Il Nuovo Cimento A (1965-1970)

, Volume 70, Issue 1, pp 1–11

# On current-field identities

• H. Genz
• J. Katz
Article

## Summary

The spectral representation for the vacuum expectation value of the axial-vector-current commutator is used to discuss the current-field identities given byg A a μ α (x)=fπμϕα(x)+A μ α (x). Herea μ α (x) denotes the axial-vector-meson field,gA is a properly chosen normalization constant, and the pion field ϕα(x) is defined by the relation$$\partial ^\mu A_\mu ^\alpha (x) \equiv \widehat\varphi ^\alpha (x) = f_\pi m_\pi ^2 \varphi ^\alpha (x)$$, whereA μ α (x) denotes the axial vector current. We find that
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
and that these expressions vanish if and only if the pion field ϕα(x) is a free field. We also note that
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (\mathop x\limits_r ),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
and that these expressions vanish if and only if the axial vector currentA μ α (x) is conserved. The consequences for canonical realizations of current-field identities and PCAC are given. We also find that the vacuum expectation value of the σ-term is nonvanishing unless the current is conserved and that a nonconserved charge cannot annihilate the vacuum (as stated by Coleman's theorem).

## Keywords

Spectral Function Vacuum Expectation Derivative Coupling Axial Vector Current Pion Field

# О ток-полевых тождествах

## Резюме

Спектральное представление для вакуумной ожидаемой величины коммутатора аксиально-векторного тока используется для обсуждения ток-полевых тождеств, заданных соотношениемg A a μ α (x)=fπμϕα(x)+A μ α (x). Здесьa μ α (x) обозначает поле аксиально-векторного мезона,gA есть надлежашим образом выбранная перенормировочная постоянная; пионное поле ϕα(x) определяется соотношением$$\partial ^\mu A_\mu ^\alpha (x) \equiv \widehat\varphi ^\alpha (x) = f_\pi m_\pi ^2 \varphi ^\alpha (x)$$, гдеA μ α (x) представляет аксиально-векторный ток. Мы получаем, что
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
и что эти выражения обрашаются в нуль, если и только если пионное поле ϕα(x) представляет свободное поле. Мы также отмечаем, что
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (\mathop x\limits_r ),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
и что эти выражения обращаются в нуль, если и только если аксиально-векторный токA μ α (x) сохраняется. Приводятся следствия для канонических реализаций токполевых тождеств и РСАС. Мы также находим, что вакуумная ожидаемая величина σ-члена не обращается в нуль, если ток не сохраняется, и что несохраняющийся заряд не может уничтожить вакуум (как утверждается теоремой Колемана).

## Riassunto

Si usa la rappresentazione spettrale del valore di aspettazione del vuoto per discuter e le identità corrente-campo date dag A a μ α (x)=fπμϕα(x)+A μ α (x). Quia μ α (x) denota il campo del mesone vettoriale assiale,gA è una costante di rinormalizzazione opportunamente scelta, ed il campo pionico ϕα(x) è definito dalla relazione$$\partial ^\mu A_\mu ^\alpha (x) \equiv \widehat\varphi ^\alpha (x) = f_\pi m_\pi ^2 \varphi ^\alpha (x)$$, doveA μ α (x) denota la corrente vettoriale assiale. Si trova che
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
e che queste espressioni si annullano se e solo se il campo pionico ϕα(x) è un campo libero. Si nota anche che
$$\left\langle {\left[ {\frac{\partial }{{\partial x_k }}a_0^\alpha (\mathop x\limits_r ),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } = \left\langle {\left[ {\frac{\partial }{{\partial x_0 }}a_k^\alpha (x),\widehat\varphi {}^\beta (y)} \right]} \right\rangle _{x_0 = y_0 } \propto \delta ^{\alpha \beta } \frac{\partial }{{\partial x_k }}\delta (x - y)$$
e che queste espressioni si annullano se e solo se la corrente vettoriale assialeA μ α (x) è conservata. Si espongono le conseguenze per le realizzazioni canoniche delle identità corrente-campo e per PCAC. Si trova anche che il valore di aspettazione del vuoto del termine σ non tende a zero a meno che la corrente sia conservata e che una carica non conservata non possa annichilare il vuoto (come è affermato dal teorema di Coleman).

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