub-

jective,

and it should ultimately be determined by experiments.

Currently active search for the support of the see-saw mechanism

in the form of Majorana neutrinos is going, and we expect that this

esthetical issue will be tested soon by experiments.

may also be appropriate to examine the naturalness of the

see-saw mechanism from a different perspective. We attempt to

understand the natural appearance of the eigenstates of charge

conjugation C, Majorana fermions, using an analogue of the Bo-

goliubov

transformation when C is strongly violated by the right-

handed

neutrino mass term which has good CP symmetry. We

then recognize that the tiny neutrino mass in the see-saw mecha-

nism

is given as a difference of two large masses, precise values of

which depend on models. This suggests a view different from the

conventional one, motivating us to ask whether the see-saw mech-

anism

is “natural” in the sense emphasized, for example, in [6,7].

We show that a hitherto unrecognized ﬁne tuning of the order

Corresponding author.

E-mail

address: anca.tureanu@helsinki.ﬁ (A. Tureanu).

is required to make the see-saw mechanism work in a nat-

ural

setting.

ﬁrst recapitulate the basic properties of Majorana fermions,

namely, charge conjugation and parity. The Majorana fermions are

deﬁned by the condition

ψ(x) = C

(x) =ψ

(x),

where C = iγ

stands for the charge conjugation matrix [8];

the quantity C

(x) is directly evaluated for a given ψ(x) while

(x) is evaluated by a unitary charge conjugation operator, and

the agreement of these two expressions provides an important

consistency check in our analysis (for example, of eq. (26) below).

We start with a generic neutral Dirac fermion, which is denoted by

ν(x) for later convenience, and deﬁne the combinations

(x) =

√

[ν(x) ±ν

(x)],

which satisfy

(x) =±ψ

(x),

showing that ψ

(x) and ψ

−

(x) are Majorana ﬁelds. We treat the

fermion with ψ

−

(x) =−ψ

−

(x) also as a Majorana fermion.

is well-known [8,9] that, in theories where the fermion num-

ber

is conserved, discrete symmetries such as parity can generally

be deﬁned with an arbitrary phase freedom δ,

ν(x) → e

iδ

ν(t, −



x).

http://dx.doi.org/10.1016/j.physletb.2017.01.069

0370-2693/

SCOAP

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