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D. J. Broadhurst

Publications and source records attributed to D. J. Broadhurst.

At least 37 records · Page 2Linked to original sources
arXiv1998

A dilogarithmic 3-dimensional Ising tetrahedron

In 3 dimensions, the Ising model is in the same universality class as $ϕ^4$-theory, whose massive 3-loop tetrahedral diagram, $C^{Tet}$, was of an unknown analytical nature. In contrast, all single-scale 4-dimensional tetrahedra were reduced, in hep-th/9803091, to special values of exponentially convergent polylogarithms. Combining dispersion relations with the integer-relation finder PSLQ, we find that $C^{Tet}/2^{5/2} = Cl_2(4α) - Cl_2(2α)$, with $Cl_2(θ):=\sum_{n>0}\sin(nθ)/n^2$ and $α:=\arcsin\frac13$. This empirical relation has been checked at 1,000-digit precision and readily yields 50,000 digits of $C^{Tet}$, after transformation to an exponentially convergent sum, akin to those studied in math.CA/9803067. It appears that this 3-dimensional result entails a polylogarithmic ladder beginning with the classical formula for $π/\sqrt2$, in the manner that 4-dimensional results build on that for $π/\sqrt3$.

hep-th↗
arXiv1998

Polylogarithmic ladders, hypergeometric series and the ten millionth digits of $ζ(3)$ and $ζ(5)$

We develop ladders that reduce $ζ(n):=\sum_{k>0}k^{-n}$, for $n=3,5,7,9,11$, and $β(n):=\sum_{k\ge0}(-1)^k(2k+1)^{-n}$, for $n=2,4,6$, to convergent polylogarithms and products of powers of $π$ and $\log2$. Rapid computability results because the required arguments of ${\rm Li}_n(z)=\sum_{k>0}z^k/k^n$ satisfy $z^8=1/16^p$, with $p=1,3,5$. We prove that $G:=β(2)$, $π^3$, $\log^32$, $ζ(3)$, $π^4$, $\log^42$, $\log^52$, $ζ(5)$, and six products of powers of $π$ and $\log2$ are constants whose $d$th hexadecimal digit can be computed in time~$=O(d\log^3d)$ and space~$=O(\log d)$, as was shown for $π$, $\log2$, $π^2$ and $\log^22$ by Bailey, Borwein and Plouffe. The proof of the result for $ζ(5)$ entails detailed analysis of hypergeometric series that yield Euler sums, previously studied in quantum field theory. The other 13 results follow more easily from Kummer's functional identities. We compute digits of $ζ(3)$ and $ζ(5)$, starting at the ten millionth hexadecimal place. These constants result from calculations of massless Feynman diagrams in quantum chromodynamics. In a related paper, hep-th/9803091, we show that massive diagrams also entail constants whose base of super-fast computation is $b=3$.

math.CA↗
arXiv1998

Massive 3-loop Feynman diagrams reducible to SC* primitives of algebras of the sixth root of unity

In each of the 10 cases with propagators of unit or zero mass, the finite part of the scalar 3-loop tetrahedral vacuum diagram is reduced to 4-letter words in the 7-letter alphabet of the 1-forms $Ω:=dz/z$ and $ω_p:=dz/ (λ^{-p}-z)$, where $λ$ is the sixth root of unity. Three diagrams yield only $ζ(Ω^3ω_0)=1/90π^4$. In two cases $π^4$ combines with the Euler-Zagier sum $ζ(Ω^2ω_3ω_0)=\sum_{m> n>0}(-1)^{m+n}/m^3n$; in three cases it combines with the square of Clausen's $Cl_2(π/3)=\Im ζ(Ωω_1)=\sum_{n>0}\sin(πn/3)/n^2$. The case with 6 masses involves no further constant; with 5 masses a Deligne-Euler-Zagier sum appears: $\Re ζ(Ω^2ω_3ω_1)= \sum_{m>n>0}(-1)^m\cos(2πn/3)/m^3n$. The previously unidentified term in the 3-loop rho-parameter of the standard model is merely $D_3=6ζ(3)-6 Cl_2^2(π/3)-{1/24}π^4$. The remarkable simplicity of these results stems from two shuffle algebras: one for nested sums; the other for iterated integrals. Each diagram evaluates to 10 000 digits in seconds, because the primitive words are transformable to exponentially convergent single sums, as recently shown for $ζ(3)$ and $ζ(5)$, familiar in QCD. Those are SC$^*(2)$ constants, whose base of super-fast computation is 2. Mass involves the novel base-3 set SC$^*(3)$. All 10 diagrams reduce to SC$^*(3)\cup$SC$^* (2)$ constants and their products. Only the 6-mass case entails both bases.

hep-th↗
arXiv1997

Feynman diagrams as a weight system: four-loop test of a four-term relation

At four loops there first occurs a test of the four-term relation derived by the second author in the course of investigating whether counterterms from subdivergence-free diagrams form a weight system. This test relates counterterms in a four-dimensional field theory with Yukawa and $ϕ^4$ interactions, where no such relation was previously suspected. Using integration by parts, we reduce each counterterm to massless two-loop two-point integrals. The four-term relation is verified, with $ = 0 - 3ζ_3 + 6ζ_3 - 3ζ_3 = 0$, demonstrating non-trivial cancellation of the trefoil knot and thus supporting the emerging connection between knots and counterterms, via transcendental numbers assigned by four-dimensional field theories to chord diagrams. Restrictions to scalar couplings and renormalizable interactions are found to be necessary for the existence of a pure four-term relation. Strong indications of richer structure are given at five loops.

hep-th↗
arXiv1997

Conjectured enumeration of Vassiliev invariants

A rational Ansatz is proposed for the generating function $\sum_{j,k} β_{2j+k,2j}x^j y^k$, where $β_{m,u}$ is the number of primitive chinese character diagrams with $u$ univalent and $2m-u$ trivalent vertices. For $P_m:=\sum_{u\ge2}β_{m,u}$, the conjecture leads to the sequence $$1,1,1,2,3,5,8,12,18,27,39,55,\underline{78,108,150,207,284,388,532,726}$$ for primitive chord diagrams of degrees $m\le20$, with predictions underlined. The asymptotic behaviour $\lim_{m\to\infty}P_m/r^m= 1.06260548918755$ results, with $r=1.38027756909761$ solving $r^4=r^3+1$. Vassiliev invariants of knots are then enumerated by $$0,1,1,3,4,9,14,27,44, 80,132,232,\underline{384,659,1095,1851,3065,5128,8461,14031}$$ and Vassiliev invariants of framed knots by $$1,2,3,6,10,19,33,60,104,184,316, 548,\underline{932,1591,2686,4537,7602,12730,21191,35222}$$ These conjectures are motivated by successful enumerations of irreducible Euler sums. Predictions for $β_{15,10}$, $β_{16,12}$ and $β_{19,16}$ suggest that the action of sl and osp Lie algebras, on baguette diagrams with ladder insertions, fails to detect an invariant in each case.

q-alg↗
arXiv1996

Conjectured Enumeration of irreducible Multiple Zeta Values, from Knots and Feynman Diagrams

Multiple zeta values (MZVs) are under intense investigation in three arenas -- knot theory, number theory, and quantum field theory -- which unite in Kreimer's proposal that field theory assigns MZVs to positive knots, via Feynman diagrams whose momentum flow is encoded by link diagrams. Two challenging problems are posed by this nexus of knot/number/field theory: enumeration of positive knots, and enumeration of irreducible MZVs. Both were recently tackled by Broadhurst and Kreimer (BK). Here we report large-scale analytical and numerical computations that test, with considerable severity, the BK conjecture that the number, $D_{n,k}$, of irreducible MZVs of weight $n$ and depth $k$, is generated by $\prod_{n\ge3}\prod_{k\ge1}(1-x^n y^k) ^{D_{n,k}}=1-\frac{x^3y}{1-x^2}+\frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}$, which is here shown to be consistent with all shuffle identities for the corresponding iterated integrals, up to weights $n=44, 37, 42, 27$, at depths $k=2, 3, 4, 5$, respectively, entailing computation at the petashuffle level. We recount the field-theoretic discoveries of MZVs, in counterterms, and of Euler sums, from more general Feynman diagrams, that led to this success.

hep-th↗
arXiv1996

Compact analytical form for non-zeta terms in critical exponents at order 1/N^3

We simplify, to a single integral of dilogarithms, the least tractable O(1/N^3) contribution to the large-N critical exponent $η$ of the non-linear sigma-model, and hence $ϕ^4$-theory, for any spacetime dimensionality, D. It is the sole generator of irreducible multiple zeta values in epsilon-expansions with $D=2-2ε$, for the sigma-model, and $D=4-2ε$, for $ϕ^4$-theory. In both cases we confirm results of Broadhurst, Gracey and Kreimer (BGK) that relate knots to counterterms. The new compact form is much simpler than that of BGK. It enables us to develop 8 new terms in the epsilon-expansion with $D=3-2ε$. These involve alternating Euler sums, for which the basis of irreducibles is larger. We conclude that massless Feynman diagrams in odd spacetime dimensions share the greater transcendental complexity of massive diagrams in even dimensions, such as those contributing to the electron's magnetic moment and the electroweak $ρ$-parameter. Consequences for the perturbative sector of Chern-Simons theory are discussed.

hep-th↗
arXiv1996

Association of multiple zeta values with positive knots via Feynman diagrams up to 9 loops

It is found that the number, $M_n$, of irreducible multiple zeta values (MZVs) of weight $n$, is generated by $1-x^2-x^3=\prod_n (1-x^n)^{M_n}$. For $9\ge n\ge3$, $M_n$ enumerates positive knots with $n$ crossings. Positive knots to which field theory assigns knot-numbers that are not MZVs first appear at 10 crossings. We identify all the positive knots, up to 15 crossings, that are in correspondence with irreducible MZVs, by virtue of the connection between knots and numbers realized by Feynman diagrams with up to 9 loops.

hep-th↗
arXiv1996

Evaluations of k-fold Euler/Zagier sums: a compendium of results for arbitrary k

Euler sums (also called Zagier sums) occur within the context of knot theory and quantum field theory. There are various conjectures related to these sums whose incompletion is a sign that both the mathematics and physics communities do not yet completely understand the field. Here, we assemble results for Euler/Zagier sums (also known as multidimensional zeta/harmonic sums) of arbitrary depth, including sign alternations. Many of our results were obtained empirically and are apparently new. By carefully compiling and examining a huge data base of high precision numerical evaluations, we can claim with some confidence that certain classes of results are exhaustive. While many proofs are lacking, we have sketched derivations of all results that have so far been proved.

hep-th↗
arXiv1996

Beyond the triangle and uniqueness relations: non-zeta counterterms at large N from positive knots

Counterterms that are not reducible to $ζ_{n}$ are generated by ${}_3F_2$ hypergeometric series arising from diagrams for which triangle and uniqueness relations furnish insufficient data. Irreducible double sums, corresponding to the torus knots $(4,3)=8_{19}$ and $(5,3)=10_{124}$, are found in anomalous dimensions at ${\rm O}(1/N^3)$ in the large-$N$ limit, which we compute analytically up to terms of level 11, corresponding to 11 loops for 4-dimensional field theories and 12 loops for 2-dimensional theories. High-precision numerical results are obtained up to 24 loops and used in Padé resummations of $\varepsilon$-expansions, which are compared with analytical results in 3 dimensions. The ${\rm O}(1/N^3)$ results entail knots generated by three dressed propagators in the master two-loop two-point diagram. At higher orders in $1/N$ one encounters the uniquely positive hyperbolic 11-crossing knot, associated with an irreducible triple sum. At 12 crossings, a pair of 3-braid knots is generated, corresponding to a pair of irreducible double sums with alternating signs. The hyperbolic positive knots $10_{139}$ and $10_{152}$ are not generated by such self-energy insertions.

hep-th↗
arXiv1996

On the enumeration of irreducible k-fold Euler sums and their roles in knot theory and field theory

A generating function is given for the number, $E(l,k)$, of irreducible $k$-fold Euler sums, with all possible alternations of sign, and exponents summing to $l$. Its form is remarkably simple: $\sum_n E(k+2n,k) x^n = \sum_{d|k}μ(d) (1-x^d)^{-k/d}/k$, where $μ$ is the Möbius function. Equivalently, the size of the search space in which $k$-fold Euler sums of level $l$ are reducible to rational linear combinations of irreducible basis terms is $S(l,k) = \sum_{n<k}{\lfloor(l+n-1)/2\rfloor\choose n}$. Analytical methods, using Tony Hearn's REDUCE, achieve this reduction for the 3698 convergent double Euler sums with $l\leq44$; numerical methods, using David Bailey's MPPSLQ, achieve it for the 1457 convergent $k$-fold sums with $l\leq7$; combined methods yield bases for all remaining search spaces with $S(l,k)\leq34$. These findings confirm expectations based on Dirk Kreimer's connection of knot theory with quantum field theory. The occurrence in perturbative quantum electrodynamics of all 12 irreducible Euler sums with $l\leq 7$ is demonstrated. It is suggested that no further transcendental occurs in the four-loop contributions to the electron's magnetic moment. Irreducible Euler sums are found to occur in explicit analytical results, for counterterms with up to 13 loops, yielding transcendental knot-numbers, up to 23 crossings.

hep-th↗
arXiv1995

Unknotting the polarized vacuum of quenched QED

A knot-theoretic explanation is given for the rationality of the quenched QED beta function. At the link level, the Ward identity entails cancellation of subdivergences generated by one term of the skein relation, which in turn implies cancellation of knots generated by the other term. In consequence, each bare three-loop diagram has a rational Laurent expansion in the Landau gauge, as is verified by explicit computation. Comparable simplification is found to occur in scalar electrodynamics, when computed in the Duffin-Kemmer-Petiau formalism.

hep-ph↗
arXiv1995

Three-loop QED Vacuum Polarization and the Four--loop Muon Anomalous Magnetic Moment

Three--loop contributions to massive QED vacuum polarization are evaluated by a combination of analytical and numerical techniques. The first three Taylor coefficients, at small $q^2$, are obtained analytically, using $d$\/--dimensional recurrence relations. Combining these with analytical input at threshold, and at large $q^2$, an accurate Padé approximation is obtained, for all $q^2$. Inserting this in the one--loop diagram for the muon anomalous magnetic moment, we find reasonable agreement with four--loop, single--electron--loop, muon--anomaly contributions, recently re--evaluated by Kinoshita, using 8--dimensional Monte--Carlo integration. We believe that our new method is at least two orders of magnitude more accurate than the Monte--Carlo approach, whose uncertainties appear to have been underestimated, by a factor of 6.

hep-ph↗
arXiv1995

Knots and Numbers in $ϕ^4$ Theory to 7 Loops and Beyond

We evaluate all the primitive divergences contributing to the 7--loop $β$\/--function of $ϕ^4$ theory, i.e.\ all 59 diagrams that are free of subdivergences and hence give scheme--independent contributions. Guided by the association of diagrams with knots, we obtain analytical results for 56 diagrams. The remaining three diagrams, associated with the knots $10_{124}$, $10_{139}$, and $10_{152}$, are evaluated numerically, to 10 sf. Only one satellite knot with 11 crossings is encountered and the transcendental number associated with it is found. Thus we achieve an analytical result for the 6--loop contributions, and a numerical result at 7 loops that is accurate to one part in $10^{11}$. The series of `zig--zag' counterterms, $\{6ζ_3,\,20ζ_5,\, \frac{441}{8}ζ_7,\,168ζ_9,\,\ldots\}$, previously known for $n=3,4,5,6$ loops, is evaluated to 10 loops, corresponding to 17 crossings, revealing that the $n$\/--loop zig--zag term is $4C_{n-1} \sum_{p>0}\frac{(-1)^{p n - n}}{p^{2n-3}}$, where $C_n=\frac{1}{n+1}{2n \choose n}$ are the Catalan numbers, familiar in knot theory. The investigations reported here entailed intensive use of REDUCE, to generate ${\rm O}(10^4)$ lines of code for multiple precision FORTRAN computations, enabled by Bailey's MPFUN routines, running for ${\rm O}(10^3)$ CPUhours on DecAlpha machines.

hep-ph↗
arXiv1994

Matching QCD and HQET heavy--light currents at two loops and beyond

Heavy--light QCD currents are matched with HQET currents at two loops and leading order in $1/m$. A single formula applies to all current matchings. As a by--product, a master formula for the two--loop anomalous dimension of the QCD current $\bar{q}γ^{[μ_1}\ldotsγ^{μ_n]}q$ is obtained, yielding a new result for the tensor current. The dependence of matching coefficients on $γ_5$ prescriptions is elucidated. Ratios of QCD matrix elements are obtained, independently of the three--loop anomalous dimension of HQET currents. The two--loop coefficient in $f_{{\rm B}^*}/f_{\rm B} =1-2α_{\rm s}(m_b)/3π-K_bα_{\rm s}^2/π^2 +{\rm O}(α_{\rm s}^3,1/m_b)$ is \[K_b=\frac{83}{12}+\frac{4}{81}π^2+\frac{2}{27}π^2\log2-\frac19ζ(3) -\frac{19}{54}N_l+Δ_c=6.37+Δ_c\] with $N_l=4$ light flavours, and a correction, $Δ_c=0.18\pm0.01$, that takes account of the non--zero ratio $m_c/m_b=0.28\pm0.03$. Fastest apparent convergence would entail $α_{\rm s}(μ)$ at $μ=370$~MeV. ``Naive non--abelianization'' of large--$N_l$ results, via $N_l\to N_l-\frac{33}{2}$, gives reasonable approximations to exact two--loop results. All--order results for anomalous dimensions and matching coefficients are obtained at large $β_0=11-\frac23N_l$. Consistent cancellation between infrared-- and ultraviolet--renormalon ambiguities is demonstrated.

hep-ph↗
arXiv1994

Two-Loop Gluon-Condensate Contributions To Heavy-Quark Current Correlators: Exact Results And Approximations

The coefficient functions of the gluon condensate $ $, in the correlators of heavy-quark vector, axial, scalar and pseudoscalar currents, are obtained analytically, to two loops, for all values of $z=q^2/4m^2$. In the limiting cases $z\to0$, $z\to1$, and $z\to-\infty$, comparisons are made with previous partial results. Approximation methods, based on these limiting cases, are critically assessed, with a view to three-loop work. High accuracy is achieved using a few moments as input. A {\em single} moment, combined with only the {\em leading} threshold and asymptotic behaviours, gives the two-loop corrections to better than 1% in the next 10 moments. A two-loop fit to vector data yields $<\frac{α_{\rm s}}πG^2>\approx0.021$ GeV$^4$.

hep-ph↗
arXiv1993

Connections between Deep-Inelastic and Annihilation Processes at Next-to-Next-to-Leading Order and Beyond

We have discovered 7 intimate connections between the published results for the radiative corrections, $\Ck$, to the Gross--Llewellyn Smith (GLS) sum rule, in deep-inelastic lepton scattering, and the radiative corrections, $\Cr$, to the Adler function of the flavour-singlet vector current, in $\ee$ annihilation. These include a surprising relation between the scheme-independent single-electron-loop contributions to the 4-loop QED $β$\/-function and the zero-fermion-loop abelian terms in the 3-loop GLS sum rule. The combined effect of all 7 relations is to give the factorization of the 2-loop $β$\/-function in \[\Ds\equiv\Ck\Cr-1=\frac{\Be}{\Aq}\left\{S_1\Cf\Aq+\left[S_2\Tf\Nf +\Sa\Ca+\Sf\Cf\right]\Cf\Aq^2\right\}+O(\Aq^4)\,,\] where $\Aq=\al(μ^2=Q^2)/4π$ is the $\MS$ coupling of an arbitrary colour gauge theory, and \[S_1=-\Df{21}{2}+12\Ze3\,;\quad S_2=\Df{326}{3}-\Df{304}{3}\Ze3\,;\quad \Sa=-\Df{629}{2}+\Df{884}{3}\Ze3\,;\quad \Sf=\Df{397}{6}+136\Ze3-240\Ze5\] specify the sole content of $\Ck$ that is not already encoded in $\Cr$ and $\Be=Q^2\rd\Aq/\rd Q^2$ at $O(\Aq^3)$. The same result is obtained by combining the radiative corrections to Bjorken's polarized sum rule with those for the Adler function of the non-singlet axial current. We suggest possible origins of $β$ in the `Crewther discrepancy', $\Ds$, and determine $\Ds/(\Be/\Aq)$, to all orders in $\Nf\Aq$, in the large-$\Nf$ limit, obtaining the {\em entire\/} series of coefficients of which $S_1$ and $S_2$ are merely the first two members.

hep-ph↗