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电荷共轭变换是联系粒子与其反粒子的变换, 费米子按电荷共轭变换性质分为两类, 狄拉克(Dirac)型和马约拉纳(Majorana)型. Majorana[1]于1937年预言电中性费米子可用实值波函数描述,其反粒子就是自身. 自中微子被发现后, 在理论和实验方面关于中微子可能是Majorana粒子的探索一直没停止过[2], 但迄今为止, 无论是理论还是实验方面都没有确定答案. 中微子是否是Majorana粒子对于解决中微子质量等级问题, 探索超出标准模型新物理都很重要.
无中微子的双β衰变是中微子为Majorana粒子的特征, 寻找无中微子双β衰变事例就成了甄别中微子费米子类型的粒子物理首选手段, 然而至今为止尚未发现这样的事例, 也无法否定其存在[3]. 因而寻求弱相互作用之外鉴别中微子费米子类型的机制研究一直没有停止过, 20世纪90年代中期开始出现用费米子对引力场的响应来分辨Dirac和Majorana费米子的研究[4-8]. 2006年Papini等[6]提出设想, 旋转引力源的Lense-Thirring效应会拖曳参考系标架场旋转, 其Dirac和Majorana费米子的自旋翻转矩阵元的空间依赖不同, 能够区分中微子类型. 但是该研究基于一次量子化波函数来处理Majorana粒子是有问题的, 满足电荷共轭变换不变的波函数不可能是自由粒子波函数, Papini等[6]讨论的波包[9]在天文尺度下传播时, 由于波包弥散, 其意义是不明确的. 由于电荷共轭不变的波函数不是确定能动量本征态的自由粒子态, Majorana粒子在一次量子化框架内无法自洽的描述, 处理Majorana粒子必须考虑到电荷共轭不变性, 只有在二次量子化框架内才能得到自洽的处理.
Lai和Xue[10]用二次量子化的框架研究了渐近平直的史瓦西(Schwarzschild)时空和带挠率Schwarzschild度规时空对自由Dirac费米子和Majorana费米子的量子散射, 发现Schwarzschild时空中的散射无法区分两种费米子, 而带矢量挠率的黎曼-嘉当(Riemann-Cartan)时空的散射能够区分Dirac和Majorana费米子. Majorana费米子由于其电荷共轭不变性, 挠率的矢量部分不会对Majorana费米子散射产生影响, 只有轴矢挠率会影响Majorana费米子的散射, 而Dirac费米子会同时受到矢量挠率和轴矢挠率的影响.
广义相对论是无挠的引力理论, 引力表现为黎曼时空的弯曲, 时空联络为Levi-Civita联络, 完全由度规确定. 但是自广义相对论提出后关于有挠引力的探讨从来没有停止过, 理论上关于挠率在引力理论中的角色有两种观点. 一种是将黎曼时空推广为带挠率的时空, 即Riemann-Cartan时空来描述引力. 另一种观点认为挠率提供了引力区别与时空曲率的另一种等价描述[11,12], 例如绝对平行引力中, 时空曲率为零, 引力用挠率描述. 文献[10]中关于挠率对于Majorana费米子和Dirac费米子的散射矩阵和散射截面的影响对于以上两种观点同样成立, 但是对于将挠率认为是引力等效描述的第二种观点, 意味着引力场对应的矢量挠率对费米子的散射可以区分Dirac费米子和Majorana费米子,这就为用中微子的引力量子散射效应来区分中微子的费米子类型提供了理论基础.
挠率按其在洛伦兹变换下的变换性质可分解为矢量、轴矢量和纯张量三部分, 挠率与费米场的耦合在Riemann-Cartan框架中, 最小耦合只有轴矢挠率与费米子存在耦合, 但是物质场的重整化效应使得非最小耦合具有普适性, 矢量挠率和轴矢量挠率都存在与旋量场的耦合. 在绝对平行引力框架中, 矢量挠率和轴矢挠率也都存在与旋量场的耦合, 挠率场对Dirac费米子与Majorana费米子散射效应的差异来自矢量挠率[10]. 可以将对挠率按洛伦兹变换性质进行的分解推广到无挠的自旋联络上, 与挠率类似Levi-Civita联络分解为三部分, 与旋量场发生耦合的是类似矢量挠率和轴矢量挠率的两部分. 本文在一般引力场的度规描述框架中, 分别计算了Dirac费米子和Majorana费米子对Levi-Civita联络中分解出的类似矢量挠率和轴矢量挠率的两部分的散射振幅, 发现由于Majorana费米子的电荷共轭对称性, 自旋联络的其中一部分对Majorana费米子的散射振幅没有贡献, 而对Dirac费米子的散射振幅有贡献. 将这个结果应用于旋转引力源产生的克尔(Kerr)度规场中, 发现Kerr时空对Majorana费米子和Dirac费米子的散射振幅不同, 尤其是自旋翻转散射矩阵元不同, 不同的散射振幅有不同的散射截面, 这个结果与绝对平行引力框架中, Kerr挠率场对Majorana费米子和Dirac费米子散射的差别预期是相互印证的. 并且当引力源的角动量为零, Kerr时空退化为Schwarzschild时空, 其对Majorana费米子和Dirac费米子的散射差异也消失, 印证Schwarzschild时空散射不能区分中微子的费米子类型的结论[10].
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Lai和Xue[10]用二次量子化的框架研究了自由Dirac费米子和Majorana费米子在渐近平直Schwarzschild时空中的量子散射. 在弯曲时空背景中, 费米子拉格朗日密度
$\mathcal{L}$ 为$ \mathcal{L} = \bar \psi \left[ {{\text{i}}{\gamma ^\mu }\left( x \right){D_\mu } - m} \right]\psi \text{, } $ 其中m为粒子的质量;
$\psi $ 是旋量粒子的二次量子化形式, Dirac费米子和Majorana费米子的二次量子化形式并不相同, 其具体形式会在后文给出;$\bar \psi = {\psi ^\dagger }{\gamma ^0}$ ;$ {\gamma ^\mu }\left( x \right) = {{\boldsymbol{\gamma}} ^a}{e_a}^\mu \left( x \right)\text{, } $ ${{\boldsymbol{\gamma}} ^a}$ 为伽马矩阵;${e_a}^\mu $ 为局域4标架场, 它在时空中的每一点将时空坐标和局域自由降落坐标系联系起来. 本文中希腊字母上下标表示时空指标, 如$\mu , \nu $ 等, 其取值为0, 1, 2, 3, 上下标可以使用时空度规${g_{\mu \nu }}$ 进行升降, 例如${\gamma _\mu } = {g_{\mu \nu }}{\gamma ^\nu }$ ; 本文中拉丁字母上下标表示局域自由降落坐标系的指标, 如a, b等, 其取值为0, 1, 2, 3, 上下标使用局域自由降落坐标系的度规, 即闵氏度规${\eta _{ab}} = {\text{diag}}( + 1, - 1, - 1, - 1)$ 进行升降, 例如${\gamma _a} = {\eta _{ab}}{\gamma ^b}$ . 另外本文采用爱因斯坦求和约定, 即相同的上下标表求和并略去求和符号, 例如${\gamma _a} = {\eta _{ab}}{\gamma ^b} = {\eta _{a0}}{\gamma ^0} + {\eta _{a1}}{\gamma ^1} + {\eta _{a2}}{\gamma ^2} + {\eta _{a3}}{\gamma ^3}$ ;$ {D_\mu }\psi = {\partial _\mu }\psi - \dfrac{{\text{i}}}{{\text{2}}}{A^{ab}}_\mu {S_{ab}}\psi \text{, } $ ${A^{ab}}_\mu $ 为洛伦兹联络,${S_{ab}}$ 为洛伦兹生成元的旋量表示,$ {S_{ab}} = \dfrac{1}{2}{\sigma _{ab}} = \dfrac{{\text{i}}}{{\text{4}}}[{\gamma _a},{\gamma _b}] . $ 在Riemann-Cartan空间中,
$ {D_\mu } = {\partial _\mu } - \dfrac{{\text{i}}}{4}{{\tilde A}^{ab}}_{\;\;\,\mu} {\sigma _{ab}} + {\text{i}}{\eta _1}{A_\mu }{\gamma _5} + {\eta _2}{V_\mu }\text{, } $ 式中,
${\tilde A^{ab}}_{\quad \mu }$ 为Levi-Civita联络; 轴矢挠率${A_\mu } = {g_{\mu \alpha }}{A^\alpha } = {1}/({{3!}}){g_{\mu \alpha }}{\varepsilon ^{\alpha \nu \rho \lambda }}{T_{\nu \rho \lambda }}$ , 其中$ {\varepsilon ^{\alpha \nu \rho \lambda }} $ 为四维Levi-Civita符号,${\varepsilon ^{0123}} = 1$ ; 矢量挠率${V_\mu } = {T^\nu }_{\nu \mu }$ ,${T^\rho }_{\mu \nu }$ 为挠率;${\eta _1}$ 和${\eta _2}$ 为挠率与旋量耦合常数, 在Riemann-Cartan时空挠率与旋量最小耦合情形${\eta _1} = - ({3}/{4})$ 和${\eta _2} = 0$ . 在弱场近似下, 一阶小量为$ {h_{\mu \nu }} = {g_{\mu \nu }} - {\eta _{\mu \nu }}. $ 在一阶近似下
$ \sqrt{-g}\cong 1-\dfrac{1}{2}{h}_{ \mu }^{\mu } \text{, } $ 其中g是度规的行列式. 4标架场可取为
$ {e_a}^\mu \cong {\delta _a}^\mu - \dfrac{1}{2}{h_a}^\mu . $ 费米子作用量
$S = \displaystyle\int {{{\text{d}}^4}x} \sqrt { - g} \mathcal{L}$ 展开到一阶为$ S={S}_{0}+{S}_{\mathrm{int}}={\displaystyle \displaystyle\int {\text{d}}^{4}x\bar{\psi }\left[\text{i}/ \partial -m-\dfrac{\text{i}}{\text{2}}{\gamma }^{a}{h}_{a}{}^{\mu }{\partial }_{\mu }+\dfrac{1}{4}{\gamma }^{a}{\varGamma }_{a}-\dfrac{1}{2}{h}_{\mu }{}^{\mu }\left(\text{i}/ \partial-m\right)\right]\psi } \text{, } $ 其中
$/ \partial={\gamma }^{a}{\delta }_{a}{}^{\mu }{\partial }_{\;\mu }$ ;${S_0}$ 为闵氏时空背景中的费米子作用量;${\varGamma _a} = {e_a}^\mu {A^{bc}}_\mu {\sigma _{bc}}$ ;${S_{{\rm int} }}$ 是弯曲时空背景对费米子作用导致的相互作用作用量,$ {S}_{\text{int}}={\displaystyle \displaystyle\int {\text{d}}^{\text{4}}x}\bar{\psi }\left[-\dfrac{\text{i}}{\text{2}}{\gamma }^{a}{h}_{a}{}^{\mu }{\partial }_{\mu }+\dfrac{1}{4}{\gamma }^{a}{\varGamma }_{a}-\dfrac{1}{2}{h}_{\mu }{}^{\mu }\left(\text{i}/ \partial-m\right)\right]\psi . $ 记初态
$\left| i \right\rangle $ 的粒子动量为${p_i}$ , 自旋为${s_i}$ ; 末态$\left| f \right\rangle $ 的粒子动量为$ {q_f} $ , 自旋为$ {m_f} $ ,$\left| i \right\rangle \to \left| f \right\rangle $ 的跃迁振幅为$ \left\langle {f\left| {\boldsymbol{S}} \right|i} \right\rangle = \left\langle {f\left| {T{\exp}\left[ { - {\text{i}}\displaystyle\int {{{\text{d}}^{\text{4}}}} x{\mathcal{H}_{\text{I}}}\left( x \right)} \right]} \right|i} \right\rangle = \left\langle {f\left| {T{\exp}\left[ {{\text{i}}\displaystyle\int {{{\text{d}}^4}} x{\mathcal{L}_{{\text{int}}}}\left( x \right)} \right]} \right|i} \right\rangle \text{, } $ 这里,
${\boldsymbol{S}}$ 为散射矩阵,$ {\mathcal{H}_{\text{I}}} $ 是相互作用哈密顿量, T表示编时乘积.对于无挠率的黎曼时空, 使用Dirac场算符的二次量子化形式
$ {\psi _{\text{D}}}\left( x \right) = \displaystyle\int {\dfrac{{{{\text{d}}^3}p}}{{{{\left( {2{\text{π}}} \right)}^3}\sqrt {2{E_p}} }}} \sum\limits_{s = 1,2} {\left[ {{a_{p,s}}{u^s}\left( p \right){{\text{e}}^{ - {\text{i}}px}} + b_{p,s}^\dagger {v^s}\left( p \right){{\rm e} ^{{\text{i}}px}}} \right]} \text{, } $ 其中,
${a_{p, s}}$ 和$b_{p, s}^\dagger$ 分别是粒子的湮灭算符和反粒子的产生算符;${u^s}\left( p \right)$ 和${v^s}\left( p \right)$ 分别为Dirac方程平面波正能解和负能解中的四分量旋量. 通过Majorana费米子场算符的二次量子化形式:$ {\psi _{\text{M}}}\left( x \right) = \displaystyle\int {\dfrac{{{{\text{d}}^3}p}}{{{{\left( {2{\text{π}}} \right)}^3}\sqrt {2{E_p}} }}} \sum\limits_{s = 1,2} {\left[ {{a_{p,s}}{u^s}\left( p \right){{\text{e}}^{ - {\text{i}}px}} + a_{p,s}^\dagger {v^s}\left( p \right){{\text{e}}^{{\text{i}}px}}} \right]} \text{, } $ 可以分别得到在引力场中的散射振幅最低阶:
$ {M_{\text{D}}} = {\text{i}}\left\langle {f\left| {{S_{{\text{int}}}}} \right|i} \right\rangle = 2{\text{πi}}\delta ({E_{{q_f}}} - {E_{{p_i}}}){\bar u^{{m_f}}}({q_f})\left[ {\dfrac{1}{4}{\gamma ^a}{\varGamma _a}(k) - \dfrac{1}{2}{h_a}^\mu (k){\gamma ^a}{{p_i}_\mu} } \right]{u^{{s_i}}}({p_i})\text{, } $ $ \begin{split} {M_{\text{M}}} =\;& {\text{πi}}\delta \left( {{E_{{q_f}}} - {E_{{p_i}}}} \right){{\bar u}^{{m_f}}}\left( {{q_f}} \right)\left[ \dfrac{1}{4}{\gamma ^a}{{{\varGamma }}_a}\left( k \right) - \dfrac{1}{2}{h_a}^\mu (k){\gamma ^a}{{p_{i}}_ \mu } \right]{u^{{s_i}}}\left( {{p_i}} \right) \\ &- {\text{πi}}\delta \left( {{E_{{q_f}}} - {E_{{p_i}}}} \right){{\bar v}^{{s_i}}}({p_i})\left[ {\dfrac{1}{4}{\gamma ^a}{\varGamma _a}(k) + \dfrac{1}{2}{h_a}^\mu (k){\gamma ^a}{q_f}_\mu } \right]{v^{{m_f}}}({q_f}),\quad \end{split} $ 其中,
${E_{{{p}_i}}} = \sqrt {{{\left| {{{\boldsymbol{p}}_i}} \right|}^2} + {m^2}}$ 是初态能量,${E_{{q_f}}}$ 为末态能量; 转移四动量$ k = {q_f} - {p_i} $ ;${{{\varGamma }}_a}\left( k \right)$ 和$ {h_a}^\mu (k) $ 分别为上文中${{{\varGamma }}_a}\left( x \right)$ 和$ {h_a}^\mu (x) $ 的傅里叶变换.对Schwarzschild引力场, 取空间各向同性广义坐标, 度规形式为
$ {\text{d}}{s^2} = \dfrac{{{{(1 - GM/2r)}^2}}}{{{{(1 + GM/2r)}^2}}}{\text{d}}{t^2} - {\left( {1 + \dfrac{{GM}}{{2r}}} \right)^4}\left( {{\text{d}}{r^2} + {r^2}{\text{d}}{\theta ^2} + {r^2}{{\sin }^2}\theta {\text{d}}{\varphi ^2}} \right) \text{, } $ 其中, G为引力常量, M为引力源质量, t为坐标时间, r为原点到某点的径向坐标. 取弱场一阶近似
$ {g_{\mu \nu }} = {\eta _{\mu \nu }} + {h_{\mu \nu }} = {\eta _{\mu \nu }} + 2\phi \left( r \right){\delta _{\mu \nu }} \text{, } $ 其中
$ \phi \left( r \right) = - GM/r $ . 对Dirac和Majorana粒子皆有$ {M}_{\text{D}}={M}_{\text{M}}=-2\pi \text{i}\delta ({E}_{{q}_{f}}-{E}_{{p}_{i}})\phi (k){\bar{u}}_{s}({q}_{f}){/ {\tilde{p}}}_{i}{u}_{r}({p}_{i}) \text{, } $ 其中,
$ {\tilde p_{i \mu}} = \left( {{p_{{i_0}}}, - {p_{{i_1}}}, - {p_{{i_2}}}, - {p_{{i_3}}}} \right) $ ,$ \phi (k) $ 为$ \phi (r) $ 的傅里叶变换.对具有Schwarzschild度规的Riemann-Cartan时空, 时空对Dirac费米子的散射振幅为
$ {M}_{\text{D}}^{\text{T}}=2\text{πi}\delta ({E}_{{q}_{f}}-{E}_{{p}_{i}}){\bar{u}}^{{m}_{f}}({q}_{f})\left[\dfrac{1}{4}{/ {\tilde{\varGamma}}}(k)+\text{i}{\eta }_{2}{/ {V}}(k)-{\eta }_{1}{/ {A}}(k){\gamma }_{5}-\dfrac{1}{2}{/ {\tilde{p}}}_{i}\right]{u}^{{s}_{i}}({p}_{i}) \text{, } $ 其中
${\;/ {\tilde{\varGamma}}}(k)={\gamma }^{a}{\tilde{\varGamma }}_{a}(k)$ ,$ {/ {V}}(k)={\gamma }^{a}{V}_{a}(k) $ ,${\;/ {A}}(k)={\gamma }^{a}{A}_{a}(k)$ ,${\tilde \varGamma _a}(k)$ ,$ {V_a}(k) $ ,$ {A_a}(k) $ 分别是${\tilde \varGamma _a}(x)$ ,$ {V_a}(x) $ ,$ {A_a}(x) $ 的傅里叶变换. 挠率的矢量部分会贡献一个矢量流耦合, 而轴矢量部分会贡献一个轴矢量流耦合. 同样, 可以写出Majorana费米子的散射振幅:$ \begin{split} {M}_{\text{M}}^{\text{T}}=\;&\text{πi}\delta ({E}_{{q}_{f}}-{E}_{{p}_{i}}){\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\left[\dfrac{1}{4}{/ {\tilde{\varGamma}}}(k)+{\rm i}{\eta }_{2}{/ {V}}(k)-{\eta }_{1}{/ {A}}(k){\gamma }_{5}-\dfrac{1}{2}{/ {\tilde{p}}}_{i}\right]{u}^{{s}_{i}}\left({p}_{i}\right)\\ &-\text{πi}\delta ({E}_{{q}_{f}}-{E}_{{p}_{i}}){\bar{v}}^{{s}_{i}}({p}_{i})\left[\dfrac{1}{4}{/ {\tilde{\varGamma}}}(k)+{\rm i}{\eta }_{2}{/ {V}}(k)-{\eta }_{1}{/ {A}}(k){\gamma }_{5}+\dfrac{1}{2}{/ {\tilde{q}}}_{f}\right]{v}^{{m}_{f}}({q}_{f})\\ =\;&\text{πi}\delta ({E}_{{q}_{f}}-{E}_{{p}_{i}}){\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\left[\dfrac{1}{4}{/ {\tilde{\varGamma}}}(k)-\dfrac{1}{2}{{h}_{a}}^{\mu }{\gamma }^{a}{({p}_{i})}_{\mu }\right]{u}^{{s}_{i}}\left({p}_{i}\right)\\ &-\text{πi}\delta ({E}_{{q}_{f}} -{E}_{{p}_{i}}) {\bar{v}}^{{s}_{i}}({p}_{i}) \left[\dfrac{1}{4}{/ {\tilde{\varGamma}}}(k) +\dfrac{1}{2}{{h}_{a}}^{\mu }{\gamma }^{a}{({q}_{f})}_{\mu }\right] {v}^{{m}_{f}}({q}_{f}) \\ &+2\text{πi}\delta ({E}_{{q}_{f}} -{E}_{{p}_{i}}){\bar{u}}^{{m}_{f}}\left({q}_{f}\right)(-{\eta }_{1}{\gamma }^{\mu }{A}_{\mu }{\gamma }^{5}){u}^{{s}_{i}}\left({p}_{i}\right), \end{split} $ 矢量挠率部分不会出现在Majorana费米子的散射振幅中, 只有轴矢挠率部分会对Majorana费米子散射振幅有贡献[10].
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文献[10]中对于挠率的分析采用了挠率对洛伦兹群不可约表示的分解, 将挠率分解为矢量部分和轴矢部分, 由于Majorana旋量的电荷共轭性质, 导致挠率的矢量部分在散射振幅中没有贡献. 我们发现如对无挠的自旋联络进行类似的分解, 则Majorana费米子和Dirac费米子在无挠引力中的散射振幅的形式也会有区别.
采用4标架场, 在弱引力背景中, 对费米子
$ \mathcal{L}=\bar{\psi }\left(\text{i}{\gamma }^{a}{e}_{a}^{ \mu }{D}_{\mu }-m\right)\psi =\bar{\psi }\left(\text{i}{\gamma }^{a}{n}_{a}^{ \mu }{\partial _\mu }-m+\dfrac{1}{2}{\gamma }^{a}{{A}^{bc}}_{a}{S}_{bc}-\dfrac{\text{i}}{\text{2}}{\gamma }^{a}{h}_{a}^{ \mu }{\partial _\mu }\right)\psi \text{, } $ 其中
${S_{bc}}$ 是旋量表示下的洛伦兹生成元,$ {S_{bc}} = \dfrac{{\text{i}}}{{\text{4}}}\left[ {{\gamma _b},{\gamma _c}} \right];$ ${n_a}^\mu$ 是平直时空的标架, 取笛卡尔坐标时标架可取为${\delta _a}^\mu$ , 使得$ {{e}_{a}}^{\mu }={{n}_{a}}^{\mu }-\dfrac{1}{2}{{h}_{a}}^{\mu } . $ 从(21)式中可以分离出相互作用的部分:
$ \mathcal{L}_{\text{int}}=\dfrac{1}{2}\bar{\psi }\left({\gamma }^{a}{{A}^{bc}}_{a}{S}_{bc}-\text{i}{\gamma }^{a}{h}_{a}^{ \mu }{\partial }_{\mu }\right)\psi =\dfrac{\text{i}}{\text{4}}\bar{\psi }{A}_{abc}\left(2{\eta }^{ca}{\gamma }^{b}-\text{i}{\varepsilon }^{dcab}{\gamma }_{d}{\gamma }^{5}\right)\psi -\dfrac{\text{i}}{\text{2}}\bar{\psi }{\gamma }^{a}{h}_{a}^{ \mu }{\partial }_{\mu }\psi , $ ${A_{abc}}$ 为Levi-Civita联络$ {A_{abc}} = {\eta _{ae}}{A^e}_{bc} = {\eta _{ae}}\left( f_{b\;\,c}^{\;e} + f_{c\;\,b}^{\;e} - {f^e}_{bc} \right) \text{, } $ 其中,
${f^c}_{ab}$ 是标架矢量${e_a} = {e_a}^\mu {\partial _\mu }$ 的结构系数, 满足$ \left[{e}_{a},{e}_{b}\right]={f^c}_{ab}{e}_{c}={e}_{a}{}^{\mu }{e}_{b}{}^{\nu }\left({\partial }_{\nu }{e}^{c}{}_{\mu }-{\partial }_{\mu }{e}^{c}{}_{\nu }\right){e}_{c} . $ 使用(26)式, 作用量中的联络部分可以写为
$ {A_{abc}}\left( {2{\eta ^{ca}}{\gamma ^b} - {\text{i}}{\varepsilon ^{dcab}}{\gamma _d}{\gamma ^5}} \right) = 4{e_a}^\mu {e_b}^\nu \left( {{\partial _\nu }{e^a}_\mu - {\partial _\mu }{e^a}_\nu } \right){\gamma ^b} - 2{\text{i}}{e_a}^\mu {e_c}^\nu {\partial _\nu }{e^b}_\mu {\varepsilon ^{dca}}_b{\gamma _d}{\gamma ^5} . $ 因此, 作用量的相互作用部分可以写为
$ {\mathcal{L}_{{\text{int}}}} = \bar \psi \left( {{\text{i}}{{K}_a}{\gamma ^a} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} - \dfrac{{\text{i}}}{{\text{2}}}{\gamma ^a}{h_a}^\mu {\partial _\mu }} \right)\psi \text{, } $ 其中,
$ {{K}_a} = {e_b}^\mu {e_a}^\nu \left( {{\partial _\nu }{e^b}_\mu - {\partial _\mu }{e^b}_\nu } \right) \text{, } $ $ {\varLambda _a} = {e_d}^\mu {e_c}^\nu {\partial _\nu }{e^b}_\mu {\varepsilon _{a\;\;\;b}^{\;\,cd}} . $ 借助Dirac场算符的自由粒子展开
$ {\psi _{\text{D}}}\left( x \right) = \displaystyle\int {\dfrac{{{{\text{d}}^3}p}}{{{{\left( {2{\text{π}}} \right)}^3}\sqrt {2{E_p}} }}} \sum\limits_{s = 1,2} {\left[ {{a_{p,s}}{u^s}\left( p \right){{\text{e}}^{ - {\text{i}}px}} + b_{p,s}^\dagger {v^s}\left( p \right){{\text{e}}^{{\text{i}}px}}} \right]} \text{, } $ Majorana场算符的自由粒子展开
$ {\psi _{\text{M}}}\left( x \right) = \displaystyle\int {\dfrac{{{{\text{d}}^3}p}}{{{{\left( {2{\text{π}}} \right)}^3}\sqrt {2{E_p}} }}} \sum\limits_{s = 1,2} {\left[ {{a_{p,s}}{u^s}\left( p \right){{\text{e}}^{ - {\text{i}}px}} + a_{p,s}^\dagger{v^s}\left( p \right){{\text{e}}^{{\text{i}}px}}} \right]} \text{, } $ 以及初末态的归一化
$ \left| i \right\rangle = \sqrt {2{E_{{p_i}}}} a_{{p_i},{s_i}}^{\dagger} \left| 0 \right\rangle ,\quad \left| f \right\rangle = \sqrt {2{E_{{q_f}}}} a_{{q_f},{m_f}}^{\dagger} \left| 0 \right\rangle \text{, } $ 得到Dirac费米子的微扰论最低阶散射振幅为
$ \begin{split} {M_{\text{D}}} =\;& {\text{i}}\left\langle {f\left| {{S_{{\text{int}}}}} \right|i} \right\rangle = {\text{i}} \langle {f| {\displaystyle\int {{{\text{d}}^4}x} \sqrt { - g} \bar \psi \left( {{\text{i}}{{K}_a}{\gamma ^a} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} - \dfrac{{\text{i}}}{{\text{2}}}{\gamma ^a}{h_a}^\mu {\partial _\mu }} \right)\psi } |i} \rangle \\ =\;& {\text{i}}\displaystyle\int {{{\text{d}}^4}x} \sqrt { - g} {{\bar u}^{{m_f}}}\left( {{q_f}} \right)\left( {{\text{i}}{{K}_a}{\gamma ^a} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} - \dfrac{1}{2}{\gamma ^a}{h_a}^\mu {p_{i \mu}} } \right){u^{{s_i}}}\left( {{p_i}} \right){{\text{e}}^{{\text{i}}\left( {{q_f} - {p_i}} \right)x}}, \end{split}$ 而Majorana费米子的微扰论最低阶散射振幅为
$\begin{split} {M_{\text{M}}} =\;& {\text{i}}\left\langle {f\left| {{S_{{\text{int}}}}} \right|i} \right\rangle = \dfrac{{\text{i}}}{{\text{2}}}\langle {f| {\displaystyle\int {{{\text{d}}^4}x} \sqrt { -g} {{\bar \psi }_{\rm{M}}}\left( {{\text{i}}{{K}_a}{\gamma ^a} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} - \dfrac{{\text{i}}}{{\text{2}}}{\gamma ^a}{h_a}^\mu {\partial _\mu }} \right){\psi _{\rm{M}}}} |i} \rangle \\ =\;& \dfrac{{\text{i}}}{2}\displaystyle\int {{{\text{d}}^4}x} \sqrt { - g} {{\bar u}^{{m_f}}}\left( {{q_f}} \right)\left( {{\text{i}}{{K}_a}{\gamma ^a} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} - \dfrac{1}{2}{\gamma ^a}{h_a}^\mu {p_{i \mu}} } \right){u^{{s_i}}}\left( {{p_i}} \right){{\text{e}}^{ - {\text{i}}\left( {{p_i} - {q_f}} \right)x}} \\ &- \dfrac{{\text{i}}}{{\text{2}}}\displaystyle\int {{{\text{d}}^4}x} \sqrt { - g} {{\bar v}^{{s_i}}}\left( {{p_i}} \right)\left( {{\text{i}}{{K}_a}{\gamma ^a} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} + \dfrac{1}{2}{\gamma ^a}{h_a}^\mu {q_f}_\mu } \right){v^{{m_f}}}\left( {{q_f}} \right){{\text{e}}^{{\text{i}}\left( {{q_f} - {p_i}} \right)x}}. \end{split} $ 其中
${E_p} = \sqrt {{{\left| {\boldsymbol{p}} \right|}^2} + {m^2}}$ ,${p_{i\mu }}$ 为初态粒子四动量,${q_{f\mu }}$ 为末态粒子四动量. 借助Majorana旋量的电荷共轭性质,${v_s}\left( k \right) = u_s^c\left( k \right) = C{\bar u^T}\left( k \right)$ , (35)式第二项为$ \begin{split} &\int {{{\text{d}}^4}x} \sqrt { - g} {{\bar v}^{{s_i}}}\left( {{p_i}} \right)\left( {{\text{i}}{{K}_a}{\gamma ^a} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} + \dfrac{1}{2}{\gamma ^a}{h_a}^\mu {q_{f\mu}} } \right){v^{{m_f}}}\left( {{q_f}} \right){{\text{e}}^{{\text{i}}\left( {{q_f} - {p_i}} \right)x}} \\ = \;&\int {{{\text{d}}^4}x} \sqrt { - g} {u^{{s_i}}}^T\left( {{p_i}} \right)C\left( {{\text{i}}{{K}_a}{\gamma ^a} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} + \dfrac{1}{2}{\gamma ^a}{h_a}^\mu {q_{f\mu}} } \right)C{{\bar u}^{{m_f}T}}\left( {{q_f}} \right){{\text{e}}^{{\text{i}}\left( {{q_f} - {p_i}} \right)x}} \\ = \;&\int {{{\text{d}}^4}x} \sqrt { - g} {u^{{s_i}}}^T\left( {{p_i}} \right)\left( {{\text{i}}{{K}_a}{\gamma ^{aT}} + \dfrac{1}{2}{\varLambda _a}{\gamma ^a}^T{\gamma ^{5T}} + \dfrac{1}{2}{\gamma ^{aT}}{h_a}^\mu {q_{f\mu}} } \right){{\bar u}^{{m_f}T}}\left( {{q_f}} \right){{\text{e}}^{{\text{i}}\left( {{q_f} - {p_i}} \right)x}} \\ =\;& \int {{{\text{d}}^4}x} \sqrt { - g} {{\bar u}^{{m_f}}}\left( {{q_f}} \right)\left( {{\text{i}}{{K}_a}{\gamma ^a} - \dfrac{1}{2}{\varLambda _a}{\gamma ^a}{\gamma ^5} + \dfrac{1}{2}{\gamma ^a}{h_a}^\mu {q_{f\mu}} } \right){u^{{s_i}}}\left( {{p_i}} \right){{\text{e}}^{{\text{i}}\left( {{q_f} - {p_i}} \right)x}}, \end{split} $ 把其代入Majorana费米子的散射矩阵(35)式可得
$ \begin{split} {M}_{\text{M}}=\;&\text{i}\langle f\left|{S}_{\text{int}}\right|i\rangle \\ =\;&\dfrac{\text{i}}{2}{\displaystyle \displaystyle\int {\text{d}}^{\text{4}}x}\sqrt{-g}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\left(\text{i}{K_a}{\gamma }^{a}+\dfrac{1}{2}{\varLambda }_{a}{\gamma }^{a}{\gamma }^{5}-\dfrac{1}{2}{\gamma }^{a}{h}_{a}{}^{\mu }{p}_{i}{}_{\mu }\right){u}^{{s}_{i}}\left({p}_{i}\right){\text{e}}^{-\text{i}\left({p}_{i}-{q}_{f}\right)x}\\ &-\dfrac{\text{i}}{2}{\displaystyle \displaystyle\int {\text{d}}^{\text{4}}x}\sqrt{-g}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\left(\text{i}{K_a}{\gamma }^{a}-\dfrac{1}{2}{\varLambda }_{a}{\gamma }^{a}{\gamma }^{5}+\dfrac{1}{2}{\gamma }^{a}{h}_{a}{}^{\mu }{q}_{f}{}_{\mu }\right){u}^{{s}_{i}}\left({p}_{i}\right){\text{e}}^{\text{i}\left({q}_{f}-{p}_{i}\right)x}\\ =\;&\dfrac{\text{i}}{2}{\displaystyle \displaystyle\int {\text{d}}^{\text{4}}x}\sqrt{-g}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\left({\varLambda }_{a}{\gamma }^{a}{\gamma }^{5}-\dfrac{1}{2}{\gamma }^{a}{h}_{a}{}^{\mu }{p}_{i}{}_{\mu }-\dfrac{1}{2}{\gamma }^{a}{h}_{a}{}^{\mu }{q}_{f}{}_{\mu }\right){u}^{{s}_{i}}\left({p}_{i}\right){\text{e}}^{-\text{i}\left({p}_{i}-{q}_{f}\right)x}. \end{split} $ 对比两种费米子的散射振幅可以得到
$ {M_{\text{D}}} = {M_{\text{M}}} + {\text{i}}\displaystyle\int {{{\text{d}}^4}x} \sqrt { - g} {\bar u^{{m_f}}}\left( {{q_f}} \right)\left( {{\text{i}}{{K}_a}{\gamma ^a} - \dfrac{1}{2}{\gamma ^a}{h_a}^\mu {q_f}_\mu } \right){u^{{s_i}}}\left( {{p_i}} \right){{\text{e}}^{ - {\text{i}}\left( {{p_i} - {q_f}} \right)x}}. $ 显然, 一般度规场对Dirac粒子和Majorana粒子的散射振幅是有差别的, 这个差别在于某些特殊的度规给不出非零的结果, 例如在文献[10]中的Schwarzschild度规场情形, 但一般而言可以期望对称性较低的度规场对于Dirac粒子和Majorana粒子的散射是不同的; 并且由(38)式, 这种散射行为的差别是自旋极化依赖的, 尤其是Dirac粒子和Majorana粒子自旋翻转散射矩阵元的一般形式完全不同, 可以据此计算其在具体度规场中的空间依赖形式的差别, 作为引力场散射鉴别费米子类型的依据.
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Kerr度规是由带角动量的旋转引力源产生的引力场度规, 在宇宙空间较具普适性, 小到天体、大至星系, 星系团等其远离源处的时空都可以近似用Kerr度规描述. 具体讨论Kerr度规的时空对Majorana费米子和Dirac费米子的散射振幅对于用引力散射鉴别费米子类型的研究具有特别的意义.
取具有轴对称的广义坐标, Kerr度规可写成
$\begin{split} & {\text{d}}{s^2} = {g_{\mu \nu }}{\text{d}}{x^\mu }{\text{d}}{x^\nu } \\ =\;& {g_{00}}{\text{d}}{t^2} + 2{g_{03}}{\text{d}}t{\text{d}}\varphi + {g_{11}}{\text{d}}{r^2} + {g_{22}}{\text{d}}{\theta ^2} + {g_{33}}{\text{d}}{\varphi ^2} \text{, } \end{split} $ 其中,
$ {g_{00}} = 1 - {{2GMr}}/{{{\rho ^2}}} \text{, } $ $ {g_{03}} = {g_{30}} = \dfrac{{2GMar}}{{{\rho ^2}}}{\sin ^2}\theta \text{, } $ $ {g_{11}} = - {{{\rho ^2}}}/{\varDelta } \text{, } $ $ {g_{22}} = - {\rho ^2} \text{, } $ $ {g_{33}} = - \dfrac{{{{\left( {{r^2} + {a^2}} \right)}^2} - {a^2}\varDelta {{\sin }^2}\theta }}{{{\rho ^2}}}{\sin ^2}\theta . $ 其中,
$a = {J}/{M}$ , J是引力源的自转角动量. 这里定义$ {\rho ^2}(r,\theta ) \equiv {r^2} + {a^2}{\cos ^2}\theta \text{, } $ $ \varDelta (r) \equiv {r^2} + {a^2} - 2GMr . $ 相应的Kerr渐近闵氏4标架场为
$ {{\boldsymbol{e}}^a}_\mu \equiv \left( {\begin{array}{*{20}{c}} {{\gamma _{00}}}&0&0&\eta \\ 0&{{\gamma _{11}}{\text{s}}\theta {\text{c}}\varphi }&{{\gamma _{22}}{\text{c}}\theta {\text{c}}\varphi }&{ - \beta {\text{s}}\varphi } \\ 0&{{\gamma _{11}}{\text{s}}\theta {\text{s}}\varphi }&{{\gamma _{22}}{\text{c}}\theta {\text{s}}\varphi }&{\beta {\text{c}}\varphi } \\ 0&{{\gamma _{11}}{\text{c}}\theta }&{ - {\gamma _{22}}{\text{s}}\theta }&0 \end{array}} \right)\text{, } $ 其逆矩阵为
$ {\boldsymbol e}_{a}{}^{\mu } \equiv \left( \begin{array}{cccc}{\gamma }_{00}^{-1}& 0& 0& 0\\ -\beta {g}^{03} \text{s}\varphi & {\gamma }_{11}^{-1}\text{s}\theta \text{c}\varphi & {\gamma }_{22}^{-1}\text{c}\theta \text{c}\varphi & -{\beta }^{-1}\text{s}\varphi \\ \beta {g}^{03}\text{c}\varphi & {\gamma }_{11}^{-1}\text{s}\theta \text{s}\varphi & {\gamma }_{22}^{-1}\text{c}\theta \text{s}\varphi & {\beta }^{-1}\text{c}\varphi \\ 0& {\gamma }_{11}^{-1}\text{c}\theta & -{\gamma }_{22}^{-1}\text{s}\theta & 0\end{array} \right) . $ 这里定义
$ {\beta ^2} = {\eta ^2} - {g_{33}}{, }\quad \eta = {{{g_{03}}} \mathord{\left/ {\vphantom {{{g_{03}}} {{\gamma _{00}}}}} \right. } {{\gamma _{00}}}}\text{, } $ $ {\gamma _{00}} = \sqrt {{g_{00}}} ,\quad {\gamma _{ii}} = \sqrt { - {g_{ii}}} \text{, } $ 以及简略记号
$ {\text{s}}\theta = \sin \theta ,~{\text{c}}\theta = \cos \theta ,~ {\text{s}}\varphi = \sin \varphi ,~ {\text{c}}\varphi = \cos \varphi . $ 在
$r$ 很大的远场情况, 度规(39)式可近似为$\begin{split} \;& {\text{d}}{s^2} =\left( {1 - \dfrac{{2GM}}{r}} \right){\text{d}}{t^2} + \dfrac{{4GMa}}{r}{\sin ^2}\theta {\text{d}}t{\text{d}}\varphi \\ & ~~- \left( {1 + \dfrac{{2GM}}{r}} \right){\text{d}}{r^2} - {r^2}{\text{d}}{\theta ^2} - {r^2}{\sin ^2}\theta {\text{d}}{\varphi ^2} \text{, } \end{split} $ 以及
$ {\gamma _{00}} \sim 1 - \dfrac{{GM}}{r},\;\;{\gamma _{11}} \sim 1 + \dfrac{{GM}}{r},{\mkern 1mu} \;\;\beta \sim \left( {1 + \dfrac{{2{G^2}{M^2}{a^2}{{\sin }^2}\theta }}{{{r^4}}}} \right)r\sin \theta ,\;\;\eta \sim \dfrac{{2GMa}}{r}{\sin ^2}\theta . $ 将4标架场远场近似形式代入Dirac费米子的散射振幅(34)式和(37)式中, 第一项为
$ \begin{split} {K_a}{\gamma }^{a}=\;&{\gamma }^{1}\left(\dfrac{6{G}^{2}{M}^{2}{a}^{2}}{{r}^{5}}\mathrm{cos}2\theta -\dfrac{GM}{{r}^{2}}\right)\mathrm{sin}\theta \mathrm{cos}\varphi +{\gamma }^{2}\left(\dfrac{6{G}^{2}{M}^{2}{a}^{2}\mathrm{cos}2\theta }{{r}^{5}}-\dfrac{GM}{{r}^{2}}\right)\mathrm{sin}\theta \mathrm{sin}\varphi \\& -{\gamma }^{3}\left(\dfrac{GM}{{r}^{2}}+\dfrac{12{G}^{2}{M}^{2}{a}^{2}{\mathrm{sin}}^{2}\theta }{{r}^{5}}\right)\mathrm{cos}\theta , \end{split} $ 化简得
$ {{K}_a}{\gamma ^a} = 6{G^2}{M^2}{a^2}\left[ {\left( {\dfrac{{2{z^2}}}{{{r^2}}} - 1} \right)\left( {\dfrac{{x{\gamma ^1}}}{{{r^6}}} + \dfrac{{y{\gamma ^2}}}{{{r^6}}}} \right) - 2\dfrac{{{x^2} + {y^2}}}{{{r^8}}}z{\gamma ^3}} \right] + {\gamma ^i}{\partial _i}\left( {\dfrac{{GM}}{r}} \right) . $ 对(55)式进行积分可得
$ \begin{split} &\displaystyle\int {{{\text{d}}^4}x} \sqrt { - g} {{\bar u}^{{m_f}}}\left( {{q_f}} \right){\text{i}}{{K}_a}{\gamma ^a}{u^{{s_i}}}\left( {{p_i}} \right){{\text{e}}^{{\text{i}}\left( {{q_f} - {p_i}} \right)x}} = 2{\text{π}}\delta \left( {{E_{{q_f}}} - {E_{{p_i}}}} \right)\displaystyle\int {{{\text{d}}^3}x\sqrt { - g} } {{\bar u}^{{m_f}}}\left( {{q_f}} \right){\text{i}}{{K}_a}{\gamma ^a}{u^{{s_i}}}\left( {{p_i}} \right){{\text{e}}^{ - {\text{i}}k \cdot x}} \\ = \;&2{\text{πi}}\delta \left( {{E_{{q_f}}} - {E_{{p_i}}}} \right)\displaystyle\int {{{\text{d}}^3}x} \sqrt { - g} {{\bar u}^{{m_f}}}\left( {{q_f}} \right)\bigg[ {\gamma ^i}{\partial _i}\left( {\dfrac{{GM}}{r}} \right) + \dfrac{{6{G^2}{M^2}{a^2}}}{{{r^6}}}\left( {\dfrac{{{z^2}}}{{{r^2}}} - \dfrac{{{x^2} + {y^2}}}{{{r^2}}}} \right)\left( {x{\gamma ^1} + y{\gamma ^2}} \right) \\ & - \dfrac{{12{G^2}{M^2}{a^2}z}}{{{r^6}}}\dfrac{{{x^2} + {y^2}}}{{{r^2}}}{\gamma ^3} \bigg] {u^{{s_i}}}\left( {{p_i}} \right){{\text{e}}^{ - {\text{i}}k \cdot x}}, \end{split}$ 其中,
$ k = {q_f} - {p_i} $ . 对(56)式第一项分部积分, 由散射前后能量守恒,${k_0} = 0$ , 得到$\begin{split} &{\displaystyle \displaystyle\int {\text{d}}^{3}x\sqrt{-g}}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right){\gamma }^{i}{\partial }_{i}\left(\dfrac{GM}{r}\right){u}^{{s}_{i}}\left({p}_{i}\right){\text{e}}^{-\text{i}k\cdot x} = -\text{i}{\displaystyle \displaystyle\int {\text{d}}^{3}x}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\dfrac{GM}{r}{\gamma }^{i}{k}_{i}{u}^{{s}_{i}}\left({p}_{i}\right){\text{e}}^{-\text{i}k\cdot x} \\ = \;& \text{i}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right){/ k}{u}^{{s}_{i}}\left({p}_{i}\right){\displaystyle \displaystyle\int {\text{d}}^{3}x}\sqrt{-g}\dfrac{GM}{r}{\text{e}}^{-\text{i}k\cdot x}=0\text{ }\text{, } \end{split}$ 这里利用了平面波旋量波函数满足的方程:
$ {\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\left({/ q}_{f}-{/ p}_{i}\right){u}^{{s}_{i}}\left({p}_{i}\right)=0 . $ 因此
$ \begin{split} &{\displaystyle \displaystyle\int {\text{d}}^{4}x}\sqrt{-g}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\text{i}{K_a}{\gamma }^{a}{u}^{{s}_{i}}\left({p}_{i}\right){\text{e}}^{\text{i}\left({q}_{f}-{p}_{i}\right)x}\\ =\;&{\text{iπ}}^{3}\delta \left({E}_{{q}_{f}}-{E}_{{p}_{i}}\right){G}^{2}{M}^{2}{a}^{2}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\left[\dfrac{2{k}^{2}-{k}_{z}{}^{2}}{k}\text{i}\left({k}_{x}{\gamma }^{1}+{k}_{y}{\gamma }^{2}+{k}_{z}{\gamma }^{3}\right)+\text{i}k{k}_{z}{\gamma }^{3}\right]{u}^{{s}_{i}}\left({p}_{i}\right)\\ =\;&-{\text{π}}^{3}\delta \left({E}_{{q}_{f}}-{E}_{{p}_{i}}\right){G}^{2}{M}^{2}{a}^{2}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right)k{k}_{z}{\gamma }^{3}{u}^{{s}_{i}}\left({p}_{i}\right). \end{split}$ 散射矩阵的第二项为
$ {\varLambda }_{a}{\gamma }^{a}{\gamma }^{5}=-{\gamma }_{1}{\gamma }^{5}\dfrac{6GMa}{{r}^{3}}\mathrm{sin}\theta \mathrm{cos}\theta \mathrm{cos}\varphi -{\gamma }_{2}{\gamma }^{5}\dfrac{6GMa}{{r}^{3}}\mathrm{sin}\theta \mathrm{cos}\theta \mathrm{sin}\varphi +{\gamma }_{3}{\gamma }^{5}\dfrac{2GMa}{{r}^{3}}{\mathrm{sin}}^{2}\theta -{\gamma }_{3}{\gamma }^{5}\dfrac{4GMa}{{r}^{3}}{\mathrm{cos}}^{2}\theta . $ 对(60)式积分可得
$\begin{split} &{\displaystyle \displaystyle\int {\text{d}}^{3}x\sqrt{-g}}{\varLambda }_{a}{\gamma }^{a}{\gamma }^{5}{\text{e}}^{-\text{i}k\cdot x} =-2GMa{\displaystyle \displaystyle\int {\text{d}}^{3}x\sqrt{-g}}\left[3\left(x{\gamma }_{1}+y{\gamma }_{2}\right)\dfrac{z}{{r}^{5}}-\dfrac{{x}^{2}+{y}^{2}-2{z}^{2}}{{r}^{5}}{\gamma }_{3}\right]{\gamma }^{5}{\text{e}}^{-\text{i}k\cdot x}\\ =\;&-2GMa{\displaystyle \displaystyle\int {\text{d}}^{3}x\sqrt{-g}}\left[3\left(x{\gamma }_{1}+y{\gamma }_{2}\right)\dfrac{z}{{r}^{5}}+\left(\dfrac{3{z}^{2}}{{r}^{5}}-\dfrac{1}{{r}^{3}}\right){\gamma }_{3}\right]{\gamma }^{5}{\text{e}}^{-\text{i}k\cdot x} = -GMa\dfrac{8\text{π}}{{k}^{2}}{k}_{z}{/ k}{\gamma }^{5}. \end{split} $ 散射矩阵第三项积分为
$ {\text{i}}\displaystyle\int {{{\text{d}}^4}x} \sqrt { - g} \left( { - \dfrac{1}{2}{\gamma ^a}{h_a}^\mu {p_{i \mu}} } \right){{\text{e}}^{{\text{i}}\left( {{q_f} - {p_i}} \right)x}} = - {\text{i\pi }}\delta \left( {{E_{{q_f}}} - {E_{{p_i}}}} \right)\displaystyle\int {{{\text{d}}^3}x} \sqrt { - g} {\gamma ^a}{h_a}^\mu {p_{i \mu}} {{\text{e}}^{ - {\text{i}}k \cdot x}} \text{, } $ 其空间积分可以分解为
$ \displaystyle\int {{{\text{d}}^3}x} \sqrt { - g} {\gamma ^0}{h_0}^t{p_i}_0{{\text{e}}^{ - {\text{i}}k \cdot x}} = \displaystyle\int {{{\text{d}}^3}x\sqrt { - g} } \dfrac{{2GM}}{r}{\gamma ^0}{p_i}_0{{\text{e}}^{ - {\text{i}}k \cdot x}} = - 2GM\dfrac{{4{\text{π}}}}{{{k^2}}}{\gamma ^0}{p_i}_0 \text{, } $ $\begin{split} &{\displaystyle \displaystyle\int {\text{d}}^{3}x}\sqrt{-g}{\gamma }^{1}{h}_{1}{}^{\mu }{p}_{i}{}_{\mu }{\text{e}}^{-\text{i}k\cdot x}\\ =\;& {\displaystyle \displaystyle\int {\text{d}}^{3}x}\sqrt{-g}\dfrac{4GMa}{{r}^{2}}\dfrac{y}{r}{\gamma }^{1}{p}_{i}{}_{0}{\text{e}}^{-\text{i}k\cdot x} +{\displaystyle \displaystyle\int {\text{d}}^{3}x}\sqrt{-g}\dfrac{2GM}{r}\dfrac{x}{r}{\gamma }^{1}\left({p}_{i}{}_{x}\dfrac{x}{r}+{p}_{i}{}_{y}\dfrac{y}{r}+{p}_{i}{}_{z}\dfrac{z}{r}\right){\text{e}}^{-\text{i}k\cdot x}\\ =\;&-2GM{\gamma }^{1}\left(-2a{k}_{y}\text{i}{p}_{i}{}_{0}-{p}_{i}{}_{x}+2{p}_{i}{}_{x}\dfrac{{k}_{x}^{2}}{{k}^{2}}+2{p}_{i}{}_{y}\dfrac{{k}_{x}{k}_{y}}{{k}^{2}}+2{p}_{i}{}_{z}\dfrac{{k}_{x}{k}_{z}}{{k}^{2}}\right)\dfrac{4\text{π}}{{k}^{2}}\text{, } \end{split}$ $\begin{split} &{\displaystyle \displaystyle\int {\text{d}}^{3}x}\sqrt{-g}{\gamma }^{2}{h}_{2}{}^{\mu }{p}_{i}{}_{\mu }{\text{e}}^{-\text{i}k\cdot x}\\ =\;&-{\displaystyle \displaystyle\int {\text{d}}^{3}x}\sqrt{-g}\dfrac{4GMa}{{r}^{2}}\dfrac{x}{r}{\gamma }^{2}{p}_{i}{}_{0}{\text{e}}^{-\text{i}k\cdot x} +{\displaystyle \displaystyle\int {\text{d}}^{3}x}\sqrt{-g}{\gamma }^{2}\dfrac{2GM}{r}\dfrac{y}{r}\left({p}_{i}{}_{x}\dfrac{x}{r}+{p}_{i}{}_{y}\dfrac{y}{r}+{p}_{i}{}_{z}\dfrac{z}{r}\right){\text{e}}^{-\text{i}k\cdot x}\\ =\;&2GM{\gamma }^{2}\left(-2a{k}_{x}\text{i}{p}_{i}{}_{0}-2{p}_{i}{}_{x}\dfrac{{k}_{x}{k}_{y}}{{k}^{2}}+{p}_{i}{}_{y}-2{p}_{i}{}_{y}\dfrac{{k}_{y}^{2}}{{k}^{2}}-\text{i}{p}_{i}{}_{z}\dfrac{{k}_{y}{k}_{z}}{{k}^{2}}\right)\dfrac{4\text{π}}{{k}^{2}}\text{, } \end{split} $ $ \begin{split} \;& \displaystyle\int {{{\text{d}}^3}x} \sqrt { - g} {\gamma ^3}{h_3}^\mu {p_{i \mu}} {{\text{e}}^{ - {\text{i}}k \cdot x}} = \displaystyle\int {{{\text{d}}^3}x} \sqrt { - g} \dfrac{{2GM}}{r}\dfrac{z}{r}{\gamma ^3}\left( {{p_i}_x\dfrac{x}{r} + {p_i}_y\dfrac{y}{r} + {p_i}_z\dfrac{z}{r}} \right){{\text{e}}^{ - {\text{i}}k \cdot x}}\\ =\;& 2GM{\gamma ^3}\left( {{p_i}_z - 2{p_i}_x\dfrac{{{k_x}{k_z}}}{{{k^2}}} - 2{p_i}_y\dfrac{{{k_y}{k_z}}}{{{k^2}}} - 2{p_i}_z\dfrac{{k_z^2}}{{{k^2}}}} \right)\dfrac{{4{\text{π}}}}{{{k^2}}} \text{, } \end{split}$ 其中
${p_i}_x$ ,${p_i}_y$ ,${p_i}_z$ 分别是初态动量${p_i}$ 在x, y, z方向的分量,${k_x}$ ,${k_y}$ ,${k_z}$ 为交换四动量$k$ 在x, y, z方向的分量.${p_i}_0 = {E_{{p_i}}}$ 为初态能量. 因而$ \text{i}{\displaystyle \displaystyle\int {\text{d}}^{4}x}\sqrt{-g}\left(-\dfrac{1}{2}{\gamma }^{a}{h}_{a}{}^{\mu }{p}_{i}{}_{\mu }\right){\text{e}}^{\text{i}\left({q}_{f}-{p}_{i}\right)x} =-2\text{πi}\delta \left({E}_{{q}_{f}}-{E}_{{p}_{i}}\right)GM\dfrac{4\text{π}}{{k}^{2}}\left({\gamma }^{\mu }{\tilde{p}}_{i}{}_{\mu }+2a{k}_{y}{\gamma }^{1}\text{i}{p}_{i}{}_{0}-2a{k}_{x}{\gamma }^{2}\text{i}{p}_{i}{}_{0}\right)\text{, } $ 其中
$ {\tilde p_{i \mu}} = \left( {{p_{{i_0}}}, - {p_{{i_1}}}, - {p_{{i_2}}}, - {p_{{i_3}}}} \right) $ . 由此可得Dirac费米子的散射振幅为$ \begin{split} {M}_{\text{D}}=\;&\text{i}\langle f\left|{S}_{\text{int}}\right|i\rangle =-2\text{πi}\delta \left({E}_{{q}_{f}}-{E}_{{p}_{i}}\right){G}^{2}{M}^{2}{a}^{2}{\bar{u}}^{{m}_{f}}\left({q}_{f}\right)\dfrac{{\text{π}}^{2}}{2}\left({k}_{z}k\right){\gamma }^{3}{u}^{{s}_{i}}\left({p}_{i}\right)\\ &-2\text{πi}\delta \left({E}_{{q}_{f}}-{E}_{{p}_{i}}\right)GMa\dfrac{4\text{π}}{{k}^{2}}{k}_{z}{k}_{\mu }{\bar{u}}^{{m}_{f}}\left({q}_{f}\right){\gamma }^{\mu }{\gamma }^{5}{u}^{{s}_{i}}\left({p}_{i}\right)\\ &-2\text{πi}\delta \left({E}_{{q}_{f}}-{E}_{{p}_{i}}\right){\bar{u}}^{{m}_{f}}\left({q}_{f}\right)GM\dfrac{4\text{π}}{{k}^{2}}\left({\gamma }^{\mu }{\tilde{p}}_{i}{}_{\mu }-2a{\gamma }^{1}{k}_{y}\text{i}{p}_{i}{}_{0}+2a{\gamma }^{2}{k}_{x}\text{i}{p}_{i}{}_{0}\right){u}^{{s}_{i}}\left({p}_{i}\right)\text{, } \end{split}$ 而Majorana费米子的散射振幅为
$ \begin{split} {M_{\text{M}}} =\;& {\text{i}}\left\langle {f\left| {{S_{{\text{int}}}}} \right|i} \right\rangle = - 2{\text{πi}}\delta \left( {{E_{{q_f}}} - {E_{{p_i}}}} \right)GMa\dfrac{{4{\text{π}}}}{{{k^2}}}{k_z}{k_\mu }{{\bar u}^{{m_f}}}\left( {{q_f}} \right){\gamma ^\mu }{\gamma ^5}{u^{{s_i}}}\left( {{p_i}} \right) \oplus \\ &- 2{\text{πi}}\delta \left( {{E_{{q_f}}} - {E_{{p_i}}}} \right)GM\dfrac{{4{\text{π}}}}{{{k^2}}}{{\bar u}^{{m_f}}}\left( {{q_f}} \right)\left( {{\gamma ^\mu }{{\tilde p}_i \mu} - 2a{\gamma ^1}{k_y}i{p_i}_0 + 2a{\gamma ^2}{k_x}i{p_i}_0} \right){u^{{s_i}}}\left( {{p_i}} \right). \end{split}$ 由此得到Kerr度规的引力场对Dirac费米子与对Majorana费米子的散射振幅差别
$ {M_{\text{D-M}}} = - 2{\text{πi}}\delta \left( {{E_{{q_f}}} - {E_{{p_i}}}} \right){G^2}{M^2}{a^2}{\bar u^{{m_f}}}\left( {{q_f}} \right)\dfrac{{{{\text{π}}^2}}}{2}\left( {{k_z}k} \right){\gamma ^3}{u^{{s_i}}}\left( {{p_i}} \right) \text{, } $ 显然Kerr引力源的角动量
$a = 0$ 时, Kerr度规退化为Schwarzschild度规, 对费米子的散射振幅也会退化到Schwarzschild度规对费米子的散射振幅,${M_{{\text{D}} \text- {\text{M}}}}$ 退化为0.$a \ne 0$ 时, 非零的${M_{{\text{D}} \text- {\text{M}}}}$ 既给出Kerr度规场对Dirac费米子和Majorana费米子的自旋翻转散射概率的不同, 又给出对不同费米子类型粒子的不同散射截面. 无论是极化还是非极化, 这个差别为${\left| {{M_{{\text{D}} \text- {\text{M}}}}} \right|^2} \sim {\left( {GMa} \right)^4}$ , 因而高$GMa$ 天体诸如脉冲星和旋转黑洞对中微子的散射有望用来确定中微子的费米子类型. -
Lai和Xue[10]从挠率散射的角度分析发现了引力场对不同费米子类型的费米子散射差别在于引力场的矢量挠率, 轴矢挠率对Dirac费米子和Majorana费米子的散射矩阵元相同. 在绝对平行引力这种引力的挠率等效描述框架中, 度规给出的引力场有相应的等效挠率, 所以矢量挠率给出的不同费米子类型费米子的散射差别也可从相应的度规场散射给出, Kerr度规引力场相应的挠率具有非零的矢量、轴矢和纯张量部分. 我们对于Kerr度规场的结果是对文献[10]中矢量挠率散射结果的印证. 但是度规场引力的挠率等效描述与广义相对论的等价性事实上是建立在标量粒子度规作用和挠率作用的等价性上的, 挠率与旋量的耦合的引入虽然保证了与旋量耦合的事实上是Levi-Civita联络, 然而挠率与旋量还会产生直接作用, 对于旋量, 绝对平行引力与度规引力场的等价性还需要研究, 因而度规场对费米子类型的分辨与挠率场对费米子类型的分辨是否一致需要进一步研究.
无论是本文中的Kerr引力场散射还是文献[10]中的挠率散射, 都是微扰论最低价的结果. 微扰论一阶和高阶结果需要进一步研究, 对于微扰论一阶以上的量子散射, 由于散射矩阵元的Wick收缩对Dirac费米子和Majorana费米子有不同的收缩结构, 即使对于微扰论最低阶不能分辨费米子类型的引力场, 在微扰论高阶上是否有费米子类型的依赖依然需要研究. 另外本文和文献[10]对全时空积分都用了弱引力近似, 在后继研究中将就强引力时空区域的积分贡献进行讨论.
本文确认了Kerr度规引力场散射对费米子类型的分辨作用, 为确定对费米子类型分辨起作用的其他引力场参数, 可以考虑具有更加复杂对称性的引力场对费米子的散射. 本文证明Kerr度规背景下Dirac费米子和Majorana费米子的散射振幅会有不同, 而粒子在引力中的传播可以看作粒子在传播过程中不断被引力场散射的过程, 散射振幅的不同就会体现在粒子在引力场中的运动轨迹上, 从而为通过引力场区分中微子的费米子类型提供了另外一种可能的方法.
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无中微子双β衰变至今尚未被观察到, 同时其存在无法被否定. 因此中微子是否是Majorana粒子这个问题目前尚无定论. 本文希望从引力对费米子散射的角度研究通过引力场区分中微子费米子类型的可能性. 对Levi-Civita联络按宇称变换做分解, 在引力场对费米子散射微扰论最低阶近似以及弱引力近似下, 发现一般度规的引力场对狄拉克和马约拉纳费米子量子散射矩阵元差别来自宇称变换下类似矢量的部分; 对克尔度规的引力场散射, 证实不同类型费米子的散射差别与克尔引力源的角动量相关, 其散射矩阵元正比于引力源的质量与角动量乘积的平方. 以上结果为通过引力场区分费米子类型提供了另外一种可能的方法.
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关键词:
- Majorana费米子 /
- 挠率 /
- 绝对平行引力 /
- 克尔度规
Neutrinoless double beta decay has not been observed so far, and its existence cannot be disproved. Therefore the question whether neutrinos are Majorana particles is still inconclusive. In this paper, we hope to investigate the possibility of distinguishing neutrino fermion types by gravitational fields from the perspective of gravitational scattering of fermions. The Levi-Civita connection is decomposed according to the parity transformation. Under the perturbation therory of gravitational field to fermion quantum scattering and weak gravitational approximation, it is found that the difference between Dirac and Majorana fermion quantum scattering matrix elements of general metric gravitational field comes from similar vector parts under parity transformation; the scattering of the gravitational field on the Kerr metric confirms that the difference in scattering among different types of fermions is related to the angular momentum of the Kerr gravitational source, whose scattering matrix elements are proportional to the square of the product of the mass of the gravitational source and the angular momentum. The above results provide another possible way to distinguish fermion types by gravitational fields.-
Keywords:
- Majorana fermion /
- torsion /
- teleparallel gravity /
- Kerr metric
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[1] Majorana E 1937 Nuovo Cim. 14 171
Google Scholar
[2] Furry W H 1939 Phys. Rev. 56 1184
Google Scholar
[3] Oberauer L, Ianni A, Serenelli A 2020 Solar Neutrino Physics: The Interplay between Particle Physics and Astronomy (Wiley-VCH) pp120–127
[4] Ng K L 1993 Phys. Rev. D 47 5187
Google Scholar
[5] Ng K L 1994 Nuovo Cim. B 109 1143
Google Scholar
[6] Singh D, Mobed N, Papini G 2006 Phys. Rev. Lett. 97 041101
Google Scholar
[7] Menon A, Thalapillil A M 2008 Phys. Rev. D 78 667
Google Scholar
[8] Alavi S A, Abbasnezhad A 2012 Grav. Cosmol. 22 288
Google Scholar
[9] Nieves J F, Pal P B 2007 Phys. Rev. Lett. 98 288
Google Scholar
[10] Lai J H, Xue X 2021 arXiv: 2112.10590[gr-qc]
[11] Arcos H I, Andrade V D, Pereira J G 2004 Int. J. Mod. Phys. D 13 807
Google Scholar
[12] Aldrovandi R, Pereira J G 2013 Teleparallel Gravity: An Introduction (Dordrecht: Springer)
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