Multiple Deligne values: a data mine with empirically tamed denominators
Multiple Deligne values: a data mine with empirically tamed denominators
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多个德利涅值：具有经过经验驯服的分母的数据挖掘
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发表时间：
2014
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通讯作者：
D. Broadhurst
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作者：
D. Broadhurst
多重Deligne值（MDV）是微分形式$A=d\log（x）$，$B=-d\log（1-x）$和$D=-d\log（1-\lambda x）$在区间$x\in[0，1]$上的迭代积分，其中$\lambda$是本原六次单位根。质量为11的MDV通过希格斯玻色子的自耦合反项进入粒子物理学标准模型的7圈重正化。埃里克·潘泽（Erik Panzer）最近的一项评估显示，有理系数的分母中存在令人震惊的素数50909和121577，这些分母将这个反项简化为皮埃尔·德利涅（Pierre Deligne）思想提出的林登基。奥利弗施内茨研究了这个问题，使用的方法从弗朗西斯布朗。这使得2111，14929，24137，50909和121577作为系数$\pi^{11}/\sqrt{3}$的分母的因子。这里我构造一个基，使得结果中没有大于3的分母素数出现。这是通过构建一个13，369，520个合理系数的数据挖掘来实现的，具有驯服的分解器，用于减少118，097个权重高达11的MDV。然后，仅53个基元的数值数据使得能够非常快速地评估所有这些MDV到20000位。在此Aufbau的过程中，制定并严格测试了MDV的六种方法。
Multiple Deligne values (MDVs) are iterated integrals on the interval $x\in[0,1]$ of the differential forms $A=d\log(x)$, $B=-d\log(1-x)$ and $D=-d\log(1-\lambda x)$, where $\lambda$ is a primitive sixth root of unity. MDVs of weight 11 enter the renormalization of the standard model of particle physics at 7 loops, via a counterterm for the self-coupling of the Higgs boson. A recent evaluation by Erik Panzer exhibited the alarming primes 50909 and 121577 in the denominators of rational coefficients that reduce this counterterm to a Lyndon basis suggested by ideas from Pierre Deligne. Oliver Schnetz has studied this problem, using a method from Francis Brown. This gave 2111, 14929, 24137, 50909 and 121577 as factors of the denominator of the coefficient of $\pi^{11}/\sqrt{3}$. Here I construct a basis such that no denominator prime greater than 3 appears in the result. This is achieved by building a datamine of 13,369,520 rational coefficients, with tame denominators, for the the reductions of 118,097 MDVs with weights up to 11. Then numerical data for merely 53 primitives enables very fast evaluation of all of these MDVs to 20000 digits. In the course of this Aufbau, six conjectures for MDVs are formulated and stringently tested.