E8超对称三元统一：层次结构、对称破缺与粒子光谱

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本文探讨了一种在超弦理论框架下对标准模型（SM）进行扩展的全新概念，即SU(3)-家族扩展的SUSY三元组（trinification）模型。在这个模型中，作者提出了将E8的最高级别8子群中的三个关键对称性——左右手SU(3)_L×SU(3)_R、颜色SU(3)_C和家族SU(3)_F——进行统一，这被视为超越SM标准框架的重要一步。这个模型的基础是对称性结构是基于三分规[SU(3)]_3和全局SU(3)_F。核心特点包括： 1. **统一性**：模型中，希格斯粒子和轻子的体系被统一处理，这是与SM的一个显著区别，暗示着可能存在的更深层次的物理原理。 2. **Yukawa相互作用**：模型引入了手性费米子的一般Yukawa相互作用，这可能解释了粒子的质量和生成机制，对于理解粒子物理学的底层规律至关重要。 3. **μ问题的解决**：通过模型设计，μ问题得到了自然的避免，这是一个在传统的SUSY理论中常见的问题，其解决对于模型的自洽性至关重要。 4. **规范耦合的统一**：模型预测了规范耦合的统一，这有助于预测更高的能量尺度下的物理现象，并可能揭示新的物理学规律。 5. **质子稳定性**：模型保证了质子在量子力学扰动理论中的稳定性，这对于保持宇宙稳定性和防止过快衰变具有重要意义。在最低的能量层面上，模型的基础结构是由E6的27plet以及受E8基本扇区启发的三个轨道和一个SU(3)八重态字节构成，这些构成了基础粒子的组合。这提供了理论上的基础来构建一个在低能有效理论中与SM相似的框架。文章还深入研究了模型中的对称性破裂机制，特别是在SUSY和轨道分割方面。多尺度轨道耦合随能量变化的规律，以及这种演化如何导致一个有效的低能量解决方案，是文章讨论的核心内容。通过对这些复杂的数学和物理过程的分析，研究人员希望能找到连接已知粒子世界和潜在新物理之间的桥梁。这篇论文为未来的粒子物理学研究提供了一个有前景的方向，特别是对于那些寻求突破标准模型框架并解决未解之谜的理论学家来说，它提出了一种有吸引力的可能性，即SU(3)-家族扩展的SUSY trinification模型，有望揭示出更深层次的宇宙物理法则。

LR parity also remains unbroken since v

¼ v



, which is

true as long as λ

is taken to be real.

By making the shift

L;R

→ Δ

L;R

ﬃﬃﬃ

; Δ

→ Δ

ﬃﬃﬃ

ð17Þ

and substituting μ

ﬃﬃ

, μ

ﬃﬃ

in the superpotential,

we obtain

W ⊃

B¼L;R



ﬃﬃﬃ



aa8

ﬃﬃﬃ



abc



ﬃﬃﬃ



aa8

ﬃﬃﬃ



abc

ﬃﬃﬃ

abc

þconst: ð18Þ

The quadratic terms in the superpotential vanish for

4;5;6;7

L;R;F

, since d

aa8

¼ −1=ð2

ﬃﬃﬃ

Þfor a ¼ 4, 5, 6, 7, meaning

that these fields receive no F -term contribution to their

masses (contrary to the other components of Δ

L;R

and

, which receive GUT-scale masses m

∼ λ

and

, respectively). While the global Goldstone bosons

Re½

4;5;6;7

 are present in the physical spectrum, the gauge

ones become the longitudinal polarization states of the

heavy gauge bosons related to the breaking G

333

→ G

32211

The presence of massless scalar degrees of freedom can

only be avoided in the extended model with the gauged

family symmetry. It is clear, however, that even in the case

of an approximately global SUð3Þ

with g

≪ g

there are

no massless Goldstones in the spectrum (provided that the

accidental symmetries are softly broken at low energies),

but a set of relatively light family gauge bosons very

weakly interacting with the rest of the spectrum.

By performing the shifts in Eq. (17) in the D-terms, we

obtain

⊃−if

abc

b †

→ −i

ﬃﬃﬃ

a8b

−

b †

− if

abc

b †

;

for B ¼ L; R, leading to the universal GUT-scale mass term

¼ 3g

=4 for the gauge adjoints Im½

4;5;6;7

L;R

, while

4;5;6;7

have no D-term contributions [or a small one in the

case of approximately global SUð3Þ

with g

≪ g

Hence, all components of the gauge adjoints and

1;2;3;8

receive masses of order GUT scale and are integrated out in

the low-energy EFT. The remaining

4;5;6;7

, on the other

hand, receive a much smaller mass from the soft SUSY-

breaking sector (and strongly suppressed D-terms) and stay

in the physical spectrum of the EFT. In what follows, we

shall denote by H

the superfields containing Im½

4;5;6;7

L;R



and by G

the superfields containing Re½

4;5;6;7

L;R

.

C. LR-symmetric SUSY theory

In this section we describe the details of the super-

symmetric theory left after the adjoint fields acquire VEVs.

As shown in the previous section, all components of the

gauge-adjoint chiral superfields receive masses of the order

of the GUT scale [OðvÞ] in the vacuum given by Eq. (17).

This means that to study the low-energy predictions of the

theory, we need to integrate out Δ

L;R;C

, as well as

components 1, 2, 3, and 8 of Δ

For the gauge sector of the SHUT model, h

L;R

naturally triggers an SUð3Þ

L;R

→ SUð2Þ

L;R

×Uð1Þ

L;R

breaking also for the tritriplets (whose interactions with

L;R

are mediated via V

L;R

gauge bosons). For the global

SUð3Þ

sector, there is no coupling of

to the tritriplets

and, thus, the SUð3Þ

symmetry remains intact (or approxi-

mate in the case of g

≪ g

) in the tritriplet sector, resulting

in G

32211f3g

rather than G

32211f21g

. Integrating out Δ

L;R;C

and components 1, 2, 3, and 8 of Δ

, therefore leaves us

with a supersymmetric theory based on the symmetry

group G

32211f3g

, with a chiral superfield content given

by Δ

4–7

and by the branching of L, Q

, and Q

Writing the trinification tritriplets in terms of G

32211f3g

representations, one gets

ðL

; ð19Þ

ðQ

¼ð

;

ðQ

¼ð

⊤i

; ð20Þ

where the vertical and horizontal lines denote the separation

of the original tritriplets into SU(2) doublets and singlets

after the first SSB step. We will refer to the lepton and

quark SUð2Þ

L;R

doublets as E

L;R

and q

L;R

. With this, we

find that the most general superpotential consistent with

32211f3g

W ¼ ε

ijk

þ y

ðH

ðq

þ y

ðE

ðq

þ y

ðE

ðq

g: ð21Þ

Note, in this effective SUSY LR theory one could naively

add a mass term like ε

˜μH

[which is symmetric under

SUð2Þ

×Uð1Þ

but not under full SUð3Þ

] between the

massless components of the family adjoint superfield H

and the massless superfield G

containing the Goldstone

bosons. Such an effective μ-term is matched to zero at tree

JOS

E E. CAMARGO-MOLINA et al. PHYS. REV. D 99, 035041 (2019)

035041-6

level at the GUT scale. Due to SUSY nonrenormalization

theorems [57], in the exact SUSY limit this term cannot be

regenerated radiatively at low energies, so ˜μ is identically

zero and was not included in the superpotential given by

Eq. (21). So, the resulting superpotential contains only

fundamental superfields coming from L, Q

, and Q

and is

indeed invariant under SUð3Þ

In the GUT-scale theory, a complex λ

would be the

only source of LR-parity violation. In the low-energy

theory this should lead to y

≠ y



. Otherwise, y

¼ y



and after the matching is performed we can always make

any y

1;2;3;4

real by field redefinitions. The same argument

applies for the equality of the corresponding LR gauge

couplings for SUð2Þ

L;R

×Uð1Þ

L;R

symmetries.

Since we now have an effective LR-symmetric SUSY

model with a Uð1Þ

L;R

symmetry, there is a possibility of

having gauge kinetic mixing. The Uð1Þ

L;R

D-term con-

tribution to the Lagrangian is given by

L ⊃

ðχD

þD

Þ− κðD

− D

ÞþX

þX

;

ð22Þ

where the terms proportional to κ are the Fayet-Iliopoulos

terms, while the D-terms and the expressions for X

L;R

are

shown in Appendix D3b.

The values of the parameters fy

1;2;3;4

L;R

; χ; κg

in the LR-symmetric SUSY theory are determined by the

values of the parameters fλ

; λ

;vg in the high-scale

trinification theory at the GUT-scale boundary through a

matching procedure.

Regarding the RG evolution of the

couplings, we note that the only dimensionful parameter in

the effective theory is the Fayet-Iliopoulos parameter κ.

This means that β

∝ κ so that if κ ¼ 0 at the matching

scale (which is true, at least, at tree level), then κ will

remain zero throughout the RG flow yielding no sponta-

neous SUSY breaking. Thus, we stick to the concept of soft

SUSY breaking in what follows.

IV. SOFTLY BROKEN SUSY

In this section we describe the details of adding

soft SUSY-breaking terms before the SHUT symmetry

is broken spontaneously by adjoint field VEVs. One

of the most important results is treated in Sec. IV B,

where it is shown that the symmetry breakings below

the GUT scale are triggered solely by the soft SUSY-

breaking sector. This, in turn, allows for a strong hierarchy

between the GUT scale and the scale of the following

VEVs.

A. The soft SUSY-breaking Lagrangian

The soft SUSY-breaking scalar potential terms respect-

ing the imposed G

333f3g

symmetry are bilinear and trilinear

interactions given by

soft





þ m

a



þ c:c



þ d

abc



a

þ c:c:



þ½A

ðT



þ A

ðT



þ c:c:þðZ

ðLRCÞ

permutationsÞ



þ½A

ijk

þ c:c:; ð23Þ

for the gauge adjoints and pure tritriplet terms, and

soft

¼ m

a



þ c:c



þ d

abc



a

þ c:c:



þ½A

ðT



þ c:c: þðZ

ðLRCÞ

permutationsÞ ð24Þ

for the family adjoint. All parameters here are assumed to

be real for simplicity. We note that, although trilinear terms

with the gauge singlets (such as



above) are not in

general soft, due to the family symmetry and the fact that

aab

¼ 0, the dangerous tadpole diagrams do indeed

cancel and do not lead to quadratic divergences.

The terms in Eqs. (23) and (24), which account for the most

general soft SUSY-breaking scalar potential consistent with

333f3g

and real parameters, also respect the accidental

Uð1Þ

×Uð1Þ

symmetry of the original SUSY theory.

Ho we v er, accidental LR parity is, in general, softly broken

as long as A

≠ A

, and this breaking can then be transmitted

to the other sectors of the effective theory radiatively (e.g., via

RG ev olution and radiativ e corrections at the matching scale).

The only dimensionful parameters entering in the tree-

level tritriplet masses come from soft SUSY-breaking

Before adding soft SUSY-breaking interactions, Δ

is com-

pletely decoupled from the fundamental sector when taking

SUð3Þ

to be global, meaning that λ

and v

do not enter in

the matching conditions .

SCALE HIERARCHIES, SYMMETRY BREAKING, AND PARTICLE … PHYS. REV. D 99, 035041 (2019)

035041-7

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