Polylogs of sixth roots of unity in QFT David Broadhurst a,b Madrid, - PDF document

Polylogs of sixth roots of unity in QFT David Broadhurst a,b Madrid, 1 December 2014 Plan: 1) Enumerations by weight 2) Enumerations by weight and depth 3) Conjectures for MDVs 4) Massive Feynman diagrams 5) 7-loop counterterm a Department of Physical Sciences, Open University, Milton Keynes MK7 6AA, UK b Institut f¨ ur Mathematik und Institut f¨ ur Physik, Humboldt-Universit¨ at zu Berlin 1

1 Enumerations by weight 7 letter alphabet : √ √ let λ = exp(2 πi/ 6) = (1 + i 3) / 2, λ = (1 − i 3) / 2, A = d log( x ) B = − d log(1 − x ) C = − d log(1 + x ) D = − d log(1 − λx ) D = − d log(1 − λx ) E = − d log(1 − λ 2 x ) 2 x ) E = − d log(1 − λ Subalphabets : { A, B } : Multiple Zeta Values (MZVs) { A, B, C } : Alternating sums { A, B, D } : Multiple Deligne Values (MDVs) { A, B, E, E } : Cube roots of unity 2

Iterated integrals : � 1 � x 1 � x 2 λ dx 1 dx 2 dx 3 Z ( DAB ) ≡ 1 − λx 1 x 2 1 − x 3 0 0 0 Netsted sums : expand − d log(1 − λ n x ) = dx � ( λ n x ) k x k> 0 to obtained nested sums of the form   z k j  z 1 , z 2 , . . . , z d d � � j  ≡ S k a j a 1 , a 2 , . . . , a d k 1 >k 2 >...>k d > 0 j =1 j where z 6 j = 1 and a j is a positive integer. Thus, for example,   n λ m ∞ m − 1  λ, λ λ � �  ≡ Z ( DAB ) = S n 2 1 , 2 m m =1 n =1 3

Shuffle product : � Z ( U ) Z ( V ) = Z ( W ) W ∈S ( U,V ) where S ( U, V ) is the set of all words W that result from shuffling the words U and V . Thus, for example, Z ( AB ) Z ( CD ) = Z ( ABCD ) + Z ( ACBD ) + Z ( ACDB ) + Z ( CABD ) + Z ( CADB ) + Z ( CDAB ) preserves the order of letters in U = AB and V = CD . Stuffle product : the full 7-letter alphabet { A, B, C, D, D, E, E } has stuffle algebra, resulting from shuffling the arguments of nested sums, with extra stuff when indices of summation coincide. For example            1  λ  1 , λ  λ, 1  λ  = S  + S  + S Z ( AB ) Z ( D ) = S  S  2 1 2 , 1 1 , 2 3 = Z ( ABD ) + Z ( DAD ) + Z ( AAD ) Alphabets { A, B } , { A, B, C } , { A, B, E, E } { A, B, C, D, D, E, E } have a double shuffle algeba, but the Deligne alphabet { A, B, D } is not closed under stuffles: Z ( AD ) Z ( D ) = Z ( ADE ) + Z ( DAE ) + Z ( AAE ). 4

Weight : number of letters in a word, W . � Dimension D w of vector space for Q -linear expansions Z ( W ) = n Q n V n of words of weight w in terms of a (conjectured) basis { V n | n = 1 . . . D w } . � w> 0 D w x w . Generating function : G ( x ) = 1 + { A, B } : G ( x ) = 1 / (1 − x 2 − x 3 ), with seeds π 2 and ζ (3), giving Padovan numbers. [Hoffman, Zagier, Terasoma, Brown] { A, B, C } : G ( x ) = 1 / (1 − x − x 2 ), with seeds log(2) and π 2 , giving Fibonacci numbers: D w = F w +1 . [Broadhurst, Deligne] { A, B, D } : G ( x ) = 1 / (1 − x − x 2 ), with seeds π and Cl 2 ( π/ 3), also giving Fibonacci numbers. [Deligne] { A, B, E, E } : G ( x ) = 1 / (1 − 2 x ), with seeds π and log(3), giving D w = 2 w . [Deligne] { A, B, C, D, D, E, E } : G ( x ) = 1 / (1 − 3 x + x 2 ), with seeds π , log(2), log(3) and due attention to weight 2, giving Fibonacci numbers: D w = F 2 w +2 . [Broadhurst, November 2014] PS: 4th roots: 1 / (1 − 2 x ); 8th roots: 1 / (1 − 3 x ). 5

2 Enumerations of primitives by weight and depth Depth : number of letters not equal to A . A word W is a primitive in a given alphabet if Z ( W ) can not be expressed as a Q -linear combination of terms of lesser depth, powers of (2 πi ), or their products. Example [Broadhurst, 1996]: At weight w = 12 and depth d = 4, � Z ( A 3 BA 3 BABAB ) = ζ (4 , 4 , 2 , 2) = 1 / ( j 4 k 4 l 2 m 2 ) j>k>l>m> 0 is a primitive MZV, but is not primitive in the { A, B, C } alphabet, because 2 5 · 3 3 Z ( A 3 BA 3 BABAB ) − 2 14 Z ( A 8 CA 2 B ) = 2 5 · 3 2 ζ 4 (3) + 2 6 · 3 3 · 5 · 13 ζ (9) ζ (3) + 2 6 · 3 3 · 7 · 13 ζ (7) ζ (5) + 2 7 · 3 5 ζ (7) ζ (3) ζ (2) + 2 6 · 3 5 ζ 2 (5) ζ (2) − 2 6 · 3 3 · 5 · 7 ζ (5) ζ (4) ζ (3) − 2 8 · 3 2 ζ (6) ζ 2 (3) − 13177 · 15991 ζ (12) 691 + 2 4 · 3 3 · 5 · 7 ζ (6 , 2) ζ (4) − 2 7 · 3 3 ζ (8 , 2) ζ (2) − 2 6 · 3 2 · 11 2 ζ (10 , 2) � where Z ( A 8 CA 2 B ) = m>n> 0 ( − 1) m + n / ( m 9 n 3 ) has depth 2. 6

For a given alphabet, let N w,d be the dimension of the space of Q -linearly independent primitives of weight w and depth d . We then seek � � (1 − x w y d ) N w,d H ( x, y ) = w> 0 d> 0 According to the Broadhurst-Kreimer conjecture (1997), the answer is bizarre for the { A, B } alphabet of MZVs, namely x 3 x 12 1 − x 2 + y 2 (1 − y 2 ) H ( x, y ) = 1 − y (1 − x 4 )(1 − x 6 ) with a final term that (coincidentally?) counts cuspforms. It appears that only the first root of unity is so badly behaved. In all other cases that I have studied, the single sums tell all: H ( x, y ) = 1 − yH 1 ( x ) � w> 0 N w, 1 x n , given in the following list. with H 1 ( x ) = 7

Enumerations of single sums : { A, B, C } : H 1 ( x ) = x/ (1 − x 2 ) from Z ( A 2 n C ) with n ≥ 0. { A, B, D } : H 1 ( x ) = x 2 / (1 − x ) from Z ( A n D ) with n > 0. { A, B, E, E } : H 1 ( x ) = x/ (1 − x ) from Z ( A n E ) with n ≥ 0. { A, B, D, E } : same as { A, B, E, E } { A, B, C, D, D, E, E } : H 1 ( x ) = x + x/ (1 − x ) from Z ( C ) and Z ( A n E ) with n ≥ 0. PS: 4th roots: x/ (1 − x ); 8th roots: 2 x/ (1 − x ). Reminder : For the { A, B } alphabet of MZVs, H 1 ( x ) = x 3 / (1 − x 2 ) does not tell all. This was first observed by Ihara in the year that the wall fell: MSRI Publ 16 (1989), 299–313, Springer Verlag. 8

Lyndon words provide (conjectural) primitives in all cases except for MZVs. A Lyndon word is a word W such that for every splitting W = UV we have U coming before V , in lexicographic ordering. Here are 3 such cases, with examples up to weight 5. Alternating sums in the { A, B, C } alphabet [Broadhurst, 1997]: take Lyndon words in the { A, C } alphabet and retain those with even powers of A . With w ≤ 5 this gives C , A 2 C , A 2 C 2 , A 4 C , A 2 C 3 . MDVs in the { A, B, D } alphabet [Deligne, 2010]: take Lyndon words in the { A, D } alphabet and retain those in which D is preceded by A . With w ≤ 5 this gives AD , A 2 D , A 3 D , A 4 D , A 2 DAD . 7-letter alphabet of polylogs of 6th roots of unity [Broadhurst, 2014]: take Lyndon words in the { A, E, C } alphabet, omit A and all words in which C is preceded by A . With w ≤ 5 this gives E , C , AE , EC , AAE , AEE , AEC , EEC , ECC , AAAE , AAEE , AAEC , AEEE , AEEC , AECE , AECC , EEEC , EECC , ECCC , AAAAE , AAAEE , AAAEC , AAEAE , AAEEE , AAEEC , AAECE , AAECC , AEAEE , AEAEC , AEEEE , AEEEC , AEECE , AEECC , AECEE , AECEC , AECCE , AECCC , EEEEC , EEECC , EECEC , EECCC , ECECC , ECCCC . 9

Generalized parity conjecture [Broadhurst, 1999]: the primitives may be taken as real parts of Z ( W ) for which the parities of weight and depth of W coincide and as imaginary parts when they differ. A legal word does not begin with B or end in A . Numbers of empirical reductions to conjectured bases : MDVs of the { A, B, D } alphabet: all 118,097 legal words with w ≤ 11. { A, B, E, E } : 12,287 words with w ≤ 7. { A, B, D, E } : 12,287 words with w ≤ 7. { A, B, C, D, D, E, E } : 28,265 words with w ≤ 5 or w = 6 and d ≤ 4. PS: 4th roots: 62,499 words; 8th roots: 23,815 words. MDV datamine with 13,369,520 non-zero rational coefficients: http://physics.open.ac.uk/ � dbroadhu/cert/MDV.tar.gz explained in http://arxiv.org/pdf/1409.7204v1 10

3 Conjectures for MDVs Conjecture 1 [Fibonacci enumeration]: Let F n be the n -th Fibonacci number. Then every Q -linear combination of MDVs of weight w is re- ducible to a Q -linear combination of F w +1 basis terms between which there is no Q -linear relation. Conjecture 2 [enumeration of primitives]: The dimension N w,d of the space of primitive MDVs of weight w and depth d is generated by (1 − x w y d ) N w,d = 1 − x 2 y � � 1 − x. w> 1 d> 0 Conjecture 3 [generalized parity]: The primitives of Conjecture 2 may be taken as real parts of MDVs for which the parities of weight and depth coincide and as imaginary parts of MDVs for which those parities differ. Comments : Motivically plausible, but beyond present reach. Conjecture 3 refers to Conjecture 2 which implies Conjecture 1 which requires that ζ 2 3 /ζ 3 2 is irrational, which no-one knows how to prove. 11

Conjecture 4 [sum rule at odd weight]: At odd weight w > 1, there exists a unique Z -linear reducible combination ( w − 1) / 2 � C w,k ℑ Z ( A w − 2 k − 1 DA 2 k − 1 B ) , X w = (1) k =1 with C w, 1 > 0 and integer coefficients C w,k whose greatest common divisor is unity. Moreover, all of the coefficients are non-zero, X w is free of products of primitives and hence X w /π w reduces to a rational number. Comments: The impact of such reductions in QFT will be discussed later. The datamine contains the relevant rationals up to w = 31. The numerator of X 31 /π 31 contains the 49-digit prime 1052453969156963777695781293476878259787114222411 and the 81-digit prime 5398660771478298532475166018701166835343 \ 25958155228637043335803543859216008062953 Erik Panzer has found systems of equations that determine sets of coef- ficients C w,k for which X w /π w is a rational number, undetermined by his method. I have shown that none of his C w,k vanishes for w ≤ 601. 12