0 引言

在标准模型(SM)中，人们提出Higgs机制来实现电弱对称性破缺.大型强子对撞机(LHC)实验结果表明，有一个质量约为125 GeV的Higgs粒子存在^[1-2].由于二次发散辐射修正的存在，为了使Higgs质量稳定在电弱标度, 需要极其精细的微调，这就导致了理论上的不自然性，LRTH模型可以解决此类问题.在LRTH模型^[3-6]中，整体对称群为U(4)×U(4), 规范子群为SU(2)_L×SU(2)_R×U(1)_B-L.在Higgs玻色子获得真空期望值后，整体对称群U(4)×U(4) 破缺成U(3)×U(3), 同时SU(2)_R×U(1)_B-L破缺成SM的U(1)_Y.随着LHC积累更多的数据^[7]以及未来国际线性对撞机(ILC)^[8]的建立，在现象学上对各种新物理模型进行计算，将成为连接理论与实验的重要一环.

费曼规则是各种计算的基础，但现有的LRTH模型的费曼规则是在幺正规范下给出的^[2]，不包含鬼粒子和规范固定项，这在某种程度上可以简化计算.但是在进行实际的唯象计算时发现，规范粒子传播子的分子中的动量项使得已经认可的圈图计算公式发散而没有意义.如在幺正规范下，W玻色子的传播子为$\frac{{{g_{\mu v}}-{p_\mu }{p_v}/m_W^2}}{{{p^2}-m_W^2}} $, 其中p、m_W分别为W粒子的动量和质量.可以看出，分子中后一项$\frac{{{p_\mu }{p_v}}}{{m_W^2\left( {{p^2}-m_W^2} \right)}} $中动量的幂次为零，当它在圈中时，还要乘以积分因子d⁴p, 当p外推到无穷大时，很容易造成发散项无法消除.如果可以引入任意规范下的费曼规则，就能把这一项改写为$\left( {1-\frac{1}{\xi }} \right)\frac{{{p_\mu }{p_v}/m_W^2}}{{{p^2}-m_W^2}} $，选取合适的ξ值就可以消去这一项.因此，本文在更普遍的R_ξ(ξ取任意值)规范下, 导出了LRTH模型的拉格朗日密度函数.

1 LRTH模型的拉格朗日密度

在量子场论中，拉格朗日密度决定着场的运动学和动力学.对于LRTH模型，其拉格朗日密度由Higgs场、规范玻色子场、费米子场、规范固定项、鬼场、汤川耦合项、一圈Coleman-Weinberg势和软对称破缺项μ组成.在R_ξ规范下，讨论以下四项：

$ \ell = {\ell _{\rm{G}}} + {\ell _{{\rm{GF}}}} + {\ell _{\rm{H}}} + {\ell _{{\rm{FPG}}}}, $

(1)

式中：${\ell _{\rm{G}}}、{\ell _{{\rm{GF}}}}、{\ell _{\rm{H}}} $和${\ell _{{\rm{FPG}}}} $分别为规范场、规范固定项、Higgs场以及鬼粒子的拉格朗日密度.

1.1 规范场的拉格朗日密度

规范场的拉格朗日密度^[2]为

$ {\ell _{\rm{G}}} = - \frac{1}{2}{\rm{tr}}{\left( {{F_{\mu \nu }}} \right)_L}{\left( {{F^{\mu \nu }}} \right)_L} - \frac{1}{2}{\rm{tr}}{\left( {{F_{\mu \nu }}} \right)_R}{\left( {{F^{\mu \nu }}} \right)_R} - \frac{1}{4}{\rm{tr}}{\left( {{F_{\mu \nu }}} \right)_{B - L}}{\left( {{F^{\mu \nu }}} \right)_{B - L}}, $

(2)

式中：(F_μν)_L，R和(F_μν)_B-L分别为SU(2)_{L, R}和U(1)_B-L的场强张量.F_μν^a=∂_μA_ν^a-∂_νA_μ^a-gf^abcA_μ^bA_ν^c.

在H和$ \boldsymbol{\hat H}$获得真空期望值后, 规范子群SU(2)_L×SU(2)_R×U(1)_B-L破缺到U(1)_EM.这样就产生了6个有质量的规范玻色子W^±、W_H^±、Z、Z_H, 以及1个无质量的光子γ.荷电规范玻色子的左右手态W_L^±和W_R^±没有混合，即

$ {W^ \pm } = W_L^ \pm ,W_H^ \pm = W_R^ \pm . $

(3)

中性规范玻色子Z_H、Z和γ则是W_L⁰、W_R⁰和W₁的线性组合：

$ \left( {\begin{array}{*{20}{c}} {{Z_H}}\\ Z\\ \gamma \end{array}} \right) = \mathit{\boldsymbol{U}}\left( {\begin{array}{*{20}{c}} {W_R^0}\\ {W_L^0}\\ {{W_1}} \end{array}} \right),\mathit{\boldsymbol{U}} \sim \left( {\begin{array}{*{20}{c}} {\frac{{\sqrt {\cos 2{\theta _W}} }}{{\cos {\theta _W}}}}&{\frac{{\sqrt {\cos 2{\theta _W}} {{\sin }^2}{\theta _W}}}{{{{\cos }^3}{\theta _W}}}\frac{{m_W^2}}{{m_{{W_H}}^2}}}&{ - \frac{{\sin {\theta _W}}}{{\cos {\theta _W}}}}\\ { - \frac{{{{\sin }^2}{\theta _W}}}{{\cos {\theta _W}}}}&{\cos {\theta _W}}&{ - \frac{{\sin {\theta _W}\sqrt {\cos 2{\theta _W}} }}{{\cos {\theta _W}}}}\\ {\sin {\theta _W}}&{\sin {\theta _W}}&{\sqrt {\cos 2{\theta _W}} } \end{array}} \right). $

(4)

把式(3) 和(4) 代入(2), 可得规范场的拉格朗日密度为

$ \begin{array}{l} {\ell _{\rm{G}}} = - \frac{1}{4}\left[ {{\partial _\mu }\left( {W_\nu ^ + + W_\nu ^ - } \right){\partial ^\mu }\left( {{W^{\nu + }} + {W^{\nu - }}} \right) - {\partial _\mu }\left( {W_\nu ^ + + W_\nu ^ - } \right){\partial ^\nu }\left( {{W^{\mu + }} + {W^{\mu - }}} \right) + } \right.\\ {\rm{i}}g{\partial _\mu }\left( {W_\nu ^ + + W_\nu ^ - } \right)\left( {{W^{\mu + }} - {W^{\mu - }}} \right)\left( {\frac{{\sqrt {\cos 2{\theta _W}} s_W^2}}{{c_W^3}}\frac{{m_W^2}}{{m_{{W_H}}^2}}Z_H^\nu + {c_W}{Z^\nu } + {s_W}{\gamma ^\nu }} \right) - \\ {\rm{i}}g{\partial _\mu }\left( {W_\nu ^ + + W_\nu ^ - } \right)\left( {\frac{{\sqrt {\cos 2{\theta _W}} s_W^2}}{{c_W^3}}\frac{{m_W^2}}{{m_{{W_H}}^2}}Z_H^\mu + {c_W}{Z^\mu } + {s_W}{\gamma ^\mu }} \right)\left( {{W^{\nu + }} - {W^{\nu - }}} \right) + \\ \frac{{{\rm{i}}g}}{2}\left( {W_\mu ^ + + W_\mu ^ - } \right){\partial ^\mu }\left( {{W^{\nu + }} - {W^{\nu - }}} \right)\left( {\frac{{\sqrt {\cos 2{\theta _W}} s_W^2}}{{c_W^3}}\frac{{m_W^2}}{{m_{{W_H}}^2}}Z_H^\nu + {c_W}{Z^\nu } + {s_W}{\gamma ^\nu }} \right) - \\ \frac{{{\rm{i}}g}}{2}\left( {W_\mu ^ + - W_\mu ^ - } \right)\left( {\frac{{\sqrt {\cos 2{\theta _W}} s_W^2}}{{c_W^3}}\frac{{m_W^2}}{{m_{{W_H}}^2}}Z_H^\nu + {c_W}{Z^\nu } + {s_W}{\gamma ^\nu }} \right){\partial ^\nu }\left( {{W^{\mu + }} - {W^{\mu - }}} \right) - \\ \left. {\frac{{{g^2}}}{2}\left( {W_\mu ^ + - W_\mu ^ - } \right)\left( {\frac{{\sqrt {\cos 2{\theta _W}} s_W^2}}{{c_W^3}}\frac{{m_W^2}}{{m_{{W_H}}^2}}Z_H^\nu + {c_W}{Z^\nu } + {s_W}{\gamma ^\nu }} \right){\partial ^\nu }\left( {{W^{\mu + }} - {W^{\mu - }}} \right)} \right] + \cdots \end{array} $

(5)

1.2 Higgs场的拉格朗日密度和规范固定项的形式

Higgs场的拉格朗日密度可表示为

$ {\ell _{\rm{H}}} = {\left( {{D_\mu }\mathit{\boldsymbol{H'}}} \right)^\dagger }{D^\mu }\mathit{\boldsymbol{H'}} + {\left( {{D_\mu }\mathit{\boldsymbol{\hat H'}}} \right)^\dagger }{D^\mu }\mathit{\boldsymbol{\hat H'}}, $

(6)

式中：D_μ=∂_μ-ig₂T_iA_μⁱ-ig₁T₇n_B-LB_μ(i=1, 2, …, 7), 其中g₁和g₂分别为规范群U(1)_B-L和SU(2)_{L, R}的耦合常数, n_B-L是群U(1)_B-L的荷, T_i是规范群SU(2)_L×SU(2)_R×U(1)_B-L的生成元, 可取${\boldsymbol{T}_k} = \left( {\begin{array}{*{20}{c}} {\frac{1}{2}{\boldsymbol{\sigma }_k}}&0 \\ 0&0 \end{array}} \right) $, ${\boldsymbol{T}_j} = \left( {\begin{array}{*{20}{c}} 0&0 \\ 0&{\frac{1}{2}{\boldsymbol{\sigma }_j}} \end{array}} \right) $, ${\boldsymbol{T}_7} = \frac{1}{2}\left( {\begin{array}{*{20}{c}} 1&0 \\ 0&1 \end{array}} \right) $，其中k=1, 2, 3, j=4, 5, 6, σ_i(i=1, 2, 3) 是泡利矩阵.

LRTH模型中有两种Higgs场^[2-5]:H′和$\boldsymbol{\hat H} $，H′=H+ν₁和$\boldsymbol{\hat H' = \hat H + }{\boldsymbol{\nu }_2} $, 其中ν₁、ν₂分别为场H、$\boldsymbol{\hat H} $的真空期望值.故Higgs场的拉格朗日密度可表示为

$ \begin{array}{l} {\ell _{\rm{H}}} = {\left[ {\left( {{\partial _\mu } - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i} - {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{\nu }}_1}} \right)} \right]^\dagger }\\ \left[ {\left( {{\partial _\mu } - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i - {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{\nu }}_1}} \right)} \right] + \\ {\left[ {\left( {{\partial _\mu } - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i} - {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{\nu }}_2}} \right)} \right]^\dagger }\\ \left[ {\left( {{\partial ^\mu } - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i - {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{\nu }}_2}} \right)} \right] = \\ {\partial _\mu }{\mathit{\boldsymbol{H}}^\dagger }{\partial ^\mu }\mathit{\boldsymbol{H}} + {\partial _\mu }{\mathit{\boldsymbol{H}}^\dagger }\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{\nu }}_1}} \right) + \\ {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{\nu }}_1}} \right)^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right) \times {\partial ^\mu }\mathit{\boldsymbol{H}} + \\ {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{\nu }}_1}} \right)^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\\ \left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{\nu }}_1}} \right) + \\ {\partial _\mu }{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\partial ^\mu }\mathit{\boldsymbol{\hat H}} + {\partial _\mu }{{\mathit{\boldsymbol{\hat H}}}^\dagger }\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{\nu }}_2}} \right) + \\ {\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{\nu }}_2}} \right)^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right) \times {\partial ^\mu }\mathit{\boldsymbol{\hat H}} + \\ {\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{\nu }}_2}} \right)^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\\ \left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{\nu }}_2}} \right). \end{array} $

(7)

其中包含有Higgs场和规范场的两点直接作用，这是不自然的非物理项：

$ \begin{array}{l} - {\partial _\mu }{\mathit{\boldsymbol{H}}^\dagger }\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right){\mathit{\boldsymbol{\nu }}_1} + \mathit{\boldsymbol{\nu }}_1^\dagger \left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right){\partial ^\mu }\mathit{\boldsymbol{H}} - \\ {\partial _\mu }{\mathit{\boldsymbol{H}}^\dagger }\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right){\mathit{\boldsymbol{\nu }}_2} + \mathit{\boldsymbol{\nu }}_2^\dagger \left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right){\partial ^\mu }\mathit{\boldsymbol{\hat H}}. \end{array} $

这些非物理项的出现主要是由于规范场A_μ、B_μ不是物理上直接可以观察到的场量，其含有的非物理自由度需要引入洛伦兹条件来限制.为了同时保证理论的规范性和协变性，人们采用了路径积分量子化方法，这样就可以通过引入合适的规范项消除掉由规范场引起的非物理项.选择如下规范项就可以消除这些非物理项：

$ {\ell _{{\rm{GF}}}} = \frac{1}{{2{\xi _1}}}{\left( {{F^i}\left[ {A_\mu ^i} \right]} \right)^2} + \frac{1}{{2{\xi _2}}}{\left( {F\left[ {{B_\mu }} \right]} \right)^2} + \frac{1}{{2{\xi _3}}}{\left( {{f^i}\left[ {A_\mu ^i} \right]} \right)^2} + \frac{1}{{2{\xi _4}}}{\left( {f\left[ {{B_\mu }} \right]} \right)^2}, $

(8)

式中：ξ₁、ξ₂、ξ₃、ξ₄是规范参数，可以取任意值.取ξ_i=1(i=1, 2, 3, 4) 时为费曼规范; 取ξ_i=∞(i=1, 2, 3, 4) 时为幺正规范.而式(8) 中的规范固定项的各项分别为

$ \begin{array}{l} {F^i}\left[ {A_\mu ^i} \right] = {\partial ^v}A_\mu ^i - {\rm{i}}{g_2}{\xi _1}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{H}}} \right);\\ F\left[ {{B_\mu }} \right] = {\partial ^v}{B_\mu } - {\rm{i}}{g_1}{\xi _2}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger \mathit{\boldsymbol{H}}} \right);\\ {f^i}\left[ {A_\mu ^i} \right] = {\partial ^v}A_\mu ^i - {\rm{i}}{g_2}{\xi _3}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2} - \mathit{\boldsymbol{\nu }}_2^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{\hat H}}} \right);\\ f\left[ {{B_\mu }} \right] = {\partial ^v}{B_\mu } - {\rm{i}}{g_1}{\xi _4}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{v}}_2} - \mathit{\boldsymbol{\nu }}_2^\dagger \mathit{\boldsymbol{\hat H}}} \right). \end{array} $

(9)

把式(9) 代入式(8)，可以看到规范固定项的一般形式为

$ \begin{array}{l} {\ell _{{\rm{GF}}}} = \frac{1}{{2{\xi _1}}}\left[ {{{\left( {{\partial ^v}A_\mu ^i} \right)}^2} - 2{\rm{i}}{g_2}{\xi _1}{\partial ^v}A_\mu ^i\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{H}}} \right)\\ - g_2^2\xi _1^2{{\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{H}}} \right)}^2}} \right] + \\ \frac{1}{{2{\xi _2}}}\left[ {{{\left( {{\partial ^v}{B_\mu }} \right)}^2} - 2{\rm{i}}{g_1}{\xi _2}{\partial ^v}{B_\mu }\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger \mathit{\boldsymbol{H}}} \right)\\ - g_1^2\xi _2^2{{\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger \mathit{\boldsymbol{H}}} \right)}^2}} \right] + \\ \frac{1}{{2{\xi _3}}}\left[ {{{\left( {{\partial ^v}A_\mu ^i} \right)}^2} - {\rm{i}}{g_2}{\xi _3}{\partial ^v}A_\mu ^i\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2} - \mathit{\boldsymbol{\nu }}_2^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{\hat H}}} \right)\\ - g_2^2\xi _3^2{{\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2} - \mathit{\boldsymbol{\nu }}_2^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{\hat H}}} \right)}^2}} \right] + \\ \frac{1}{{2{\xi _4}}}\left[ {{{\left( {{\partial ^v}{B_\mu }} \right)}^2} - {\rm{i}}{g_1}{\xi _4}{\partial ^v}{B_\mu }\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{v}}_2} - \mathit{\boldsymbol{\nu }}_2^\dagger \mathit{\boldsymbol{\hat H}}} \right)\\ - g_1^2\xi _4^2{{\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{v}}_2} - \mathit{\boldsymbol{\nu }}_2^\dagger \mathit{\boldsymbol{\hat H}}} \right)}^2}} \right]. \end{array} $

(10)

其中这些项:

$ \begin{array}{l} - {\rm{i}}{g_2}{\partial ^\mu }A_\mu ^i\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{H}}} \right) - {\rm{i}}{g_1}{\partial ^\mu }{B_\mu }\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger \mathit{\boldsymbol{H}}} \right) - \\ {\rm{i}}{g_2}{\partial ^v}A_\mu ^i\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2} - \mathit{\boldsymbol{\nu }}_1^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{\hat H}}} \right) - {\rm{i}}{g_1}{\partial ^\mu }{B_\mu }\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{v}}_1} - \mathit{\boldsymbol{\nu }}_1^\dagger \mathit{\boldsymbol{\hat H}}} \right) = \\ - {\rm{i}}{\mathit{\boldsymbol{H}}^\dagger }\left( {{g_2}{\mathit{\boldsymbol{T}}^i}{\partial ^\mu }A_\mu ^i + {g_1}{\partial ^\mu }{B_\mu }} \right){\mathit{\boldsymbol{v}}_1} + {\rm{i}}h_1^\dagger \left( {{g_2}{\mathit{\boldsymbol{T}}^i}{\partial ^\mu }A_\mu ^i + {g_1}{\partial ^\mu }{B_\mu }} \right)\mathit{\boldsymbol{H}} - \\ {\rm{i}}{{\mathit{\boldsymbol{\hat H}}}^\dagger }\left( {{g_2}{\mathit{\boldsymbol{T}}^i}{\partial ^\mu }A_\mu ^i + {g_1}{\partial ^\mu }{B_\mu }} \right){\mathit{\boldsymbol{v}}_1} + {\rm{i}}\mathit{\boldsymbol{v}}_1^\dagger \left( {{g_2}{\mathit{\boldsymbol{T}}^i}{\partial ^\mu }A_\mu ^i + {g_1}{\partial ^\mu }{B_\mu }} \right)\mathit{\boldsymbol{\hat H}}, \end{array} $

恰好可以抵消掉${\ell _{\rm{H}}} $中的非物理项.

消除非物理项的式(8) 所示的Higgs场的拉格朗日密度可分为以下四部分：

$ {\ell _{\rm{H}}} = {\ell _{{{\rm{H}}_1}}} + {\ell _{{{\rm{H}}_2}}} + {\ell _{{{\rm{H}}_3}}} + {\ell _{{{\rm{H}}_4}}}, $

(11)

其中：

$ \begin{array}{l} {\ell _{{{\rm{H}}_1}}} = {\partial _\mu }{\mathit{\boldsymbol{H}}^\dagger }{\partial ^\mu }\mathit{\boldsymbol{H}} + {\partial _\mu }{\mathit{\boldsymbol{H}}^\dagger }\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\mathit{\boldsymbol{H + }}\\ {\mathit{\boldsymbol{H}}^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right){\partial _\mu }\mathit{\boldsymbol{H}} + \\ {\mathit{\boldsymbol{H}}^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\mathit{\boldsymbol{H + }}\\ {\mathit{\boldsymbol{H}}^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right){\mathit{\boldsymbol{v}}_1} + \\ \mathit{\boldsymbol{v}}_1^\dagger \left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\mathit{\boldsymbol{H}},\\ {\ell _{{{\rm{H}}_2}}} = \mathit{\boldsymbol{v}}_1^\dagger \left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right){\mathit{\boldsymbol{v}}_1}, \end{array} $

(12)

$ \begin{array}{l} {\ell _{{{\rm{H}}_3}}} = {\partial _\mu }{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\partial ^\mu }\mathit{\boldsymbol{\hat H + }}{\partial _\mu }{{\mathit{\boldsymbol{\hat H}}}^\dagger }\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\mathit{\boldsymbol{\hat H + }}\\ {{\mathit{\boldsymbol{\hat H}}}^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right){\partial _\mu }\mathit{\boldsymbol{\hat H}} + \\ {{\mathit{\boldsymbol{\hat H}}}^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\mathit{\boldsymbol{\hat H}} + \\ {{\mathit{\boldsymbol{\hat H}}}^\dagger }\left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right){\mathit{\boldsymbol{v}}_1} + \\ \mathit{\boldsymbol{v}}_1^\dagger \left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right)\mathit{\boldsymbol{\hat H}}, \end{array} $

$ {\ell _{{{\rm{H}}_4}}} = \mathit{\boldsymbol{v}}_2^\dagger \left( {{\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_\mu ^i + {\rm{i}}{g_1}{n_{B - L}}{B_\mu }} \right)\left( { - {\rm{i}}{g_2}{\mathit{\boldsymbol{T}}_i}A_i^\mu - {\rm{i}}{g_1}{n_{B - L}}{B^\mu }} \right){\mathit{\boldsymbol{v}}_2}. $

LRTH引入的两个Higgs场H和$\boldsymbol{\hat H} $, 在全局对称性的变换分别为(4, 1) 和(1, 4), 变换后^[2]可以表示为

$ \mathit{\boldsymbol{H}} = \left( {\begin{array}{*{20}{c}} {{h_1} + {\rm{i}}{h_2}}\\ {{h_3} + {\rm{i}}{h_4}}\\ {{h_5} + {\rm{i}}{h_6}}\\ {{h_7} + {\rm{i}}{h_8}} \end{array}} \right),\;\;\;\;\mathit{\boldsymbol{\hat H = }}\left( {\begin{array}{*{20}{c}} {{{\hat h}_1} + {\rm{i}}{{\hat h}_2}}\\ {{{\hat h}_3} + {\rm{i}}{{\hat h}_4}}\\ {{{\hat h}_5} + {\rm{i}}{{\hat h}_6}}\\ {{{\hat h}_7} + {\rm{i}}{{\hat h}_8}} \end{array}} \right),\;\;\;{\mathit{\boldsymbol{v}}_1} = \left( {\begin{array}{*{20}{c}} 0\\ {{\rm{i}}f\sin x}\\ 0\\ {f\cos x} \end{array}} \right),\;\;{\mathit{\boldsymbol{v}}_2} = \left( {\begin{array}{*{20}{c}} 0\\ 0\\ 0\\ {\hat f} \end{array}} \right). $

(13)

在U(4)→U(3) 破缺以后, 包括7个Goldstone玻色子场的这两个Higgs场可以重新表示为

$ \mathit{\boldsymbol{H}} = {\rm{i}}\frac{{\sin \sqrt \chi }}{{\sqrt \chi }}{{\rm{e}}^{{\rm{i}}\frac{N}{{2f}}}}\left( {\begin{array}{*{20}{c}} {{h_1}}\\ {{h_2}}\\ C\\ {N - {\rm{i}}f\sqrt \chi \cot \sqrt \chi } \end{array}} \right),\;\;\mathit{\boldsymbol{\hat H = }}{\rm{i}}\frac{{\sin \sqrt {\hat \chi } }}{{\hat \chi }}{{\rm{e}}^{{\rm{i}}\frac{{\hat N}}{{2\hat f}}}}\left( {\begin{array}{*{20}{c}} {{{\hat h}_1}}\\ {{{\hat h}_2}}\\ {\hat C}\\ {\hat N - {\rm{i}}\hat f\sqrt {\hat \chi } \cot \sqrt {\hat \chi } } \end{array}} \right), $

(14)

其中$\chi = \left( {h_1^\dagger {h_1} + h_2^\dagger {h_2} + {C^*}C + {N^2}} \right)/{f^2} $, $\hat \chi $的形式与之类似.

正如文献[3]所给出的, 再重新参数化这些场以后，Higgs场剩下了8个有意义的物理场：一个中性的赝标量场φ⁰, 一对荷电标量场φ^±, SM物理的Higgs场h, 以及一个SU(2)_L二重态$\hat h = \left( {\hat h_1^ +, \hat h_2^0} \right) $.为得到这种情形下的费曼规则，只需要考虑采用Higgs场重新参数化后的表达形式.

1.3 鬼场的拉格朗日密度

鬼场的拉格朗日密度可表示为

$ \begin{array}{l} {\ell _{{\rm{FPG}}}} = \int {\left[ {c_i^\dagger \left( x \right){M^{ij}}\left( {x,y} \right){c_j}\left( y \right) + {c^\dagger }\left( x \right)M\left( {x,y} \right)c\left( y \right) + c_i^\dagger \left( x \right){M^i}\left( {x,y} \right)c\left( y \right) + } \right.} \\ {c^\dagger }\left( x \right){M^i}\left( {x,y} \right){c_i}\left( y \right) + c_i^\dagger \left( x \right){{\hat M}^{ij}}\left( {x,y} \right){c_j}\left( y \right) + {c^\dagger }\left( x \right)\hat M\left( {x,y} \right)c\left( y \right) + \\ \left. {c_i^\dagger \left( x \right){{\hat M}^i}\left( {x,y} \right)c\left( y \right) + {c^\dagger }\left( x \right){{\hat M}^i}\left( {x,y} \right){c_i}\left( y \right)} \right]{{\rm{d}}^4}x{{\rm{d}}^4}y, \end{array} $

(15)

式中：c_i(x)和c_i^†(x)是与规范场A_μⁱ对应的鬼场，而c(x)和c^†(x)则是与规范场B_μ相对应的规范场.群变换可表示为

$ \begin{array}{l} {M^{ij}}\left( {x,y} \right) = \frac{{\delta {F^i}\left[ {A_\mu ^i\left( x \right)} \right]}}{{\delta {\theta _j}\left( y \right)}};\\ {M^i}\left( {x,y} \right) = \frac{{\delta {F^i}\left[ {A_\mu ^i\left( x \right)} \right]}}{{\delta \theta \left( y \right)}} = \frac{{\delta F\left[ {{B_\mu }\left( x \right)} \right]}}{{\delta {\theta _i}\left( y \right)}};\\ M\left( {x,y} \right) = \frac{{\delta F\left[ {{B_\mu }\left( x \right)} \right]}}{{\delta \theta \left( y \right)}};\\ {{\hat M}^{ij}}\left( {x,y} \right) = \frac{{\delta {f^i}\left[ {A_\mu ^i\left( x \right)} \right]}}{{\delta {\theta _j}\left( y \right)}};\\ {{\hat M}^i}\left( {x,y} \right) = \frac{{\delta {f^i}\left[ {A_\mu ^i\left( x \right)} \right]}}{{\delta \theta \left( y \right)}} = \frac{{\delta f\left[ {{B_\mu }\left( x \right)} \right]}}{{\delta {\theta _i}\left( y \right)}};\\ \hat M\left( {x,y} \right) = \frac{{\delta f\left[ {{B_\mu }\left( x \right)} \right]}}{{\delta \theta \left( y \right)}}, \end{array} $

(16)

式中：θⁱ(x)和θ(x)分别为SU(2) 和U(1) 群的群参数.在F[A_μⁱ]、F[B_μ]、f[A_μⁱ]和f[B_μ]规范下SU(2)×U(1) 的场变换可由生成元方程得

$ A_\mu ^{iu} = {g_2}A_\mu ^i + {g_2}{\varepsilon ^{ijk}}A_\mu ^k - {\partial _\mu }{\theta ^i};\\ B_\mu ^u = {g_1}{B_\mu } - {\partial _\mu }\theta ;\\ {\mathit{\boldsymbol{H}}^u} = \mathit{\boldsymbol{H}} - {\rm{i}}\left( {{g_2}{\theta ^i}{\mathit{\boldsymbol{T}}^i} + {g_1}\theta } \right)\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right). $

(17)

将式(9) 和(17) 代入式(16)，对应可得

$ \begin{array}{l} {M^{ij}}\left( {x,y} \right) = - \left\{ {{\delta _{ij}}{\partial ^2} - {g_2}{\varepsilon _{ijk}}A_\mu ^k{\partial ^\mu } - g_2^2{\xi _1}\left[ {{{\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{T}}^j}{\mathit{\boldsymbol{v}}_1} + } \right.} \right.\\ \left. {\left. {\mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{T}}^j}\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)} \right]} \right\}{\delta ^4}\left( {x - y} \right);\\ {M^i}\left( {x,y} \right) = {g_2}{g_1}{\xi _2}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1} + \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{H}} + 2\mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1}} \right){\delta ^4}\left( {x - y} \right);\\ M\left( {x,y} \right) = - \left[ {{\partial ^2} - g_1^2{\xi _2}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{v}}_1} + \mathit{\boldsymbol{v}}_1^\dagger \mathit{\boldsymbol{H}} + 2\mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{v}}_1}} \right)} \right]{\delta ^4}\left( {x - y} \right);\\ {{\hat M}^{ij}}\left( {x,y} \right) = - \left\{ {{\delta _{ij}}{\partial ^2} - {g_2}{\varepsilon ^{ijk}}A_\mu ^k{\partial ^\mu } - g_2^2{\xi _3}\left[ {{{\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{v}}_2}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{T}}^j}{\mathit{\boldsymbol{v}}_2} - } \right.} \right.\\ \left. {\left. {\mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{T}}^j}\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{v}}_2}} \right)} \right]} \right\}{\delta ^4}\left( {x - y} \right);\\ {{\hat M}^i}\left( {x,y} \right) = {g_2}{g_1}{\xi _4}\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2} + \mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{\hat H}} + 2\mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2}} \right){\delta ^4}\left( {x - y} \right);\\ \hat M\left( {x,y} \right) = - \left[ {{\partial ^2} - g_1^2{\xi _4}\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{v}}_2} + \mathit{\boldsymbol{v}}_2^\dagger \mathit{\boldsymbol{H}} + 2\mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{v}}_2}} \right)} \right]{\delta ^4}\left( {x - y} \right). \end{array} $

(18)

因此, 鬼场的拉格朗日密度可表示为

$ \begin{array}{l} {\ell _{{\rm{FPG}}}} = c_i^\dagger \left\{ {{\delta _{ij}}{\partial ^2} - {g_2}{\varepsilon _{ijk}}A_\mu ^k{\partial ^\mu } - g_2^2{\xi _1}\\ \left[ {{{\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{T}}^j}{\mathit{\boldsymbol{v}}_1} + \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{T}}^j}\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)} \right]} \right\}{c_j} + \\ {c^\dagger }\left[ {{\partial ^2} - g_1^2{\xi _2}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{v}}_1} + \mathit{\boldsymbol{v}}_1^\dagger \mathit{\boldsymbol{H}} + 2\mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{v}}_1}} \right)} \right]\\ c - c_i^\dagger {g_2}{g_1}{\xi _2}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1} + \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{H}} + 2\mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1}} \right)c - \\ {c^\dagger }{g_2}{g_1}{\xi _2}\left( {{\mathit{\boldsymbol{H}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1} + \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{H}} + 2\mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_1}} \right){c_i} - \\ c_i^\dagger \left\{ {{\delta _{ij}}{\partial ^2} - {g_2}{\varepsilon ^{ijk}}A_\mu ^k{\partial ^\mu } - g_2^2{\xi _3}\left[ {{{\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{v}}_2}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{T}}^j}{\mathit{\boldsymbol{v}}_2} - } \right.} \right.\\ \left. {\left. {\mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{T}}^j}\left( {\mathit{\boldsymbol{\hat H}} + {\mathit{\boldsymbol{v}}_2}} \right)} \right]} \right\}\\ {c_j} + {c^\dagger }\left[ {{\partial ^2} - g_1^2{\xi _4}\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{v}}_2} + \mathit{\boldsymbol{v}}_2^\dagger \mathit{\boldsymbol{\hat H}} + 2\mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{v}}_2}} \right)} \right]c - \\ c_i^\dagger {g_2}{g_1}{\xi _4}\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2} + \mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{T}}^i}\mathit{\boldsymbol{\hat H}} + 2\mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2}} \right)\\ c - {c^\dagger }{g_2}{g_1}{\xi _4}\left( {{{\mathit{\boldsymbol{\hat H}}}^\dagger }{\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2} + \mathit{\boldsymbol{v}}_2^\dagger {{\mathit{\boldsymbol{\hat T}}}^i}\mathit{\boldsymbol{\hat H}} + 2\mathit{\boldsymbol{v}}_2^\dagger {\mathit{\boldsymbol{T}}^i}{\mathit{\boldsymbol{v}}_2}} \right){c_i}. \end{array} $

(19)

式(19) 中的每一项都可表示为${\ell _{gi}} $, 其中${\ell _{g1}} = c_i^\dagger {\delta _{ij}}{\partial ^2}{c_j} $, ${\ell _{g2}} =-g_2^2{\xi _1}c_i^\dagger {\left( {\boldsymbol{H} + {\boldsymbol{\nu }_1}} \right)^\dagger }{\boldsymbol{T}^i}{\boldsymbol{T}^j}{\boldsymbol{\nu }_1}{c_j} $, ${\ell _{g3}} =-{g_2}{\varepsilon ^{ijk}}c_i^\dagger A_\mu ^k{\partial ^\mu }{c_j} $, $ {\ell _{g4}} = {c^\dagger }{\partial ^2}c, \cdots $

把这些鬼场变化到质量本征态，有

$ {c_ + } = \frac{1}{{\sqrt 2 }}\left( {{c_1} - {\rm{i}}{c_2}} \right),{c_ - } = \frac{1}{{\sqrt 2 }}\left( {{c_1} + {\rm{i}}{c_2}} \right), $

$ {c_{{H_ + }}} = \frac{1}{{\sqrt 2 }}\left( {{c_4} - {\rm{i}}{c_5}} \right),{c_{{H_ - }}} = \frac{1}{{\sqrt 2 }}\left( {{c_4} + {\rm{i}}{c_5}} \right), $

$ {c_{{Z_H}}} = \frac{{\sqrt {\cos 2{\theta _W}} }}{{\cos {\theta _W}}}{c_6} + \frac{{\cos 2{\theta _W}{{\sin }^2}{\theta _W}}}{{{{\cos }^3}{\theta _W}}}\frac{{m_W^2}}{{m_{{W_H}}^2}}{c_3} - \frac{{\sin {\theta _W}}}{{\cos {\theta _W}}}c, $

$ {c_Z} = - \frac{{{{\sin }^2}{\theta _W}}}{{\cos {\theta _W}}}{c_6} + \cos {\theta _W}{c_3} - \frac{{\sin {\theta _W}\sqrt {\cos 2{\theta _W}} }}{{\cos {\theta _W}}}c, $

$ {c_\gamma } = \sin {\theta _W}{c_6} + \sin {\theta _W}{c_3} + \sqrt {\cos 2{\theta _W}} c. $

可得

$ \begin{array}{l} {\ell _{{g_1}}} = c_1^\dagger {\partial ^2}{c_1} + c_2^\dagger {\partial ^2}{c_2} + c_3^\dagger {\partial ^2}{c_3} + c_4^\dagger {\partial ^2}{c_4} + c_5^\dagger {\partial ^2}{c_5} + c_6^\dagger {\partial ^2}{c_6} = \\ {c_ - }{\partial ^2}{c_ + } + {c_ + }{\partial ^2}{c_ - } + \frac{{\cos 2{\theta _W}{{\sin }^2}{\theta _W}}}{{{{\cos }^3}{\theta _W}}}\frac{{m_W^2}}{{m_{{W_H}}^2}}c_{{Z_H}}^\dagger {\partial ^2}{c_{{Z_H}}} + \\ \frac{{\sqrt {\cos 2{\theta _W}} {{\sin }^2}{\theta _W}}}{{{{\cos }^2}{\theta _W}}}\frac{{m_W^2}}{{m_{{W_H}}^2}}c_{{Z_H}}^\dagger {\partial ^2}{c_Z} + \frac{{\sqrt {\cos 2{\theta _W}} {{\sin }^3}{\theta _W}}}{{{{\cos }^3}{\theta _W}}}\frac{{m_W^2}}{{m_{{W_H}}^2}}c_{{Z_H}}^\dagger {\partial ^2}{c_\gamma } + \\ \frac{{\sqrt {\cos 2{\theta _W}} {{\sin }^2}{\theta _W}}}{{{{\cos }^2}{\theta _W}}}\frac{{m_W^2}}{{m_{{W_H}}^2}}c_Z^\dagger {\partial ^2}{c_{{Z_H}}} + {\cos ^2}{\theta _W}c_Z^\dagger {\partial ^2}{c_{{Z_H}}} + \frac{1}{2}\sin 2{\theta _W}c_Z^\dagger {\partial ^2}{c_\gamma } + \\ \frac{{\sqrt {\cos 2{\theta _W}} {{\sin }^3}{\theta _W}}}{{{{\cos }^3}{\theta _W}}}\frac{{m_W^2}}{{m_{{W_H}}^2c_\gamma ^\dagger {\partial ^2}{c_{{Z_H}}}}} + \frac{1}{2}\sin 2{\theta _W}c_\gamma ^\dagger {\partial ^2}{c_{{Z_H}}} + {\sin ^2}{\theta _W}c_\gamma ^\dagger {\partial ^2}{c_\gamma } + {c_{{H_ - }}}{\partial ^2}{c_{{H_ + }}} + \\ {c_{{H_ + }}}{\partial ^2}{c_{{H_ - }}} + \frac{{\cos 2{\theta _W}}}{{{{\sin }^2}{\theta _W}}}c_{{Z_H}}^\dagger {\partial ^2}{c_{{Z_H}}} - \frac{{\sqrt {\cos 2{\theta _W}} {{\sin }^2}{\theta _W}}}{{{{\cos }^2}{\theta _W}}}c_{{Z_H}}^\dagger {\partial ^2}{c_Z} + \\ \sqrt {\cos 2{\theta _W}} \tan {\theta _W}c_{{Z_H}}^\dagger {\partial ^2}{c_\gamma } - {\tan ^2}{\theta _W}\sqrt {\cos 2{\theta _W}} c_Z^\dagger {\partial ^2}{c_{{Z_H}}} + {\tan ^2}{\theta _W}{\sin ^2}{\theta _W}c_{{Z_H}}^\dagger {\partial ^2}{c_Z} - \\ \frac{{{{\sin }^3}{\theta _W}}}{{\cos {\theta _W}}}c_Z^\dagger {\partial ^2}{c_\gamma } + \tan {\theta _W}\sqrt {\cos 2{\theta _W}} c_\gamma ^\dagger {\partial ^2}{c_{{Z_H}}} - \frac{{{{\sin }^3}{\theta _W}}}{{\cos {\theta _W}}}c_\gamma ^\dagger {\partial ^2}{c_Z} + {\sin ^2}{\theta _W}c_\gamma ^\dagger {\partial ^2}{c_\gamma }, \end{array} $

$ \begin{array}{l} {\ell _{{g_2}}} = - g_2^2{\xi _1}\left[ {c_1^\dagger {{\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^1}{\mathit{\boldsymbol{T}}^1}{\mathit{\boldsymbol{v}}_1}{c_1} + c_2^\dagger {{\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^2}{\mathit{\boldsymbol{T}}^2}{\mathit{\boldsymbol{v}}_1}{c_2} + c_3^\dagger {{\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^3}{\mathit{\boldsymbol{T}}^3}{\mathit{\boldsymbol{v}}_1}{c_3} + } \right.\\ c_4^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^4}{\mathit{\boldsymbol{T}}^4}{\mathit{\boldsymbol{v}}_1}{c_4} + c_5^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^5}{\mathit{\boldsymbol{T}}^5}{\mathit{\boldsymbol{v}}_1}{c_5} + c_6^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^6}{\mathit{\boldsymbol{T}}^6}{\mathit{\boldsymbol{v}}_1}{c_6} + \\ c_1^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^1}{\mathit{\boldsymbol{T}}^2}{\mathit{\boldsymbol{v}}_1}{c_2} + c_1^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^1}{\mathit{\boldsymbol{T}}^3}{\mathit{\boldsymbol{v}}_1}{c_3} + c_2^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^2}{\mathit{\boldsymbol{T}}^3}{\mathit{\boldsymbol{v}}_1}{c_3} + \\ c_2^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^2}{\mathit{\boldsymbol{T}}^1}{\mathit{\boldsymbol{v}}_1}{c_1} + c_3^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^3}{\mathit{\boldsymbol{T}}^1}{\mathit{\boldsymbol{v}}_1}{c_1} + c_3^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^3}{\mathit{\boldsymbol{T}}^2}{\mathit{\boldsymbol{v}}_1}{c_2} + \\ c_4^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^4}{\mathit{\boldsymbol{T}}^5}{\mathit{\boldsymbol{v}}_1}{c_5} + c_4^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^4}{\mathit{\boldsymbol{T}}^6}{\mathit{\boldsymbol{v}}_1}{c_6} + c_5^\dagger {\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)^\dagger }{\mathit{\boldsymbol{T}}^5}{\mathit{\boldsymbol{T}}^6}{\mathit{\boldsymbol{v}}_1}{c_6} + \\ \left. {c_5^\dagger {{\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^5}{\mathit{\boldsymbol{T}}^4}{\mathit{\boldsymbol{v}}_1}{c_4} + c_6^\dagger {{\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^6}{\mathit{\boldsymbol{T}}^4}{\mathit{\boldsymbol{v}}_1}{c_4} + c_6^\dagger {{\left( {\mathit{\boldsymbol{H}} + {\mathit{\boldsymbol{v}}_1}} \right)}^\dagger }{\mathit{\boldsymbol{T}}^6}{\mathit{\boldsymbol{T}}^5}{\mathit{\boldsymbol{v}}_1}{c_5}} \right], \end{array} $

$ \begin{array}{l} {\ell _{{g_3}}} = - {g_2}{\varepsilon ^{i,j,k}}c_i^\dagger A_\mu ^k{\partial ^\mu }{c_j} = \\ - {g_2}\left[ {{\varepsilon ^{1,2,3}}c_1^\dagger A_\mu ^3{\partial ^\mu }{c_2} + {\varepsilon ^{1,3,2}}c_1^\dagger A_\mu ^2{\partial ^\mu }{c_3} + {\varepsilon ^{2,1,3}}c_2^\dagger A_\mu ^3{\partial ^\mu }{c_1} + {\varepsilon ^{2,3,1}}c_2^\dagger A_\mu ^1{\partial ^\mu }{c_3} + } \right.\\ {\varepsilon ^{3,1,2}}c_3^\dagger A_\mu ^2{\partial ^\mu }{c_1} + {\varepsilon ^{3,2,1}}c_3^\dagger A_\mu ^1{\partial ^\mu }{c_2} + {\varepsilon ^{4,5,6}}c_4^\dagger A_\mu ^6{\partial ^\mu }{c_5} + {\varepsilon ^{4,6,5}}c_4^\dagger A_\mu ^5{\partial ^\mu }{c_6} + \\ \left. {{\varepsilon ^{5,4,6}}c_5^\dagger A_\mu ^6{\partial ^\mu }{c_4} + {\varepsilon ^{5,6,4}}c_5^\dagger A_\mu ^4{\partial ^\mu }{c_6} + {\varepsilon ^{6,4,5}}c_6^\dagger A_\mu ^5{\partial ^\mu }{c_4} + {\varepsilon ^{6,5,4}}c_6^\dagger A_\mu ^4{\partial ^\mu }{c_5}} \right], \end{array} $

$ {\ell _{{g_4}}} = {c^\dagger }{\partial ^2}c, $

$ \begin{array}{l} {\ell _{{g_5}}} = - 2{g_2}{g_1}{\xi _2}\\ \left[ {c_1^\dagger \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^1}{\mathit{\boldsymbol{v}}_1}c + c_2^\dagger \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^2}{\mathit{\boldsymbol{v}}_1}c + c_3^\dagger \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^3}{\mathit{\boldsymbol{v}}_1}c + c_4^\dagger \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^4}{\mathit{\boldsymbol{v}}_1}c + c_5^\dagger \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^5}{\mathit{\boldsymbol{v}}_1}c + c_6^\dagger \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^6}{\mathit{\boldsymbol{v}}_1}c} \right] = \\ - 2{g_2}{g_1}{\xi _2}\left[ {c_3^\dagger \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^3}{\mathit{\boldsymbol{v}}_1}c + c_6^\dagger \mathit{\boldsymbol{v}}_1^\dagger {\mathit{\boldsymbol{T}}^6}{\mathit{\boldsymbol{v}}_1}c} \right]. \end{array} $

有了上面的拉格朗日密度, 就可以得到LRTH模型中的费曼规则和耦合形式.从这些拉格朗日密度可以看出，鬼粒子的费曼规则也包含在推导之中，而文献[3]给出的费曼规则不包含鬼粒子，这是因为它采用幺正规范，不需要引入辅助的鬼粒子，这使得它的计算在某些程度上更简单.但是在进行实际的唯象计算时发现，规范粒子传播子的分子中的动量项使得已经认可的圈图计算公式发散而没有意义，所以有必要把这些费曼规则推广到任意规范下，然后在实际应用中就可以选取合适的规范使计算简单可行.

2 规范粒子在幺正规范、任意规范以及费曼规范下的传播子比较

通过前面的推导，可以直接写出三点、四点的顶角耦合形式.对于各个粒子的传播子函数，也可以经过简单计算给出.在幺正规范下，W玻色子的传播子为$\frac{{{g_{\mu \nu }}-{p_\mu }{p_\nu }/m_W^2}}{{{p^2}-m_W^2}} $, 而分子中后面的项使得常用的圈图公式失效，所以寻求其他规范来处理这一项.

下面推导出任意规范下规范场的传播子形式.把(5) 式的第一、第二项以及(10) 式的(∂^νA_μ)²(相应地，A_μ也和式(3) 和(4) 一样混合为6个规范玻色子W^±、W_H^±、Z、Z_H以及无质量的光子γ)结合起来给出了规范粒子的传播子项.如对荷正电的W传播子，拉格朗日密度函数中含两点W粒子的项可改写为$\frac{1}{2}{W_\mu }{K^{\mu \nu }}{W_\nu } $，其中${K^{\mu \nu }} = {g^{\mu \nu }}{\partial ^2}-\left( {1-\frac{1}{\xi }} \right){\partial ^\mu }{\partial ^\nu } $.

定义K^μν的逆函数D^μν，有

$ \int {{{\rm{d}}^4}z{K^{\mu \lambda }}\left( {x - z} \right){g^{\lambda \rho }}{D^{\rho v}}\left( {z - y} \right)} = {g^{\mu v}}{\delta ^4}\left( {x - y} \right), $

可求出

$ {D^{\mu v}}\left( x \right) = \int {\frac{{{{\rm{d}}^4}k}}{{{{\left( {2{\rm{ \mathsf{ π} }}} \right)}^4}}}\frac{{{{\rm{e}}^{ - {\rm{i}}kx}}}}{{{k^2} + {\rm{i}}\varepsilon }}\left[ { - {g^{\mu v}} + \left( {1 + \xi } \right)\frac{{{k^\mu }{k^v}}}{{{k^2} - {m^2}}}} \right]} . $

此即规范场W在坐标空间中的传播子，经过一个傅里叶变换，即可得出在任意规范下动量空间的传播子形式，即

$ {D^{\mu v}}\left( k \right) = \frac{{ - {g^{\mu v}} + \left( {1 + \xi } \right)\frac{{{k^\mu }{k^v}}}{{{m^2}}}}}{{{k^2} - {m^2} + {\rm{i}}\varepsilon }}. $

为了使圈图计算函数有效，可以选取费曼规范，令ξ=1，这样分子中后一项就消失了.

3 结论

新的Higgs场的性质与SM的Higgs的性质相同，这是由左右手分立对称性保证的，即LRTH模型.LRTH模型可以较好地解决电弱理论的小等级问题.由于此模型中现有的费曼规则会引起圈图计算工具失效，本文在此类模型中得到在任意规范下的费曼规则，给出了包括规范场、Higgs场、鬼场以及规范固定项等的拉格朗日密度函数，为便利计算提供了可能.