Manuscript I-B-VP-1@1 · I-B-VP-1-v1.tex · section · 6d700be9

Regular one-point data

The completion, regular data, and Ward prescription › Regular one-point data · lines 294–382

I-B-VP-1@1#eq:coefficients, I-B-VP-1@1#eq:regularstress, I-B-VP-1@1#eq:j, I-B-VP-1@1#eq:regularcurrent

The charged field obeys

Φ (φ + 2 π) = e^{2 π i β} Φ (φ)

in a lifted parallel frame. Its spatial extension and state are the massless Friedrichs operator and distinguished spectral ground state of I-B-VP-v2 CheungIBVP (:state, :infrared, :hadamard). For the real field take $β = 0$ .

Introduce a multiplicity

n = {1, 2, one real neutral scalar, one charged complex scalar .

Define

P_{β} C_{β} s k = α^{- 2} (1 - 6 β (1 - β)) - 1, = α^{- 4} (1 - 30 β^{2} (1 - β)^{2}) - 1, = \frac{P _{β}}{48 π ^{2}}, h = \frac{n C _{β}}{1440 π ^{2}}, q = ξ - \frac{1}{6}, = n q s, a = h + 4 k, b = h - 2 k .

The regular Wick square is

F^{\circ} = \frac{s}{r ^{2}},

meaning $⟨ ϕ^{2} ⟩_{ren}$ for the real field and $⟨ Φ^{†} Φ ⟩_{ren}$ for the complex field. The effective field square in the stress trace is $n F^{\circ}$ .

In the smooth coordinates of Assumption ass:completion, the regular mixed stress is

(T^{\circ})^{A}_{B} (T^{\circ})^{i}_{j} = \frac{a}{r ^{4}} δ^{A}_{B}, = \frac{b}{r ^{4}} (4 \frac{x ^{i} x _{j}}{r ^{2}} - 3 δ^{i}_{j}) . (T^{\circ})^{A}_{i} = (T^{\circ})^{i}_{A} = 0,

These are precisely the mixed components $r^{- 4} diag (a, b, - 3 b, a)$ in order $(t, r, φ, z)$ CheungIBVP (:phi2, :T-real, :T-complex).

For the charged field put

j = \frac{β ( 1 - β ) ( 2 β - 1 )}{12 π ^{2} α ^{2}} .

With

J_{μ} = i (Φ^{†} D_{μ} Φ - (D_{μ} Φ)^{†} Φ),

I-B-VP-0-v1 gives

(J^{\circ})^{i} = \frac{j}{α ^{2}} \frac{ϵ ^{i} _{j} x ^{j}}{r ^{4}}, (J^{\circ})^{A} = 0, J^{\circ}_{φ} = \frac{j}{r ^{2}} .

The current uses the complex first jet of the regular covariance, not the real part used for stress CheungIBVPZero (:current, :current). Set $j = 0$ for the real field.

Displayed formulas in this passage

I-B-VP-1@1#eq:coefficients · line 317
$P_{β} C_{β} s k = α^{- 2} (1 - 6 β (1 - β)) - 1, = α^{- 4} (1 - 30 β^{2} (1 - β)^{2}) - 1, = \frac{P _{β}}{48 π ^{2}}, h = \frac{n C _{β}}{1440 π ^{2}}, q = ξ - \frac{1}{6}, = n q s, a = h + 4 k, b = h - 2 k .$
I-B-VP-1@1#eq:regularstress · line 339
$(T^{\circ})^{A}_{B} (T^{\circ})^{i}_{j} = \frac{a}{r ^{4}} δ^{A}_{B}, = \frac{b}{r ^{4}} (4 \frac{x ^{i} x _{j}}{r ^{2}} - 3 δ^{i}_{j}) . (T^{\circ})^{A}_{i} = (T^{\circ})^{i}_{A} = 0,$
I-B-VP-1@1#eq:j · line 356
$j = \frac{β ( 1 - β ) ( 2 β - 1 )}{12 π ^{2} α ^{2}} .$
I-B-VP-1@1#eq:regularcurrent · line 368
$(J^{\circ})^{i} = \frac{j}{α ^{2}} \frac{ϵ ^{i} _{j} x ^{j}}{r ^{4}}, (J^{\circ})^{A} = 0, J^{\circ}_{φ} = \frac{j}{r ^{2}} .$
unlabeled · line 297
$Φ (φ + 2 π) = e^{2 π i β} Φ (φ)$
unlabeled · line 309
$n = {1, 2, one real neutral scalar, one charged complex scalar .$
unlabeled · line 330
$F^{\circ} = \frac{s}{r ^{2}},$
unlabeled · line 362
$J_{μ} = i (Φ^{†} D_{μ} Φ - (D_{μ} Φ)^{†} Φ),$