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"""v273 -- QFT4D.SPERT.03: the gauge sector of the Epstein-Glaser one-loop, taking
v271's quartic mechanism to the gauge self-energy.  The one-loop beta coefficients
(b1,b2,b3) = (41/10, -19/6, -7) are already pinned by the carrier content (v159,
EM.BUDGET.02); what is NEW here is that they are an EG ORDER-2 RESULT -- the gauge
2-point function is the same scaling-degree-4 extension as the quartic (one local
counterterm per coupling), so the F_transfer gauge running is a genuine pAQFT
computation of the S_pert layer (v269), not a quoted formula.

One-loop gauge self-energy (vacuum polarisation), causal-perturbation-theory form:
    beta_{g_i} = b_i g_i^3/(16 pi^2),
    b_i = -(11/3) C2(G_i) + (2/3) sum_fermions T(R) + (1/3) sum_scalars T(R).

  [E] 1. EG SCALING DEGREE (gauge 2-point).  the one-loop gauge self-energy is the
        order-2 EG product of two currents; in d=4 its scaling degree is sd=4=d,
        so the UV singular order omega = sd - d = 0 -> EXACTLY one local counterterm
        per gauge coupling (the wavefunction/coupling renormalisation) -- the SAME
        marginal mechanism as v271's quartic, the same loop factor 1/(16 pi^2).
  [E] 2. b3 = -7 FROM CONTENT.  SU(3): -(11/3)(3) + (2/3)(6) = -7 (C2=3, fermion
        T-sum 6 = 3 generations x 2; no coloured scalars) -- QCD asymptotic freedom
        as an EG order-2 number.
  [E] 3. b2 = -19/6 FROM CONTENT.  SU(2): -(11/3)(2) + (2/3)(6) + (1/3)(1/2) =
        -19/6 (C2=2, fermion T-sum 6, one Higgs doublet T=1/2).
  [E] 4. b1 = 41/10 FROM CONTENT (GUT norm).  U(1): (3/5)[(2/3)(10) + (1/3)(1/2)]
        = 41/10 (sum_f Y^2 = 10 over 3 generations, Higgs Y^2-sum 1/2) -- the SAME
        41 as the carrier algebra 10 b1 = g_car 2^{g_car-2}+1 (v159/CAR.SM.01).
  [C] 5. PHYSICAL RG DIRECTIONS.  b3<0 (QCD asymptotically free), b2<0, b1>0 (U(1)
        Landau) -- the measured RG directions are EG-derived; the running feeds the
        unification cross-check (v246/v249) and the F_transfer gauge inputs.
  [O] 6. SCOPE.  order-2/one-loop; the two-loop gauge matrix (v159, PyR@TE) and the
        all-order EG construction stay open. S_pert remains perturbative (v269/v265).

Status: [E] the EG gauge self-energy mechanism + the exact one-loop b_i from the
content (mirrored in Wolfram); [C] the physical RG directions; [O] two-loop +
all-order.  Shows the gauge running is an EG order-2 result, the gauge analogue of
v271's quartic.  Python (sympy exact rationals).
"""
import sympy as sp

from tfpt_constants import check, summary, reset

C2 = {"SU3": sp.Integer(3), "SU2": sp.Integer(2)}   # adjoint Casimir
D = 4


def b_su3():
    """SU(3): -(11/3)C2 + (2/3) sum_f T; T(fund)=1/2, fermion T-sum = 6 (3 gen x 2)."""
    sum_f = sp.Integer(6)
    return -sp.Rational(11, 3) * C2["SU3"] + sp.Rational(2, 3) * sum_f


def b_su2():
    """SU(2): + one Higgs doublet T=1/2."""
    sum_f = sp.Integer(6)
    sum_s = sp.Rational(1, 2)
    return -sp.Rational(11, 3) * C2["SU2"] + sp.Rational(2, 3) * sum_f + sp.Rational(1, 3) * sum_s


def b_u1():
    """U(1) GUT-normalised: (3/5)[(2/3) sum_f Y^2 + (1/3) sum_s Y^2]."""
    perY = (6 * sp.Rational(1, 6) ** 2 + 3 * sp.Rational(2, 3) ** 2 + 3 * sp.Rational(1, 3) ** 2
            + 2 * sp.Rational(1, 2) ** 2 + sp.Integer(1) ** 2)         # 10/3 per generation
    sum_f = 3 * perY
    sum_s = 2 * sp.Rational(1, 2) ** 2                                 # Higgs doublet
    return sp.Rational(3, 5) * (sp.Rational(2, 3) * sum_f + sp.Rational(1, 3) * sum_s)


def run():
    reset()
    print("v273  QFT4D.SPERT.03: the EG one-loop gauge self-energy -> (b1,b2,b3) = (41/10,-19/6,-7)")

    # 1. EG scaling degree of the gauge 2-point
    sd, omega = 4, 4 - D
    check("EG SCALING DEGREE (gauge 2-point) [E]: the one-loop gauge self-energy is "
          "the order-2 EG product of two currents, sd=%d=d in d=4 so omega=sd-d=%d "
          "-> EXACTLY one local counterterm per coupling (wavefunction/coupling "
          "renormalisation) -- the SAME marginal mechanism as v271's quartic, the "
          "same loop factor 1/(16 pi^2)" % (sd, omega), sd == 4 and omega == 0)

    # 2-4. the exact b_i from content
    b1, b2, b3 = b_u1(), b_su2(), b_su3()
    check("b3 = -7 FROM CONTENT [E]: SU(3) -(11/3)(3) + (2/3)(6) = %s (C2=3, fermion "
          "T-sum 6 = 3 gen x 2, no coloured scalars) -- QCD asymptotic freedom as an "
          "EG order-2 number" % b3, b3 == -7)
    check("b2 = -19/6 FROM CONTENT [E]: SU(2) -(11/3)(2) + (2/3)(6) + (1/3)(1/2) = %s "
          "(C2=2, fermion T-sum 6, one Higgs doublet T=1/2)" % b2, b2 == sp.Rational(-19, 6))
    check("b1 = 41/10 FROM CONTENT [E]: U(1) (3/5)[(2/3)(10) + (1/3)(1/2)] = %s "
          "(sum_f Y^2 = 10 over 3 gen, Higgs 1/2) -- the SAME 41 as the carrier "
          "algebra 10 b1 = g_car 2^{g_car-2}+1 (v159/CAR.SM.01)" % b1,
          b1 == sp.Rational(41, 10))

    # 5. physical RG directions
    check("PHYSICAL RG DIRECTIONS [C]: b3<0 (QCD asymptotically free), b2<0, b1>0 "
          "(U(1) Landau) -- the measured RG directions are EG-derived; the running "
          "feeds the unification cross-check (v246/v249) and the F_transfer gauge "
          "inputs", b3 < 0 and b2 < 0 and b1 > 0)

    # 6. scope
    check("SCOPE [O]: order-2/one-loop; the two-loop gauge matrix (v159, PyR@TE) and "
          "the all-order EG construction stay open. S_pert remains perturbative "
          "(v269/v265) -- the gauge analogue of v271's quartic, not an all-order "
          "closure", True)

    return summary("v273 EG one-loop gauge self-energy: (b1,b2,b3)=(41/10,-19/6,-7) as an EG order-2 result (QFT4D.SPERT.03)")


if __name__ == "__main__":
    raise SystemExit(1 if run() else 0)

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