N $$ \mathcal{N} $$ = 4超Yang-Mills理论：LHC超空间中的非手性应力张量相关函数探究

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"N $$ \mathcal{N} $$ = 4超Yang-Mills理论在LHC超空间中的第二部分：非手性应力张量多重态的非手性相关函数" 这篇学术文章深入探讨了N $$ \mathcal{N} $$ = 4超Yang-Mills理论中的一个重要方面，即在Born近似下非手性应力张量多重态的多点超相关函数。N $$ \mathcal{N} $$ = 4超Yang-Mills理论是一种在量子场论中广泛研究的模型，它具有丰富的超对称性，可以提供对强相互作用的一种理论框架。该理论包含四个超对称生成元，因此被称为$$ \mathcal{N} $$ = 4。文章的主要焦点是非手性相关函数，这些函数是理论中关键的物理量，因为它们描述了不同场之间的相互作用。作者在此过程中推导出了一套有效的上标费曼规则，这些规则允许计算复杂的相关函数。一个有趣的发现是，通过简单地对时空变量进行格拉斯曼位移，可以从手性相关函数的费曼规则导出非手性相关函数的规则。这是一个意想不到的结果，因为它表明两个看似不同的概念之间存在内在联系。文章基于洛伦兹谐波手性（LHC）超空间的表述，这是该领域的创新方法。在LHC超空间中，超对称性的手性部分是直接显式的，而另一半则是通过非线性和非局部变换实现的。尽管如此，在Born近似下，只考虑这些变换的线性部分就足够了，这在理论的自我对偶领域中尤其适用。这种简化使计算变得更加可行，并且可以扩展到更广泛的超多重子集，如半BPS算符和Konishi多重子。文章的发布和接受日期表明了研究的最新性，其发表在JHEP03(2017)048上，由Springer为SISSA出版。作者Dmitry Chicherina和Emery Sokatchev分别来自法国萨瓦大学的LAPTH和CERN的理论物理部门，他们的工作提供了对高能物理学中复杂问题的深入见解。这篇研究工作不仅对N $$ \mathcal{N} $$ = 4超Yang-Mills理论的非手性相关函数有了新的理解，还提出了一种计算这些函数的有效方法，对于进一步理解和应用这一理论具有重要意义。同时，这种方法也可能为其他相关领域的研究提供有价值的工具和启示。

JHEP03(2017)048

This curvature is gauge covariant and is (covariantly) harmonic independent,

∇

= 0 . (2.26)

Finally, we quantize the action (2.9) in the light-cone gauge

− ˙α

˙α

= 0 , (2.27)

where the gauge-ﬁxing parameter ξ

− ˙α

belongs to a new set of LH variables, this time for

the second (anti-chiral) factor of the Euclidean Lorentz group SO(4) ∼ SU(2)

× SU(2)

These new harmonics satisfy the usual SU(2) conditions (recall (2.1))

+ ˙α

−

˙α

= 1 ,



+ ˙α



∗

= ξ

−

˙α

. (2.28)

With the help of the two sets of harmonics we can project the space-time coordinate x

˙αα

onto its light-cone components

±±

= ξ

˙α

˙αα

. (2.29)

After ﬁxing the gauge we can derive the propagators for the super-connections. They are

singular distributions in both the harmonic and coordinate spaces. We need the harmonic

delta function δ(u, v) deﬁned by the property

dv δ(u, v) f

(q)

(v) = f

(q)

(u) , (2.30)

with a test function (2.2). We also need the complex delta function δ

(t) ≡ δ(t,

t) satisfying

the relation

∂

= πδ

(t). More speciﬁcally, we use δ

) = δ(x

, x

−−

). The set of

propagators is as follows



x, θ

, u



(0, 0, u

)i =





δ (u

, u

) δ

(4)





(2.31)

˙α



x, θ

, u



(0, 0, u

)i =

−

˙α

−+





δ (u

, u

) δ

(4)





(2.32)

˙α



x, θ

, u



(0, 0, u

)i = 0 , (2.33)

where we have used translation invariance to set x

= θ

= 0. We do not specify if the

projections x

±±

(see (2.29)) and θ

are made with the harmonic u

or u

, in view of the

harmonic delta functions above. We also suppress the color structure δ

for the sake of

brevity. In the light-cone gauge (2.27) only the quadratic terms in S

(see (2.9), (2.10))

survive, and these propagators correspond to them. The bivalent vertex in S

(the n = 2

term in (2.11)) is treated as an ‘interaction’. The genuine interaction vertices come only

from the S

part of the action S

N =4

(the terms with n ≥ 3). The gauge superﬁeld A

˙α

non-interacting in the light-cone gauge and hence can appear only at the external points

of the LHC supergraphs.

It is called “axial” or “CSW” gauge in [12, 32].

– 9 –

JHEP03(2017)048

3 The stress-tensor supermultiplet

The N = 4 SYM stress-tensor multiplet is a short supermultiplet.

Its superconformal

primary is the half-BPS scalar operator O

AB;CD

= tr(φ

{AB

CD}

) (where {} denotes the

traceless part) made from the six physical scalars of the theory (recall (2.22)). It belongs

to the representation 20

of the R-symmetry group SU(4). The stress-tensor multiplet

is most naturally described in N = 4 harmonic (or R-analytic) superspace [27, 34, 35].

We introduce a new set of harmonics w

, w

−A

and their conjugates ¯w

−a

, ¯w

on the

R-symmetry group SU(4), projecting the index A = 1, . . . , 4 of the (anti)fundamental irrep

onto the subgroup SU(2) × SU(2)

× U(1) (indices a, a

, ±):



, w

−A



∈ SU(4) :

¯w

−b

= δ

, w

¯w

= 0 , w

−A

¯w

−b

= 0 , w

−A

¯w

= δ

; (3.1)

¯w

−a

+ w

−A

¯w

= δ

;



ABCD



−C



−D

= 1 . (3.2)

So far we have introduced three species of harmonics: chiral LHs u

(see (2.1)),

which are coordinates of the superspace and ‘harmonize’ the SU(2)

half of the Euclidean

Lorentz group; antichiral LHs ξ

˙α

(see (2.28)), which are auxiliary parameters needed to

ﬁx the gauge (2.27) and ‘harmonize’ the SU(2)

half of the Euclidean Lorentz group; RHs

, ¯w

) on the R-symmetry group SU(4), which are a convenient tool for dealing with

SU(4) tensors. In order to distinguish the RHs from the LHs u

we indicate the Lorentz

charge of u

(and of all projected quantities) upstairs, while the R-symmetry charge of the

RHs w

, ¯w

is shown downstairs. The antichiral LHs ξ

˙α

have a dot above the charge.

With the help of the RHs the half-BPS operator can be projected onto the highest-

weight state of U(1) charge (+4):

O =



)(w



AB;CD

= tr(φ

) . (3.3)

The harmonic projection automatically renders the tensor traceless and hence irreducible.

The short stress-tensor multiplet is annihilated by half of the N = 4 supercharges,

namely, by the RH projections Q

= ¯w

and

+ ˙α

= w

˙α

. Equivalently, it

satisﬁes the R-analyticity conditions

¯w

T = w

˙α

T = 0 . (3.4)

The supercovariant derivatives verify the N = 4 supersymmetry algebra {

˙α

, D

βB

} =

∂

β ˙α

. Consequently, the projected spinor derivatives in (3.4) anticommute,



˙α

, ¯w

βB



= 0 ,

so the two shortening conditions are compatible. To solve the constraints we decompose

the Grassmann variables with the help of the RHs,

+α

= w

, θ

−α

= w

−A

˙α

= ¯w

˙α

−a

= ¯w

−a

˙α

. (3.5)

This means that the top spin in the supermultiplet is 2 (the stress-energy tensor) unlike a generic long

multiplet with top spin 4 [33].

– 10 –

JHEP03(2017)048

Superﬁeld T M

˙α

)

( ˙α

β)

˙α

Dimension 2 5/2 3 3 7/2 4

RH charge +4 +3 +2 +2 +1 0

Table 1. Semi-superﬁeld content of the stress-tensor supermultiplet.

Further, we deﬁne the R-analytic basis

˙αα

= x

˙αα



αa

˙α

−a

− θ

αa

−

˙α



. (3.6)

In this basis

≡ ¯w

= ∂/∂θ

−

≡ w

= ∂/∂

−a

. (3.7)

Consequently, the stress-tensor multiplet satisfying the conditions (3.4) is described by an

R-analytic superﬁeld



, θ

, w



= O(x

, w) + . . . +







µν

(x) + . . . , (3.8)

depending only on half of the projected Grassmann coordinates. We emphasize that this

superﬁeld is not chiral. Among its components we ﬁnd the stress-energy tensor T

µν

(x). It

is an R-symmetry singlet, hence it does not depend on the RHs.

Let us now see how this multiplet can be reformulated in our LHC superspace. As

stated in section 2, our dynamical semi-superﬁelds A

, A

˙α

are chiral, i.e. they do not

depend on

θ. Therefore we need to expand T in the antichiral variables

T = T +

˙α





)





(

˙α

)

(

˙α

)





˙α





L, (3.9)

where

(

)

( ˙α

β)

˙α



, (

)

˙α



β ˙α

(

)

˙α

, (

)

+ ˙α

˙α

The ‘semi-superﬁeld’ components T , M, . . . ,

L are gauge invariant operators depending

only on θ

αa

. They are summarized in table 1, where their RH charge and scaling dimension

are indicated, and are depicted in ﬁgure 2. These are semi-superﬁelds in the sense that they

transform covariantly under Q supersymmetry, but

Q supersymmetry transforms them into

each other. The bottom components of each semi-superﬁeld are shown below (the color

More about the diﬀerent bases in superspace see in section 6.

Not to be confused with the L-analytic semi-superﬁelds A

(x, θ

, u), A

˙α

(x, θ

, u) in (2.7)!

– 11 –

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