规范场的LSZ缩减：共振与非对角传播子研究

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本文主要探讨了在Landau规范下，具有精确和自发对称性的规范理论中，玻色子场的混合现象。焦点在于应用Lehmann-Symanzik-Zimmermann (LSZ) 减小技术，这是一种处理量子场论中粒子间相互作用的有效方法，特别是对于计算S矩阵元素中的散射过程。在规范场理论中，当存在不同规范标度之间相互耦合时，会涉及到混合现象，如Z0-光子和GZ-Majoron在标准模型和单线态Majoron模型中的混合。LSZ形式主义在此情境下显得尤为重要，它允许我们处理场的归一化条件和物理量的定义，确保理论在不同能量尺度下的可比性。文章的核心部分涉及对极点残差的分解，不仅限于实数p²（即能量的二次方），还扩展到了复数区域。这有助于理解粒子间的共振行为，特别是对于形成共振态的情况，这在解析散射振幅时至关重要。作者提供了一个实用的方法，用于寻找计算S矩阵所需的有效“平方根”残差，简化了实际计算的复杂性。此外，作者构造了渐近场的伪福克空间，这是一个关键的概念，它描绘了物理系统在无限远处的行为，是理解粒子状态及相互作用的基础。文章深入阐述了这个空间的BRST同调结构，这是量子场论中的一个核心概念，用于处理规范不变性和对称性的保真性。通过Slavnov-Taylor等价关系的分析，作者展示了这些通用结果在具体模型中的实用性。例如，通过分析标准模型中Z0-光子混合和单线态Majoron模型中的GZ-Majoron混合，展示了如何通过单循环图来应用这些理论工具。这篇文章提供了规范场中LSZ技术的具体应用，包括如何处理混合现象、计算残差、构造伪福克空间以及处理规范不变性问题，对于理解和计算高能物理中的基本粒子相互作用具有重要的实践指导意义。

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UNCORRECTED PROOF

JID:NUPHB AID:14419 /FLA [m1+; v1.287; Prn:13/08/2018; 10:41] P.8 (1-43)

8 A. Lewandowski / Nuclear Physics B ••• (••••) •••–•••

The most general form of the renormalized

1PI two-point function of vector ﬁelds is





μν

αβ

(−q,q) ≡−η

μν



1 −M

)



αβ

). (17)

The general form of the 1PI two-point function of scalar ﬁelds, which must be considered in

parallel to that of vector ﬁelds, if some of gauge symmetries are broken by VEVs of scalars,

is still given by (5). Even though the mixed vector-scalar two point function





αj

(−q, q) is, in

general, non-vanishing, the Landau gauge condition ensures that the mixed propagator vanishes



jβ

(q, −q) = 0. (18)

Thus, in addition to the propagator (6)of scalar ﬁelds, for practical purposes it sufﬁces to consider

the propagator of vector ﬁelds which takes (see Section 5) the form:



βδ

νρ

(q, −q) =−i



νρ

−







1 −M

)



−1



βδ

. (19)

Since the “denominator” of (19) has the same structure as that of (5), one can immediately write





1 −M

)



−1



βδ



V [λ

]

−m

V(λ)

V [λ

]

+[non-pole part] . (20)

The complete pole part of the full propagator (19) will be given in Section 5 (the formula (99)).

The formula (20)is however all one needs to write down those terms of the asymptotic vector

ﬁeld A

which are relevant for computing S-matrix amplitudes of processes with stable spin 1

particles in the initial and/or ﬁnal states:



V [λ

]

+... . (21)

As in (12)the prime over the ﬁrst sum indicates that it runs only over the indices λ labeling

poles at real values m

V(λ)

of q

. With each independent eigenvector ζ

V [λ

]

corresponding to

such a pole associated is in (21)a free Hermitian vector ﬁeld A

built out of the spin 1, mass

V(λ)

particle annihilation and creation operators acting in the Fock space of the in (or out)

states. The operator

has the unitarity gauge structure (if m

V(λ)

=0), or the Coulomb gauge

structure (if m

V(λ)

= 0). As in the case of the asymptotic ﬁeld (12), the one-particle states cre-

ated/annihilated by A

and by A



= A

are orthogonal to each other. The ellipsis in (21)

stand for free operators creating/annihilating in the in (or out) Fock space states belonging to

Kugo–Ojima quartet representations [20]; the explicit formulae for these operators are given in

Sec. 5.5. With all these operators taken into account the (Fourier transform of the) tw

o point

function 0|T(A

(x)A

(y))|0 reproduces the behavior of (19) near all poles located on the real

axis. Using the asymptotic ﬁeld (21)in the formula (13)for the S-operator

then shows that

the amplitude of a process with a stable spin 1 particle corresponding to A

in the initial or

ﬁnal state is obtained by contracting the amputated correlation functions A

α...

(p, ...) of ﬁelds

Recall that we allow for completely arbitrary renormalization conditions that are consistent with Slavnov–Taylor

identities, see e.g. [39].

The explicit form of the operator A

in our conventions is given is Sec. 5.5 (the formula (134)).

In our conventions, Eqs. (13)–(14)are valid in the generic case, provided that indices j and k run over all components

of all ﬁelds, including vectors, fermions, (anti)ghosts and Nakanishi–Lautrup multipliers, see e.g. [6].

UNCORRECTED PROOF

JID:NUPHB AID:14419 /FLA [m1+; v1.287; Prn:13/08/2018; 10:41] P.9 (1-43)

A. Lewandowski / Nuclear Physics B ••• (••••) •••–••• 9

with the eigenvector ζ

V [λ

]

and the appropriate (canonically normalized) polarization vector

(p, m

V(λ)

) or e

(p, m

V(λ)

)



In the presence of spontaneous breaking of some gauge

symmetries, it is also necessary to

identify those terms in the decomposition (12)of the asymptotic scalar ﬁeld which create/anni-

hilate physical states. This is particularly easy if there are no Goldstone bosons of spontaneously

broken global symmetries

as then all ﬁelds 



corresponding to m

S()

= 0 create would-be

Goldstone bosons while all remaining ﬁelds are associated with physical particles. If the true

Goldstone bosons are present (e.g. in the singlet Majoron model [34], see also Section 4.2), we

need a prescription for identifying massless eigenvectors ζ

S[

]

associated with them. The gauge

symmetry implies that the eigenvectors ζ

S[

]

corresponding to the would-be Goldstone bosons

are linear combinations of vectors

v. The orthogonality condition (9) then suggests that of

all vectors ζ

S[

]

associated with poles at p

= 0, to physical massless states should correspond

vectors ζ

S[

]

such that



S[

]



1 −M

2

(0)



v = 0 , (22)

for all indices α.

In Sec. 5.5 we will show that the states of the asymptotic Fock spaces asso-

ciated with eigenvectors ζ

S[

]

obeying this condition do indeed belong to the physical subspace

of the kernel of BRST charge.

It should be also stressed that the normalization condition, which for vectors ζ

S[]

takes the

form (9), has to be slightly modiﬁed in order to avoid spurious infrared divergences. Take, for

instance, the Z-photon block of the Standard Model (SM, see e.g. [1]); the 2-by-2 matrix M



(0)

(more precisely, its ZZ entry) is IR divergent at one-loop order, however the photonic singularity

is still a pole. The IR-ﬁniteness of the whole matrix M



(0) is therefore too strong a requirement.

In Sec. 5.3 we will show that Eq. (20) holds provided M

(s) is continuous at each m

V(λ)

and

that the limit

lim

→m

V(λ)





)ξ



, (23)

exists for each ξ belonging to the eigenspace

V(λ)

) associated with m

V(λ)

The vectors ζ

V [λ

]

appearing in (20)are then elements of a basis of the eigenspace

V(λ)

)ζ

V [λ

]

V(λ)

V [λ

]

, (24)

obeying the normalization conditions

lim

→m

V(λ)





V [λ

]



1 −M

2 

)



V [λ

]



=δ

. (25)

Furthermore, if massless spin 1 particles are present, an additional assumption is necessary to

ensure that the singularity of the full propagator (19)at q

=0is a (second order) pole: the limit

Of physical spin 0 particles, only Goldstone bosons can naturally be massless.

More precisely, this fact follows from the “non-renormalization theorem” expressed by the relation (87), which is a

manifestation of an additional symmetry of the action speciﬁc for the Landau gauge [22].

At the tree-level this reduces to a well-known condition (see e.g. (1.1) in [27]).

As before, we have to assume that each generalized eigenvector of M

V(λ)

) associated with the eigenvalue m

V(λ)

is an ordinary eigenvector.

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