Feynman Integrals, Polylogarithms and Symbols
Transcript of Feynman Integrals, Polylogarithms and Symbols
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Feynman Integrals, Polylogarithms and Symbols
Vittorio Del Duca
INFN LNF
ACAT2011 7 September 2011
based on work with Claude Duhr & Volodya Smirnov
Wednesday, September 7, 2011
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let us consider l-loop n-point Feynman integrals
example: 1-loop n-point scalar Feynman integralwithout internal or external masses
IDn
({νi}; {Q2
i })∝
∫dDk
iπD/2
n∏
i=1
1
Dνii
D1 = k2 + i0, Dm =
k +m−1∑
j=1
kj
2
+ i0, m = 2, . . . , n
kj are external momenta, Qi2 are Mandelstam invariants
Wednesday, September 7, 2011
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suppose that we can compute analytically the Feynman integral, which is given as a combination of multiple polylogarithms
G(a, !w; z) =∫ z
0
dt
t− aG(!w; t) , G(a; z) = ln
(1− z
a
)Goncharov 98
let us consider l-loop n-point Feynman integrals
example: 1-loop n-point scalar Feynman integralwithout internal or external masses
IDn
({νi}; {Q2
i })∝
∫dDk
iπD/2
n∏
i=1
1
Dνii
D1 = k2 + i0, Dm =
k +m−1∑
j=1
kj
2
+ i0, m = 2, . . . , n
kj are external momenta, Qi2 are Mandelstam invariants
Wednesday, September 7, 2011
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suppose that the result is quite involved, which is the analytic outcome of the Feynman integral, but that we are interested in a ``simpler’’ analytic result, e.g. because either we want to study the underlying theory, or we just need a numeric result, but we hope to use simpler numeric routines of the functions involved
Wednesday, September 7, 2011
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suppose that the result is quite involved, which is the analytic outcome of the Feynman integral, but that we are interested in a ``simpler’’ analytic result, e.g. because either we want to study the underlying theory, or we just need a numeric result, but we hope to use simpler numeric routines of the functions involved
how feasible is it to ``simplify’’ the result, i.e. to re-write itin terms of either simpler functions or a simpler combination of multiple polylogarithms ?
Wednesday, September 7, 2011
![Page 6: Feynman Integrals, Polylogarithms and Symbols](https://reader034.fdocuments.us/reader034/viewer/2022042318/625cb55329171d7242148766/html5/thumbnails/6.jpg)
suppose that the result is quite involved, which is the analytic outcome of the Feynman integral, but that we are interested in a ``simpler’’ analytic result, e.g. because either we want to study the underlying theory, or we just need a numeric result, but we hope to use simpler numeric routines of the functions involved
how feasible is it to ``simplify’’ the result, i.e. to re-write itin terms of either simpler functions or a simpler combination of multiple polylogarithms ?
suppose that the only tool we have is some (complicated) functional relations which relate multiple polylogarithms
Wednesday, September 7, 2011
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suppose that the result is quite involved, which is the analytic outcome of the Feynman integral, but that we are interested in a ``simpler’’ analytic result, e.g. because either we want to study the underlying theory, or we just need a numeric result, but we hope to use simpler numeric routines of the functions involved
how feasible is it to ``simplify’’ the result, i.e. to re-write itin terms of either simpler functions or a simpler combination of multiple polylogarithms ?
using those functional relations, it may become a frustating and unyielding exercise to simplify the result
suppose that the only tool we have is some (complicated) functional relations which relate multiple polylogarithms
Wednesday, September 7, 2011
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an example:
we computed analytically the finite part of the 2-loop 6-point amplitude in N=4 Super Yang-Mills
Duhr Smirnov VDD 09
Wednesday, September 7, 2011
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an example:
we computed analytically the finite part of the 2-loop 6-point amplitude in N=4 Super Yang-Mills
Duhr Smirnov VDD 09
the result is relevant to understand the weak-coupling structure of scattering amplitudes in N=4 SYM
Wednesday, September 7, 2011
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an example:
we computed analytically the finite part of the 2-loop 6-point amplitude in N=4 Super Yang-Mills
the result was given in terms of O(103) multiple polylogarithms G(u1, u2, u3)
Duhr Smirnov VDD 09
the result is relevant to understand the weak-coupling structure of scattering amplitudes in N=4 SYM
Wednesday, September 7, 2011
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an example:
we computed analytically the finite part of the 2-loop 6-point amplitude in N=4 Super Yang-Mills
the result was given in terms of O(103) multiple polylogarithms G(u1, u2, u3)
Duhr Smirnov VDD 09
the result is relevant to understand the weak-coupling structure of scattering amplitudes in N=4 SYM
it turned out to be unfeasible to ``simplify’’ the result throughfunctional relations which related the multiple polylogarithms
Wednesday, September 7, 2011
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N=4 Super Yang-Millsmaximal supersymmetric theory (without gravity)conformally invariant, β fn. = 0
spin 1 gluon4 spin 1/2 gluinos6 spin 0 real scalars
Wednesday, September 7, 2011
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N=4 Super Yang-Millsmaximal supersymmetric theory (without gravity)conformally invariant, β fn. = 0
spin 1 gluon4 spin 1/2 gluinos6 spin 0 real scalars
‘t Hooft limit: Nc →∞ with λ = g2Nc fixed
only planar diagrams
Wednesday, September 7, 2011
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N=4 Super Yang-Millsmaximal supersymmetric theory (without gravity)conformally invariant, β fn. = 0
spin 1 gluon4 spin 1/2 gluinos6 spin 0 real scalars
‘t Hooft limit: Nc →∞ with λ = g2Nc fixed
only planar diagrams
AdS/CFT duality Maldacena 97
large-λ limit of 4dim CFT ↔ weakly-coupled string theory
(aka weak-strong duality)
Wednesday, September 7, 2011
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MHV amplitudes in planar N=4 SYM
at any order in the coupling, a colour-ordered MHV amplitudein N=4 SYM can be written as tree-level amplitude timeshelicity-free loop coefficient M (L)
n = M (0)n m(L)
n
Wednesday, September 7, 2011
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MHV amplitudes in planar N=4 SYM
at any order in the coupling, a colour-ordered MHV amplitudein N=4 SYM can be written as tree-level amplitude timeshelicity-free loop coefficient M (L)
n = M (0)n m(L)
n
m(1)n =
∑
pq
F 2me(p, q, P, Q)
at 1 loop
n ≥ 6
Wednesday, September 7, 2011
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MHV amplitudes in planar N=4 SYM
at any order in the coupling, a colour-ordered MHV amplitudein N=4 SYM can be written as tree-level amplitude timeshelicity-free loop coefficient M (L)
n = M (0)n m(L)
n
m(1)n =
∑
pq
F 2me(p, q, P, Q)
at 1 loop
n ≥ 6
at 2 loops, iteration formula for the n-pt amplitude
Anastasiou Bern Dixon Kosower 03
m(2)n (ε) =
12
[m(1)
n (ε)]2
+ f (2)(ε) m(1)n (2ε) + Const(2) + R
Wednesday, September 7, 2011
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MHV amplitudes in planar N=4 SYM
at any order in the coupling, a colour-ordered MHV amplitudein N=4 SYM can be written as tree-level amplitude timeshelicity-free loop coefficient M (L)
n = M (0)n m(L)
n
m(1)n =
∑
pq
F 2me(p, q, P, Q)
at 1 loop
n ≥ 6
at 2 loops, iteration formula for the n-pt amplitude
Anastasiou Bern Dixon Kosower 03
m(2)n (ε) =
12
[m(1)
n (ε)]2
+ f (2)(ε) m(1)n (2ε) + Const(2) + R
at all loops, ansatz for a resummed exponent
Bern Dixon Smirnov 05
m(L)n = exp
[ ∞∑
l=1
al(f (l)(ε) m(1)
n (lε) + Const(l) + E(l)n (ε)
)]+ R
Wednesday, September 7, 2011
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MHV amplitudes in planar N=4 SYM
at any order in the coupling, a colour-ordered MHV amplitudein N=4 SYM can be written as tree-level amplitude timeshelicity-free loop coefficient M (L)
n = M (0)n m(L)
n
m(1)n =
∑
pq
F 2me(p, q, P, Q)
at 1 loop
n ≥ 6
at 2 loops, iteration formula for the n-pt amplitude
Anastasiou Bern Dixon Kosower 03
m(2)n (ε) =
12
[m(1)
n (ε)]2
+ f (2)(ε) m(1)n (2ε) + Const(2) + R
at all loops, ansatz for a resummed exponent
Bern Dixon Smirnov 05
m(L)n = exp
[ ∞∑
l=1
al(f (l)(ε) m(1)
n (lε) + Const(l) + E(l)n (ε)
)]+ R
remainderfunction
Wednesday, September 7, 2011
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the ansatz checked for the 3-loop 4-pt amplitude2-loop 5-pt amplitude Cachazo Spradlin Volovich 06
Bern Czakon Kosower Roiban Smirnov 06
Bern Dixon Smirnov 05
Wednesday, September 7, 2011
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the ansatz checked for the 3-loop 4-pt amplitude2-loop 5-pt amplitude Cachazo Spradlin Volovich 06
Bern Czakon Kosower Roiban Smirnov 06
Bern Dixon Smirnov 05
the ansatz fails on 2-loop 6-pt amplitude Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08Alday Maldacena 07; Bartels Lipatov Sabio-Vera 08
Wednesday, September 7, 2011
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the ansatz checked for the 3-loop 4-pt amplitude2-loop 5-pt amplitude Cachazo Spradlin Volovich 06
Bern Czakon Kosower Roiban Smirnov 06
Bern Dixon Smirnov 05
the ansatz fails on 2-loop 6-pt amplitude Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08Alday Maldacena 07; Bartels Lipatov Sabio-Vera 08
R(2)n = m(2)
n (ε)− 12
[m(1)
n (ε)]2− f (2)(ε) m(1)
n (2ε)− Const(2)
at 2 loops, the remainder function characterises the deviation from the ansatz
R(2)6 known analytically Duhr Smirnov VDD 09
Wednesday, September 7, 2011
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the ansatz checked for the 3-loop 4-pt amplitude2-loop 5-pt amplitude Cachazo Spradlin Volovich 06
Bern Czakon Kosower Roiban Smirnov 06
Bern Dixon Smirnov 05
for n = 4, 5, R is a constantfor n ≥ 6, R is an unknown function of conformally invariant cross ratios
the ansatz fails on 2-loop 6-pt amplitude Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08Alday Maldacena 07; Bartels Lipatov Sabio-Vera 08
R(2)n = m(2)
n (ε)− 12
[m(1)
n (ε)]2− f (2)(ε) m(1)
n (2ε)− Const(2)
at 2 loops, the remainder function characterises the deviation from the ansatz
R(2)6 known analytically Duhr Smirnov VDD 09
Wednesday, September 7, 2011
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the ansatz checked for the 3-loop 4-pt amplitude2-loop 5-pt amplitude Cachazo Spradlin Volovich 06
Bern Czakon Kosower Roiban Smirnov 06
Bern Dixon Smirnov 05
for n = 4, 5, R is a constantfor n ≥ 6, R is an unknown function of conformally invariant cross ratios
for n = 6, the conformally invariant cross ratios are
thus x2k,k+r = (pk + . . . + pk+r−1)2
u1 =x2
13x246
x214x
236
u2 =x2
24x215
x225x
214
u3 =x2
35x226
x236x
225
1
2
6
3
4
5
pi = xi − xi+1xi are variables in a dual space s.t.
the ansatz fails on 2-loop 6-pt amplitude Bern Dixon Kosower Roiban Spradlin Vergu Volovich 08Alday Maldacena 07; Bartels Lipatov Sabio-Vera 08
R(2)n = m(2)
n (ε)− 12
[m(1)
n (ε)]2− f (2)(ε) m(1)
n (2ε)− Const(2)
at 2 loops, the remainder function characterises the deviation from the ansatz
R(2)6 known analytically Duhr Smirnov VDD 09
Wednesday, September 7, 2011
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2-loop 6-point remainder function R6(2)
the remainder function R6(2) is explicitly dependent
on the cross ratios u1, u2, u3
Duhr Smirnov VDD 09
Wednesday, September 7, 2011
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2-loop 6-point remainder function R6(2)
the remainder function R6(2) is explicitly dependent
on the cross ratios u1, u2, u3
Duhr Smirnov VDD 09
it is of uniform transcendental weight 4
transcendental weights: w(ln x) = w(π) = 1 w(Li2(x)) = w(π2) = 2
Wednesday, September 7, 2011
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2-loop 6-point remainder function R6(2)
the remainder function R6(2) is explicitly dependent
on the cross ratios u1, u2, u3
Duhr Smirnov VDD 09
it is in agreement with the numeric calculation byAnastasiou Brandhuber Heslop Khoze Spence Travaglini 09
it is of uniform transcendental weight 4
transcendental weights: w(ln x) = w(π) = 1 w(Li2(x)) = w(π2) = 2
Wednesday, September 7, 2011
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2-loop 6-point remainder function R6(2)
the remainder function R6(2) is explicitly dependent
on the cross ratios u1, u2, u3
Duhr Smirnov VDD 09
it is in agreement with the numeric calculation byAnastasiou Brandhuber Heslop Khoze Spence Travaglini 09
straightforward computation
it is of uniform transcendental weight 4
transcendental weights: w(ln x) = w(π) = 1 w(Li2(x)) = w(π2) = 2
Wednesday, September 7, 2011
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2-loop 6-point remainder function R6(2)
the remainder function R6(2) is explicitly dependent
on the cross ratios u1, u2, u3
Duhr Smirnov VDD 09
it is in agreement with the numeric calculation byAnastasiou Brandhuber Heslop Khoze Spence Travaglini 09
straightforward computation
finite answer, but in intermediate steps many divergencesoutput is punishingly long: O(103) multiple polylogarithms G(u1, u2, u3)
it is of uniform transcendental weight 4
transcendental weights: w(ln x) = w(π) = 1 w(Li2(x)) = w(π2) = 2
Wednesday, September 7, 2011
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Goncharov Spradlin Vergu Volovich 10
R(2)6,WL(u1, u2, u3) =
3∑
i=1
(L4(x+
i , x−i )− 12Li4(1− 1/ui)
)
− 18
(3∑
i=1
Li2(1− 1/ui)
)2
+J4
24+
π2
12J2 +
π4
72
yet, our result was given in terms of polylogarithms
Wednesday, September 7, 2011
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Goncharov Spradlin Vergu Volovich 10
x±i = uix±
where
x± =u1 + u2 + u3 − 1±
√∆
2u1u2u3
∆ =( u1 + u2 + u3 − 1)2 − 4u1u2u3
L4(x+, x−) =3∑
m=0
(−1)m
(2m)!!log(x+x−)m(!4−m(x+) + !4−m(x−)) +
18!!
log(x+x−)4
!n(x) =12
(Lin(x)− (−1)nLin(1/x)) J =3∑
i=1
(!1(x+i )− !1(x−i ))
R(2)6,WL(u1, u2, u3) =
3∑
i=1
(L4(x+
i , x−i )− 12Li4(1− 1/ui)
)
− 18
(3∑
i=1
Li2(1− 1/ui)
)2
+J4
24+
π2
12J2 +
π4
72
yet, our result was given in terms of polylogarithms
Wednesday, September 7, 2011
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Goncharov Spradlin Vergu Volovich 10
x±i = uix±
where
x± =u1 + u2 + u3 − 1±
√∆
2u1u2u3
∆ =( u1 + u2 + u3 − 1)2 − 4u1u2u3
L4(x+, x−) =3∑
m=0
(−1)m
(2m)!!log(x+x−)m(!4−m(x+) + !4−m(x−)) +
18!!
log(x+x−)4
!n(x) =12
(Lin(x)− (−1)nLin(1/x)) J =3∑
i=1
(!1(x+i )− !1(x−i ))
R(2)6,WL(u1, u2, u3) =
3∑
i=1
(L4(x+
i , x−i )− 12Li4(1− 1/ui)
)
− 18
(3∑
i=1
Li2(1− 1/ui)
)2
+J4
24+
π2
12J2 +
π4
72
yet, our result was given in terms of polylogarithms
not a new, independent, computationjust a manipulation of our result
Wednesday, September 7, 2011
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Goncharov Spradlin Vergu Volovich 10
x±i = uix±
where
x± =u1 + u2 + u3 − 1±
√∆
2u1u2u3
∆ =( u1 + u2 + u3 − 1)2 − 4u1u2u3
L4(x+, x−) =3∑
m=0
(−1)m
(2m)!!log(x+x−)m(!4−m(x+) + !4−m(x−)) +
18!!
log(x+x−)4
!n(x) =12
(Lin(x)− (−1)nLin(1/x)) J =3∑
i=1
(!1(x+i )− !1(x−i ))
R(2)6,WL(u1, u2, u3) =
3∑
i=1
(L4(x+
i , x−i )− 12Li4(1− 1/ui)
)
− 18
(3∑
i=1
Li2(1− 1/ui)
)2
+J4
24+
π2
12J2 +
π4
72
yet, our result was given in terms of polylogarithms
not a new, independent, computationjust a manipulation of our result
answer is short and simpleintroduces symbols in TH physics
Wednesday, September 7, 2011
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SymbolsFn. F of deg(F) = n : fn. with log cuts, s.t. Disc = 2πi × f, with deg(f) = n-1
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SymbolsFn. F of deg(F) = n : fn. with log cuts, s.t. Disc = 2πi × f, with deg(f) = n-1
deg(const) = 0 ➙ deg(π) = 0ln x : cut along [-∞, 0] with Disc = 2πi ➙ deg(ln x) = 1Li2(x) : cut along [1,∞] with Disc = -2πi ln x ➙ deg(Li2(x)) = 2
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SymbolsFn. F of deg(F) = n : fn. with log cuts, s.t. Disc = 2πi × f, with deg(f) = n-1
deg(const) = 0 ➙ deg(π) = 0ln x : cut along [-∞, 0] with Disc = 2πi ➙ deg(ln x) = 1Li2(x) : cut along [1,∞] with Disc = -2πi ln x ➙ deg(Li2(x)) = 2
take a fn. defined as an iterated integral of logs of rational functions Ri
Tk =
∫ b
ad lnR1 ◦ · · · ◦ d lnRk =
∫ b
a
(∫ t
ad lnR1 ◦ · · · ◦ d lnRk−1
)d lnRk(t)
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SymbolsFn. F of deg(F) = n : fn. with log cuts, s.t. Disc = 2πi × f, with deg(f) = n-1
deg(const) = 0 ➙ deg(π) = 0ln x : cut along [-∞, 0] with Disc = 2πi ➙ deg(ln x) = 1Li2(x) : cut along [1,∞] with Disc = -2πi ln x ➙ deg(Li2(x)) = 2
take a fn. defined as an iterated integral of logs of rational functions Ri
Tk =
∫ b
ad lnR1 ◦ · · · ◦ d lnRk =
∫ b
a
(∫ t
ad lnR1 ◦ · · · ◦ d lnRk−1
)d lnRk(t)
Sym[Tk] = R1 ⊗ · · ·⊗Rk
is defined on the tensor product of the group of rational functions, modulo constants
the symbol
· · ·⊗R1R2 ⊗ · · · = · · ·⊗R1 ⊗ · · · + · · ·⊗R2 ⊗ · · ·
· · ·⊗ (cR1)⊗ · · · = · · ·⊗R1 ⊗ · · ·
Zagier, Goncharov
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SymbolsFn. F of deg(F) = n : fn. with log cuts, s.t. Disc = 2πi × f, with deg(f) = n-1
deg(const) = 0 ➙ deg(π) = 0ln x : cut along [-∞, 0] with Disc = 2πi ➙ deg(ln x) = 1Li2(x) : cut along [1,∞] with Disc = -2πi ln x ➙ deg(Li2(x)) = 2
take a fn. defined as an iterated integral of logs of rational functions Ri
Tk =
∫ b
ad lnR1 ◦ · · · ◦ d lnRk =
∫ b
a
(∫ t
ad lnR1 ◦ · · · ◦ d lnRk−1
)d lnRk(t)
Li1(z) = − ln(1− z) Lik(z) =
∫ z
0d ln t Lik−1(t)
k-1 times
Sym[Lik(z)] = −(1− z)⊗ z ⊗ · · ·⊗ z︸ ︷︷ ︸
Sym[Tk] = R1 ⊗ · · ·⊗Rk
is defined on the tensor product of the group of rational functions, modulo constants
the symbol
· · ·⊗R1R2 ⊗ · · · = · · ·⊗R1 ⊗ · · · + · · ·⊗R2 ⊗ · · ·
· · ·⊗ (cR1)⊗ · · · = · · ·⊗R1 ⊗ · · ·
Zagier, Goncharov
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Sym[lnx ln y] = x⊗ y + y ⊗ x
Disc(ln x ln y) = }2πi ln x along the y cut [-∞, 0]
2πi ln y along the x cut [-∞, 0]
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Sym[lnx ln y] = x⊗ y + y ⊗ x
Disc(ln x ln y) = }2πi ln x along the y cut [-∞, 0]
2πi ln y along the x cut [-∞, 0]
in general, if Disc(f g) = Disc(f) g + f Disc(g)
and Sym[f ] = ⊗ni=1Ri
then Sym[fg] =∑
σ
⊗ni=1Rσ(i)
where σ denotes the set of all shuffles of n+(m-n) elements
Sym[g] = ⊗mi=n+1Ri
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Sym[lnx ln y] = x⊗ y + y ⊗ x
Disc(ln x ln y) = }2πi ln x along the y cut [-∞, 0]
2πi ln y along the x cut [-∞, 0]
in general, if Disc(f g) = Disc(f) g + f Disc(g)
and Sym[f ] = ⊗ni=1Ri
then Sym[fg] =∑
σ
⊗ni=1Rσ(i)
where σ denotes the set of all shuffles of n+(m-n) elements
Sym[g] = ⊗mi=n+1Ri
e.g. Sym[g] = R3 ⊗R4
Sym[fg] = R1 ⊗R2 ⊗R3 ⊗R4 +R1 ⊗R3 ⊗R2 ⊗R4 +R1 ⊗R3 ⊗R4 ⊗R2
+ R3 ⊗R1 ⊗R2 ⊗R4 +R3 ⊗R1 ⊗R4 ⊗R2 +R3 ⊗R4 ⊗R1 ⊗R2
Sym[f ] = R1 ⊗R2
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Sym[lnx ln y] = x⊗ y + y ⊗ x
Disc(ln x ln y) = }2πi ln x along the y cut [-∞, 0]
2πi ln y along the x cut [-∞, 0]
symbols form a shuffle algebra, i.e. a vector space with a shuffle product(also iterated integrals and multiple polylogarithms form shuffle algebras)
in general, if Disc(f g) = Disc(f) g + f Disc(g)
and Sym[f ] = ⊗ni=1Ri
then Sym[fg] =∑
σ
⊗ni=1Rσ(i)
where σ denotes the set of all shuffles of n+(m-n) elements
Sym[g] = ⊗mi=n+1Ri
e.g. Sym[g] = R3 ⊗R4
Sym[fg] = R1 ⊗R2 ⊗R3 ⊗R4 +R1 ⊗R3 ⊗R2 ⊗R4 +R1 ⊗R3 ⊗R4 ⊗R2
+ R3 ⊗R1 ⊗R2 ⊗R4 +R3 ⊗R1 ⊗R4 ⊗R2 +R3 ⊗R4 ⊗R1 ⊗R2
Sym[f ] = R1 ⊗R2
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polylogarithm identities satisfied by the function fbecome algebraic identities satisfied by its symbol
Sym[lnx ln y] = x⊗ y + y ⊗ x
Disc(ln x ln y) = }2πi ln x along the y cut [-∞, 0]
2πi ln y along the x cut [-∞, 0]
symbols form a shuffle algebra, i.e. a vector space with a shuffle product(also iterated integrals and multiple polylogarithms form shuffle algebras)
in general, if Disc(f g) = Disc(f) g + f Disc(g)
and Sym[f ] = ⊗ni=1Ri
then Sym[fg] =∑
σ
⊗ni=1Rσ(i)
where σ denotes the set of all shuffles of n+(m-n) elements
Sym[g] = ⊗mi=n+1Ri
e.g. Sym[g] = R3 ⊗R4
Sym[fg] = R1 ⊗R2 ⊗R3 ⊗R4 +R1 ⊗R3 ⊗R2 ⊗R4 +R1 ⊗R3 ⊗R4 ⊗R2
+ R3 ⊗R1 ⊗R2 ⊗R4 +R3 ⊗R1 ⊗R4 ⊗R2 +R3 ⊗R4 ⊗R1 ⊗R2
Sym[f ] = R1 ⊗R2
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take f, g with deg(f) = deg(g) = n and Sym[f] = Sym[g]then f-g = h with deg(h) = n -1the symbol does not know about hinfo on the degree n-1 is lost by taking the symbol
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take f, g with deg(f) = deg(g) = n and Sym[f] = Sym[g]then f-g = h with deg(h) = n -1the symbol does not know about hinfo on the degree n-1 is lost by taking the symbol
in N=4 SYM, polynomials exhibit a uniform weightw(ln x) = 1, w(Lik(x)) = k, w(π) = 1➙ symbols fix polynomials up to factors of π times functions of lesser degree
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take f, g with deg(f) = deg(g) = n and Sym[f] = Sym[g]then f-g = h with deg(h) = n -1the symbol does not know about hinfo on the degree n-1 is lost by taking the symbol
in N=4 SYM, polynomials exhibit a uniform weightw(ln x) = 1, w(Lik(x)) = k, w(π) = 1➙ symbols fix polynomials up to factors of π times functions of lesser degree
Thus, we have a procedure to simplify a generic function of polylogarithms:
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take f, g with deg(f) = deg(g) = n and Sym[f] = Sym[g]then f-g = h with deg(h) = n -1the symbol does not know about hinfo on the degree n-1 is lost by taking the symbol
in N=4 SYM, polynomials exhibit a uniform weightw(ln x) = 1, w(Lik(x)) = k, w(π) = 1➙ symbols fix polynomials up to factors of π times functions of lesser degree
Thus, we have a procedure to simplify a generic function of polylogarithms:
find suitable variables such that arguments of multiple polylogarithmsbecome rational functions
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take f, g with deg(f) = deg(g) = n and Sym[f] = Sym[g]then f-g = h with deg(h) = n -1the symbol does not know about hinfo on the degree n-1 is lost by taking the symbol
in N=4 SYM, polynomials exhibit a uniform weightw(ln x) = 1, w(Lik(x)) = k, w(π) = 1➙ symbols fix polynomials up to factors of π times functions of lesser degree
Thus, we have a procedure to simplify a generic function of polylogarithms:
find suitable variables such that arguments of multiple polylogarithmsbecome rational functions
determine the symbol of the function
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take f, g with deg(f) = deg(g) = n and Sym[f] = Sym[g]then f-g = h with deg(h) = n -1the symbol does not know about hinfo on the degree n-1 is lost by taking the symbol
in N=4 SYM, polynomials exhibit a uniform weightw(ln x) = 1, w(Lik(x)) = k, w(π) = 1➙ symbols fix polynomials up to factors of π times functions of lesser degree
Thus, we have a procedure to simplify a generic function of polylogarithms:
find suitable variables such that arguments of multiple polylogarithmsbecome rational functions
determine the symbol of the function
through some symbol-processing procedure,find a simpler form of the integral in terms of multiple polylogarithms
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polylogarithm identities satisfied by the function fbecome algebraic identities satisfied by its symbol
let us prove the identity Li2(1− x) = −Li2(x)− lnx ln(1− x) +π2
6
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polylogarithm identities satisfied by the function fbecome algebraic identities satisfied by its symbol
let us prove the identity Li2(1− x) = −Li2(x)− lnx ln(1− x) +π2
6
Sym[Li2(1− x)] = −x⊗ (1− x)Sym[Li2(x)] = −(1− x)⊗ xproof
Sym[lnx ln(1− x)] = x⊗ (1− x) + (1− x)⊗ x
Sym[Li2(1− x)] = Sym[−Li2(x)− lnx ln(1− x)]thus
Li2(1− x) = −Li2(x)− lnx ln(1− x) + cπ2 + iπ (c′ lnx+ c′′ ln(1− x))
which determines the function up to functions of lesser degree
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polylogarithm identities satisfied by the function fbecome algebraic identities satisfied by its symbol
let us prove the identity Li2(1− x) = −Li2(x)− lnx ln(1− x) +π2
6
but the equation is real for 0 < x < 1, so c’=c’’=0
at x = 1 0 = −π2
6− 0 + cπ2 c =
1
6
Sym[Li2(1− x)] = −x⊗ (1− x)Sym[Li2(x)] = −(1− x)⊗ xproof
Sym[lnx ln(1− x)] = x⊗ (1− x) + (1− x)⊗ x
Sym[Li2(1− x)] = Sym[−Li2(x)− lnx ln(1− x)]thus
Li2(1− x) = −Li2(x)− lnx ln(1− x) + cπ2 + iπ (c′ lnx+ c′′ ln(1− x))
which determines the function up to functions of lesser degree
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let us prove the identity Li2
(1− 1
x
)= −Li2(1− x)− 1
2ln2 x
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let us prove the identity Li2
(1− 1
x
)= −Li2(1− x)− 1
2ln2 x
proof Sym[Li2(1− x)] = −x⊗ (1− x)
Sym
[Li2
(1− 1
x
)]= − 1
x⊗
(1− 1
x
)
= x⊗ x− 1
x= x⊗ (1− x)− x⊗ x
Sym[ln2 x] = 2x⊗ x
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let us prove the identity Li2
(1− 1
x
)= −Li2(1− x)− 1
2ln2 x
proof Sym[Li2(1− x)] = −x⊗ (1− x)
Sym
[Li2
(1− 1
x
)]= − 1
x⊗
(1− 1
x
)
= x⊗ x− 1
x= x⊗ (1− x)− x⊗ x
Sym[ln2 x] = 2x⊗ x
Sym
[−Li2(1− x)− 1
2ln2 x
]= x⊗ (1− x)− 1
22x⊗ x = Sym
[Li2
(1− 1
x
)]thus
which determines the function up to functions of lesser degree
Li2
(1− 1
x
)= −Li2(1− x)− 1
2ln2 x+ cπ2
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let us prove the identity Li2
(1− 1
x
)= −Li2(1− x)− 1
2ln2 x
proof Sym[Li2(1− x)] = −x⊗ (1− x)
Sym
[Li2
(1− 1
x
)]= − 1
x⊗
(1− 1
x
)
= x⊗ x− 1
x= x⊗ (1− x)− x⊗ x
Sym[ln2 x] = 2x⊗ x
Sym
[−Li2(1− x)− 1
2ln2 x
]= x⊗ (1− x)− 1
22x⊗ x = Sym
[Li2
(1− 1
x
)]thus
which determines the function up to functions of lesser degree
Li2
(1− 1
x
)= −Li2(1− x)− 1
2ln2 x+ cπ2
at x = 1 0 = −0− 0 + cπ2 c = 0
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multiple polylogarithms can also be defined through nested harmonic sums
Lim1,...,mk(u1, . . . , uk) =∞∑
nk=1
unkk
nmkk
nk−1∑
nk−1=1
. . .n2−1∑
n1=1
un11
nm11
= (−1)kGmk,...,m1
(1
uk, . . . ,
1
u1 · · ·uk
)
Gm1,...,mk(u1, . . . , uk) = G
0, . . . , 0︸ ︷︷ ︸m1−1
, u1, . . . , 0, . . . , 0︸ ︷︷ ︸mk−1
, uk; 1
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multiple polylogarithms can also be defined through nested harmonic sums
Lim1,...,mk(u1, . . . , uk) =∞∑
nk=1
unkk
nmkk
nk−1∑
nk−1=1
. . .n2−1∑
n1=1
un11
nm11
= (−1)kGmk,...,m1
(1
uk, . . . ,
1
u1 · · ·uk
)
Gm1,...,mk(u1, . . . , uk) = G
0, . . . , 0︸ ︷︷ ︸m1−1
, u1, . . . , 0, . . . , 0︸ ︷︷ ︸mk−1
, uk; 1
special values as polylogarithms and Nielsen logarithms
G(!0n;x) =1
n!lnn x G(!an;x) =
1
n!lnn
(1− x
a
)
G(!0n−1, a;x) = −Lin(xa
)G(!0n,!am;x) = (−1)m Sn,m
(xa
)Sn−1,1(x) = Lin(x)
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multiple polylogarithms can also be defined through nested harmonic sums
Lim1,...,mk(u1, . . . , uk) =∞∑
nk=1
unkk
nmkk
nk−1∑
nk−1=1
. . .n2−1∑
n1=1
un11
nm11
= (−1)kGmk,...,m1
(1
uk, . . . ,
1
u1 · · ·uk
)
Gm1,...,mk(u1, . . . , uk) = G
0, . . . , 0︸ ︷︷ ︸m1−1
, u1, . . . , 0, . . . , 0︸ ︷︷ ︸mk−1
, uk; 1
special values as polylogarithms and Nielsen logarithms
G(!0n;x) =1
n!lnn x G(!an;x) =
1
n!lnn
(1− x
a
)
G(!0n−1, a;x) = −Lin(xa
)G(!0n,!am;x) = (−1)m Sn,m
(xa
)Sn−1,1(x) = Lin(x)
when the root equals 1, multiple polylogarithms become harmonic polylogarithms (HPLs)
H(a, !w; z) =
∫ z
0dt f(a; t)H(!w; t)
Lin(x) = H(!0n−1, 1;x)
HPLs are defined through iterated integrals
f(−1; t) =1
1 + t, f(0; t) =
1
t, f(1; t) =
1
1− t
{a, !w} ∈ {−1, 0, 1}with
Sn,m(x) = H(!0n,!1m;x)
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... on to symbols
n times
Sym
[1
n!lnn x
]= x⊗ · · ·⊗ x︸ ︷︷ ︸ ≡ x⊗n
Sym[Lin(x)] = −(1− x)⊗ x⊗(n−1)
Sym[Sn,m(x)] = (−1)m(1− x)⊗m ⊗ x⊗n
Sym[H(a1, . . . , an;x)] = (−1)k(an − x)⊗ · · ·⊗ (a1 − x) {ai} ∈{ 0, 1}
k is the number of a’s equal to 1
Sym[lnx] = x
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using symbols, one can reduce the HPLs to a minimal set Buehler Duhr 11
B(1)1 (x) = lnx , B(2)
1 (x) = ln(1− x) , B(3)1 (x) = ln(1 + x)weight 1:
weight 2: B(1)2 (x) = Li2(x) , B(2)
2 (x) = Li2(−x) , B(3)2 (x) = Li2
(1− x
2
)
weight 3:
weight 4:
polylogarithms of type Li3 of various arguments
polylogarithms of type Li4 of various arguments,plus a few polylogarithms of type Li2,2, like Li2,2(-1, x) etc.Alternatively, the polylogarithms of type Li2,2 can be replacedby the HPLs: H(0,1,0,-1; x) and H(0,1,1,-1; x)
if needed numerically, any combination of HPLs up to weight 4 can be evaluated in terms of a minimal set of numerical routines
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using symbols, one can reduce the HPLs to a minimal set Buehler Duhr 11
B(1)1 (x) = lnx , B(2)
1 (x) = ln(1− x) , B(3)1 (x) = ln(1 + x)weight 1:
weight 2: B(1)2 (x) = Li2(x) , B(2)
2 (x) = Li2(−x) , B(3)2 (x) = Li2
(1− x
2
)
weight 3:
weight 4:
polylogarithms of type Li3 of various arguments
polylogarithms of type Li4 of various arguments,plus a few polylogarithms of type Li2,2, like Li2,2(-1, x) etc.Alternatively, the polylogarithms of type Li2,2 can be replacedby the HPLs: H(0,1,0,-1; x) and H(0,1,1,-1; x)
if needed numerically, any combination of HPLs up to weight 4 can be evaluated in terms of a minimal set of numerical routines
These features generalise to multiple polylogarithms
weight 1: one needs functions of type ln x
weight 2:Li3(x)
Li2(x)
weight 3:
weight 4: Li4(x), Li2,2(x,y)weight 5: Li5(x), Li2,3(x,y)weight 6: Li6(x), Li2,4(x,y), Li3,3(x,y), Li2,2,2(x,y)
Duhr Gangl Rhodes (in progress)
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symbol
integral
function
simpler function
2-loop 6-edged Wilson loop
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symbol
integral
function
simpler function
2-loop 6-edged Wilson loop
integral
symbol
(simpler) function
Drummond Henn Trnka 10
Caron-Huot 11
differential equations
discontinuities
algorithmDuhr Gangl Rhodes (in progress)
ideally
Wednesday, September 7, 2011