A051862 Perturbation expansion in quantum field theory: scalar case in 6 spacetime dimensions.

0, 1, 11, 376, 20241, 1427156, 121639250, 12007003824, 1337583507153, 165328009728652, 22404009743110566, 3299256277254713760, 524366465815117346250, 89448728780073829991976, 16301356287284530869810308, 3161258841758986060906197536, 650090787950164885954804021185

Offset: 0

Author

David Broadhurst, Dec 14 1999

Keywords

• nonn

Examples

			a(31) = 7632236320181399967333968684399053053157812979126909028545984868160 was computed using Kreimer's Hopf algebra of rooted trees. It subsumes 2.6*10^21 terms in quantum field theory.
		

Links

• Michael Borinsky, Gerald V. Dunne, and Max Meynig, Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: phi^3 QFT in 6 Dimensions, arXiv:2104.00593 [hep-th], 2021.
• Michael Borinsky, Gerald V. Dunne, and Karen Yeats, Tree-tubings and the combinatorics of resurgent Dyson-Schwinger equations, arXiv:2408.15883 [math-ph], 2024. See pp. 10, 21.
• D. J. Broadhurst and D. Kreimer, Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Padé-Borel resummation, arXiv:hep-th/9912093, 1999-2000. See also Phys. Lett. B (2000) Vol. 475, 63-70.

Crossrefs

Cf. A000699.

Programs

Sage

t = PowerSeriesRing(QQ, 't').gen()
def shadok(c):
    """
    fixed point procedure after G. Dunne talk at Kreimer's fest 2020
    """
    aa_sur_c = 2 * t * c.derivative() - c - 3
    aa = c * aa_sur_c
    bb_sur_c = 2 * t * aa.derivative() - aa - 2 * aa_sur_c
    bb = c * bb_sur_c
    cc_sur_c = 2 * t * bb.derivative() - bb - bb_sur_c
    return 3 * t / cc_sur_c
C = (-t / 2).O(2)
for k in range(10):
    C = shadok(C)
list(1 / 6 * C(-12 * t))
# F. Chapoton, Nov 19 2020

Formula

The generating procedure is described by Broadhurst and Kreimer.

Extensions

a(0)=0 and more terms from F. Chapoton, Nov 19 2020