current - ojaru.jpmathematican.ojaru.jp/note/qed_form_factor_v0a3.pdfrenormalization 1pr = 1pi + 1pr...
TRANSCRIPT
Current
• Renormalization
• Electric Charge
• Interpretation
2010年12月30日木曜日
Renormalization
= + +1PR 1PI 1PR 1PR
Γµ0,1PR = Γµ
0,1PI +−Π0(q2)
1 + Π0(q2)
�gµν −
qµqν
q2
�Γν
0,1PI
+δZ3
e0Z3
�−q2gµν + qµqν
� −i
q2(1 + Π0(q2))(ie0)Γν
0,1PI
= Γµ0,1PI +
�−Π0(q2)
1 + Π0(q2)− δZ3
Z3
11 + Π0(q2)
� �gµν −
qµqν
q2
�Γν
0,1PI
Renormalized to zero at q2=0 thanks to
�p�| (−Jµphys(x)) |p� = �p�| (−Jµ
N(x)) |p� +δZ3
Z3
1e0
�p�| ∂νF νµ0 |p�
generated by Noether current
Z3 =1
1 + Π0(0)
=1Z3
11 + Π0(q2)
− 1 q2→0−→ 0
2010年12月30日木曜日
Electric charge
�e−(p)|QN|e−(p)� =�
d3x�e(p)|J0N(x)|e(p)�
= limq→0
�d3x e−iqx�e(p)|J0
N(x)|e(p)�
= limq→0
(2π)3δ(3)(�p− �p− �q)e−iq0x0�e(p)|J0
N(0)|e(p)�
= (2π)3δ(3)(�0)× u(p)�
(−1)Z2Γ00,1PI(p, p)
�u(p)
�1 +
�1
1 + Π0(0)− 1
��
= (2π)3δ(3)(�0)×�
(−1)Z2F01 (q2 = 0) 2p0
�×
�1
1 + Π0(0)
�
�e−(p)|Qphys|e−(p)� =�
d3x�e(p)|�
J0N(x)− δZ3
Z3
1e0
∂νF ν00
�|e(p)�
= (2π)3δ(3)(�0 )×�
(−1)Z2F01 (0) 2p0
�×
�1
1 + Π0(0)− δZ3
Z3
11 + Π0(0)
�
1PI 1PR
1PR
= 1= 1
Two vertex diagrams for Noether current insertion
Counter term for physical current
�e±(p)|Qphys|e±(p)� = (±1) × (2π)32p0δ(3)(0)
= (±1) × �e±(p)|e±(p)�
2010年12月30日木曜日
Interpretation
1PI 1PR
1PI Noether current insertion
�e−(p)|QN|e−(p)� = (2π)3δ(3)(�0)× u(p)�
(−1)Z2Γ00,1PI(p, p)
�u(p)
�1 +
�1
1 + Π0(0)− 1
��
1PR
1PR Noether current insertion 1PR Counter term
1PR Noether current insertion
• 1PR Noether current insertion can be understood as, the charge operator QN itself creates the e+ e- pair in vacuum and see it as charge of an object to be measured.•We could define renormalized Noether charge by subtracting the IV divergence of 1PR-diagram in MSbar. But such a renormalization leaves 1PR contribution at q2=0. This leads strange phenomena; for instance, electron Noether charge is non-zero even for muon.•Form factor <p’|JN|p> defined by Noether current insertion receive 1PR contribution, which can be understood as running factor of QED effective charge. •Thus 1PR diagram must be completely subtracted at q2=0, which is the physical charge defined here.•The physical charge is UV finite and the one which satisfies Gauss’s law.
2010年12月30日木曜日