二维N=1 Lifshitz-Wess-Zumino模型：超对称与Lifshitz算子的和解关键

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本文主要探讨的是Lifshitz-Wess-Zumino模型，这是一种在物理学领域中结合了Lifshitz算子和超对称性的独特理论框架。Lifshitz模型通常涉及到在非平衡或非平凡流体动力学背景下，物理系统的动力学特征与空间时间维度之间的关系，其临界指数（z）描述了系统在接近相变时的行为。在这个三维Wess-Zumino模型中，作者提出了一种扩展，使得临界指数固定为z=2，这是通过引入一个关键的辅助场F来实现的。传统的Wess-Zumino模型可能存在时空混合导数的问题，这些问题在理论一致性上产生了困扰。然而，在本文中，通过施加加权重整化条件以及（N=1的）超对称性要求，这些难题得到了解决。Lifshitz算子在这个过程中起到了核心作用，它被巧妙地融入到理论的玻色子部分，从而确保了模型的自洽性和稳定性。为了验证这个新模型的正确性，作者采用Noether方法在规范形式下构建可疑代数，这有助于确保理论的可疑性质得到明确的数学支持。此外，模型在Lifshitz超空间中的表述，是对传统空间概念的一种自然扩展，进一步强调了模型的创新性和适用性。文章的核心部分还涉及对一环有效电势的计算，这一分析旨在研究对称性的破坏情况。结果表明，尽管超对称性在单环重整化过程中得以保留，但U(1)相位对称性却在临界值之上经历自发破缺，这揭示了模型中不同对称性行为的对比。在紫外(UV)行为方面，相比于相对论性的Wess-Zumino模型，通过在截止正则化方案内的重新规范化，Lifshitz-Wess-Zumino模型显示出更为改善的行为，这意味着理论在高能物理区域更加稳健和可预测。这篇论文提供了一个将Lifshitz算子与超对称性相结合的有效途径，不仅解决了理论中的问题，还揭示了新型量子场论结构的特点和可能的应用前景。这项工作对于理解非平衡物理现象，尤其是超对称在低维系统中的作用，具有重要的理论价值。

374 E.A. Gallegos / Physics Letters B 793 (2019) 372–382

δϕ =−ψ

δψ =

 F −i



∂

ϕ −ia



∂

ϕ (3)

δ F =−i



∂

ψ −ia



∂

ψ,

where  is a Grassmann x-independent parameter.

Before

proceeding with the construction of L

int

, it is necessary

to state clearly the weighted renormalization condition (wrc) [1,2].

If one writes L

int

of a given theory as L

int



, where g

la-

bel

the coupling constants and V

the interaction vertices, this con-

dition

simply says that the theory is renormalizable by weighted

power counting iff the weighted scaling dimension, the weight

for short,

[

]

of each coupling constant g

is greater or equal

to zero, i.e.,

[

]

≥ 0. Setting





=−z and





=−1,

the weight

[

]

of any object O is determined by enforcing

the action S to be weightless. In terms of the vertices, since the

Lagrangian L weighs d + z, where d denotes the spatial dimen-

sions,

the wrc asserts that a vertex V

is weighted renormalizable

iff

[

]

≤ d + z. Following the nomenclature adopted in the lit-

erature,

we call a vertex weighted marginal when this weighs

[

]

=d + z, weighted relevant when

[

]

< d + z and weighted

irrelevant when

[

]

> d + z .

should be noted that the weighted assignment in Lifshitz ﬁeld

theories is equivalent to demand the invariance of the action under

the following anisotropic scale transformations,

→ξ x

t → ξ

t, (4)

where z is the well-known critical exponent which measures the

degree of anisotropy between space and time. Moreover, notice

that the weighted scaling dimension coincides with the usual mass

one in natural units (

h = 1 = c) when z = 1.

construct L

int

we ﬁrst observe that any vertex V in it must

have the structure V ∼

(

∂

)

∂

, where N

represents the number of objects of the p type. Since the ﬁelds in

the action (2)for arbitrary z weigh

[

]

(

2 − z

)

[

]

= 1,

and

[

]

(

2 + z

)

2, one easily sees by imposing the wrc which

V is renormalizable by weighted power counting iff the condition

∂

(

2 − z

)

(

2 + z

)

≤2 + z, (5)

is satisﬁed. Here N

= N

+ N

, with X = ϕ, F , ψ . This condi-

tion

along with those which result from imposing the symmetry

requirements restrict strongly the form of L

int

. Note in particu-

lar

that the spatial rotational SO

(

)

symmetry of the Lagrangian

L demands a complete spinor/spatial index contraction as well as

that N

= N

=even number. The rigid phase U (1) symme-

try,

on the other hand, implies that N

= N

It should be noted that whether the polynomiality requirement of

the Lagrangian L,

[

]

≥0, were used, this states an upper bound

in the value of the critical z exponent, namely z = 2.

Hereafter,

we particularize the condition (5)to the case z = 2

that this becomes N

∂

+2N

≤ 4. Since N

does not ap-

pear

explicitly in this inequality, any number of scalar ϕ lines is

allowed in a given vertex V from the weighted renormalization

viewpoint. It should be noted also that the maximal number of

spatial derivatives, N

∂

= 4, can only occur in pure scalar ϕ ver-

tices:

V ∼ ∂

∂

ϕ∂

∂

ϕ, ∂

∂

ϕ∂

∂

ϕϕ, ....

The

presence of spatial derivatives in the interaction vertices,

in particular, in genuine ones (i.e. vertices with more than two

lines) made the problem of ﬁnding the most general interaction

Lagrangian L

int

extremely intricate in the component formalism.

Hence, for simplicity, we will seek L

int

with the following struc-

ture

int

ψ W

ψ +



ψ +

F +h.c.



+ U, (6)

where W

, V and U are functions of the scalar ﬁelds ϕ,

ϕ. Note

that due to the Hermiticity property of the action (2), W

, V and

U have to be real operators, whereas W

and W

complex ones.

Furthermore, it is worthwhile to note that in conformity with the

z =2wrc



∂

+2N

≤4



, W

can contain at most two spa-

tial

derivatives



∂



, U four spatial derivatives



∂



, and

V none



∂



. Since we are interested in adding higher spatial

derivatives in the kinetic part of (2), we shall look for expressions

of the form W

1,2

∼  +

1,2

(

ϕ,

)

, W

∼ 

(

ϕ or

)

(

ϕ,

)

and U ∼

ϕ

ϕ +

(

ϕ,

)

, where  =∂

∂

is the two-dimensional

spatial Laplace operator and where

U are functions only of

the scalar ﬁelds ϕ,

ϕ. With these considerations in mind, one can

easily compute the variation of L

int

under the susy transforma-

tions (3).

This can be cast in the form

δL

int

=ψ

F +



 + V ψ



F −iγ

(

∂

ϕ +∂

)

−

iaγ

(

∂

ϕ +∂

)



ψψ



∂

−∂



−

iV ψ



γ

ψ∂

ϕ +aγ

ψ∂



−

ψ∂

U + h.c., (7)

where we have introduced the notation  = W

−∂

and  =

− ∂

, and got rid of all surface terms. From this result, it

is clear that to respect susy one must demand that  =  = V =

U = 0. This in turn implies that

∂

∂ϕ

∂

. (8)

The operator W

, on the other hand, is fully determined by im-

posing

the reality condition of W

, ∂ W

/∂ϕ = ∂ W

/∂

ϕ, and the

fact that its weight according to the z = 2wrc is

[

]

= 2. This

means, as stated previously, that W

cannot contain more than

two spatial derivatives. After doing this, one gets

=mϕ +bϕ +

∞



p=1

p+1

. (9)

Notice, moreover, that susy excludes any possibility of introducing

explicitly four spatial derivatives



∂



in the off-shell formu-

lation

of the theory. Indeed, the desired Lifshitz scalar operator

ϕ

ϕ appears only in the bosonic sector of the theory after elim-

inating

the auxiliary F ﬁeld (see Eq. (12)). In the concluding part

of this section, we will conﬁrm these results and generalize them

in the Lifshitz superﬁeld formulation.

is extremely important to expose the consistency of our

model by setting explicitly up its superalgebra at classical level.

Hence, in the balance of this section, we construct the Noether

currents (and their respective charges) associated with each sym-

metry

of the model under consideration and then we set up its

superalgebra by using the canonical (anti-)commutation relations.

For

simplicity and without loss of generality, we truncate the

series in (9)at p = 1. So, the interaction Lagrangian L

int

of our

Lifshitz-Wess-Zumino (L-WZ) model, as we shall call it, reads

int

(

m +b +2g

ϕϕ

)



gϕ



mϕ +bϕ + gϕ



F +h.c.



(10)

Notice that setting a → 1 and b → 0this theory reduces to the

usual relativistic Wess-Zumino model [15,16].

The

L-WZ ﬁeld equations which result from the principle of

least action, δ S =0, are given by

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