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We consider the scattering amplitudes of five and six gravitons at tree-level in superstring theory.
Phys. Rept. 89
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D. Zagier, Values of zeta functions and their applications,
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Due to the relation D 2 R ≃ R 2 D^{2}R\!\simeq\!R^{2} some derivative terms may be converted to terms with fewer derivatives at the cost of higher orders in the Riemann tensor. Therefore, we stick to the prescription to write all terms with the highest possible number of Riemann tensors
Cited in the paper.
The set of integral linear combinations of MZVs is a ring, since the product of any two values can be expressed by a (positive) integer linear combination of the other MZVs Zagier , e.g.: ζ ( m ) ζ ( n ) = ζ ( m , n ) + ζ ( n , m ) + ζ ( m + n ) \zeta(m)\zeta(n)=\zeta(m,n)+\zeta(n,m)+\zeta(m+n) (quasi–shuffle or stuffle relation)
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There are many relations over 𝐐 {\bf Q} among MZVs, e.g. ζ ( 4 , 1 ) = 2 ζ ( 5 ) − ζ ( 2 ) ζ ( 3 ) \zeta(4,1)\!\!=\!\!2\zeta(5)-\zeta(2)\zeta(3) . For a given weight w ∈ 𝐍 w\in{\bf N} the dimension d w d_{w} of the space spanned by MZVs of weight w w is given by d w = d w − 2 + d w − 3 d_{w}=d_{w-2}+d_{w-3} ( d 1 = 0 , d 2 = d 3 = d 4 = 1 , d 5 = 2 , d 6 = 2 , d 7 = 3 , d 8 = 4 , d 9 = 5 , d 10 = 7 , … d_{1}=0,d_{2}=d_{3}=d_{4}=1,d_{5}=2,d_{6}=2,d_{7}=3,d_{8}=4,d_{9}=5,d_{10}=7,\ldots ) Zagier and can be constructed by the following basis: for w = 2 w\!\!=\!\!2 by ζ ( 2 ) \zeta(2) , for w = 3 w\!\!=\!\!3 by ζ ( 3 ) \zeta(3) , for w = 4 w\!\!=\!\!4 by ζ ( 2 ) 2 \zeta(2)^{2} , for w = 5 w\!\!=\!\!5 by ζ ( 5 ) , ζ ( 2 ) ζ ( 3 ) \zeta(5),\zeta(2)\zeta(3) , for w = 6 w\!\!=\!\!6 by ζ ( 2 ) 3 , ζ ( 3 ) 2 \zeta(2)^{3},\zeta(3)^{2} , for w = 7 w\!\!=\!\!7 by ζ ( 7 ) , ζ ( 2 ) ζ ( 5 ) , ζ ( 3 ) ζ ( 2 ) 2 \zeta(7),\zeta(2)\zeta(5),\zeta(3)\zeta(2)^{2} , for w = 8 w\!\!=\!\!8 by ζ ( 2 ) 4 , ζ ( 2 ) ζ ( 3 ) 2 , ζ ( 3 ) ζ ( 5 ) , ζ ( 5 , 3 ) \zeta(2)^{4},\zeta(2)\zeta(3)^{2},\zeta(3)\zeta(5),\zeta(5,3) , for w = 9 w=9 by ζ ( 9 ) , ζ ( 7 ) ζ ( 2 ) , ζ ( 5 ) ζ ( 2 ) 2 , ζ ( 3 ) 3 , ζ ( 3 ) ζ ( 2 ) 3 \zeta(9),\zeta(7)\zeta(2),\zeta(5)\zeta(2)^{2},\zeta(3)^{3},\zeta(3)\zeta(2)^{3} , for w = 10 w=10 by ζ ( 7 , 3 ) \zeta(7,3) , ζ ( 5 , 3 ) ζ ( 2 ) , ζ ( 7 ) ζ ( 3 ) , ζ ( 5 ) 2 , ζ ( 5 ) ζ ( 3 ) ζ ( 2 ) , ζ ( 3 ) 2 ζ ( 2 ) 2 , \zeta(5,3)\zeta(2),\zeta(7)\zeta(3),\zeta(5)^{2},\zeta(5)\zeta(3)\zeta(2),\zeta(3)^{2}\zeta(2)^{2}, ζ ( 2 ) 5 \zeta(2)^{5} , etc
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S. Stieberger and T. R. Taylor, Supersymmetry Relations and MHV Amplitudes in Superstring Theory · 2008
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J. Blumlein, D. Broadhurst, and J. Vermaseren, The Multiple Zeta Value Data Mine · 2010
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H. Elvang, D. Z. Freedman, and M. Kiermaier, SUSY Ward identities, Superamplitudes, and Counterterms · 2010
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