Polylogarithms and Multizeta Values in Massless Feynman Amplitudes

Abstract

The last two decades have seen a remarkable development of analytic methods in the study of Feynman amplitudes in perturbative quantum field theory. The present lecture offers a physicists’ oriented survey of Francis Brown’s work on singlevalued multiple polylogarithms, the associated multizeta periods and their application to Schnetz’s graphical functions and to x-space renormalization. To keep the discussion concrete we restrict attention to explicit examples of primitively divergent graphs in a massless scalar QFT.

Appendices

Appendix 1: Computation of the Integral (5)

Using conformal invariance we can send the variable x ₁ to infinity, x ₄ to zero, x ₂—to a unit 4-vector e and set

$$\displaystyle{ x_{3} = Z\,,\quad \mathrm{where}\quad Z^{2} = z\,\bar{z}\,,\ 2Ze = z +\bar{ z} }$$

(54)

so that the cross ratios (6) assume the form

$$\displaystyle{ u = Z^{2} = z\,\bar{z}\,,\quad v = (Z - e)^{2} = (z - 1)(\bar{z} - 1) }$$

(55)

in accord with (7). Then we can write, introducing spherical coordinates x = r ω, $Z =\vert z\vert \,\omega _{z}$,

$$\displaystyle\begin{array}{rcl} \mathcal{F}(u,v)& =& F(z) = \frac{1} {\pi ^{2}} \int \frac{d^{4}\,x} {x^{2}(x - e)^{2}(x - Z)^{2}} \\ & =& \frac{1} {\pi ^{2}} \int ^{\infty }r\,dr\int _{ \mathbb{S}^{3}} \frac{d^{3}\,\omega } {(r^{2} - 2r\,e \cdot \omega +1)(r^{2} +\vert z\vert ^{2} - 2r\,\vert z\vert \,\omega \,\omega _{z})}\,.{}\end{array}$$

(56)

Assuming $\vert z\vert < 1$ we can split the radial integral F into three terms $F = F_{1} + F_{2} + F_{3}$ corresponding to the domains $r <\vert z\vert$, $\vert z\vert < r < 1$ and r > 1, respectively. In the first one we can write

$$\displaystyle\begin{array}{rcl} (r^{2} - 2r\,e \cdot \omega +1)^{-1}& =& \sum _{ n=0}^{\infty }r^{n}\,C_{ n}^{1}(\omega e)\,, \\ (r^{2} +\vert z\vert ^{2} - 2r\vert z\vert \,\omega \,\omega _{ z})^{-1}& =& \frac{1} {\vert z\vert ^{2}}\sum _{m=0}^{\infty }\left (\frac{r} {\vert z\vert }\right )^{\!\!m}C_{ m}^{1}(\omega \,\omega _{ z})\ (\mathrm{for}\ r <\vert z\vert < 1){}\end{array}$$

(57)

where the hyperspherical (Gegenbauer) polynomials C _n ¹ can be written as

$$\displaystyle{ C_{n}^{1}(\cos \theta ) = \frac{\sin (n + 1)\,\theta } {\sin \theta } \,. }$$

(58)

Using further the orthogonality relation

$$\displaystyle{ \int _{\mathbb{S}^{3}}C_{n}^{1}(\omega \cdot e)\,C_{ m}^{1}(\omega \,w_{ z})\,\frac{d^{3}\omega } {\pi ^{2}} = \frac{2\,\delta _{mn}} {n + 1}\,C_{n}^{1}(\omega _{ z}\,e) }$$

(59)

where, according to (55)

$$\displaystyle{ \omega _{z} \cdot e = \frac{z +\bar{ z}} {2\,\vert z\vert } \,. }$$

(60)

Inserting in F ₁ and using (58) (or (8)) and (11) we find

$$\displaystyle{ F_{1}(z) =\int _{ 0}^{\vert z\vert }\frac{r\,dr} {\vert z\vert ^{2}} \ \sum _{n=0}^{\infty }\ \frac{2} {n + 1}\,\frac{r^{2n}} {\vert z\vert ^{n}} \,C_{n}^{1}\left (\frac{z +\bar{ z}} {2\vert z\vert } \right ) = \frac{Li_{2}(z) - Li_{2}(\bar{z})} {z -\bar{ z}} \,. }$$

(61)

The same result is obtained for F ₃(z):

$$\displaystyle{ F_{3}(z) =\int _{ 1}^{\infty }\frac{dr} {r^{3}} \ \sum _{n=0}^{\infty }\ \frac{2} {n + 1}\, \frac{\vert z\vert ^{n}} {r^{2n}}\,C_{n}^{1}\left (\frac{z +\bar{ z}} {2\vert z\vert } \right ) = \frac{Li_{2}(z) - Li_{2}(\bar{z})} {z -\bar{ z}} = F_{1}(z)\,. }$$

(62)

Finally,

$$\displaystyle{ F_{2}(z) = 2\int _{\vert z\vert }^{1}\frac{dr} {r} \ \frac{Li_{1}(z) - Li_{1}(\bar{z})} {z -\bar{ z}} =\ln z\,\bar{z}\,\frac{\ln (1 - z) -\ln (1 -\bar{ z})} {z -\bar{ z}} \,; }$$

(63)

this, together with (61), (62) completes the proof of (9) (10) for $\vert z\vert < 1$. The same expression can be obtained in a similar fashion for $\vert z\vert > 1$; alternatively, it can be deduced from the result for $\vert z\vert < 1$ using the symmetry of F(z) implied by (14). The result can also be established by verifying that it is single valued and satisfies the first equation (39) (in view of the uniqueness of SVMP,Theorem 3.1; cf. [30]).

Appendix 2: Identities Among MZV

Equation (22) which relates the MZV ζ _w (labeled by words in the two letters {0, 1}) with $\zeta (n_{1},\ldots,n_{r})$, $n_{i} = 1,2,\ldots$ becomes particularly simple for words of depth one,

$$\displaystyle{ \zeta _{0^{n_{0}}10^{n_{1}-1}} = (-1)^{n_{0}+1}\left (\begin{array}{*{10}c} n_{0} + n_{1} - 1 \\ n_{1} - 1 \end{array} \right )\zeta (n_{0}+n_{1})\,. }$$

(64)

This allows to write the depth one contribution to the generating function Z (29) in terms of multiple commutators:

$$\displaystyle{ \sum _{n=1}^{\infty }\zeta (n+1)\mathop{\underbrace{[\ldots [}}\limits _{ n}e_{0},e_{1}],e_{0}],\ldots,e_{0}] =\!\sum _{ n=1}^{\infty }\zeta (n+1)\!\sum _{ k=0}^{n}(-1)^{k+1}\left (\begin{array}{*{10}c} n\\ k \end{array} \right )e_{ 0}^{k}\,e_{ 1}\,e_{0}^{n-k}, }$$

(65)

which is another way to write down (64). It is more interesting—and more difficult— to deduce the relations among ζ _w for words of higher depth. We shall write down all such relations for depth two and weight $\vert w\vert \leq 5$. Note that the number d _n of linearly independent MZV of a given weight n can be read off the generating function conjectured by Don Zagier

$$\displaystyle{ \frac{1} {1 - t^{2} - t^{3}} =\sum _{ n=0}^{\infty }d_{ n}\,t^{n}\quad d_{ 2} = d_{3} = d_{4} = 1\quad d_{5} = d_{6} = 2,\ldots }$$

(66)

(and proven for the motivic analog of MZV by Brown [6]; in general, d _n provide an upper bound of the independent MZV).

The Euler’s relation (27) is a special case of either of the following more general relations which only involve proper (convergent) zeta series:

$$\displaystyle{ \zeta (\mathop{\underbrace{1,\ldots,1}}\limits _{n-2},2) =\zeta (n)\,, }$$

(67a)

$$\displaystyle{ \sum _{{ s_{i}\geq 1;s_{k}\geq 2 \atop \varSigma s_{i}=n} }\zeta (s_{1},\ldots,s_{k}) =\zeta (n)\,. }$$

(67b)

The “improper” (regularized) zeta value ζ(n, 1) is determined from the stuffle relation:

$$\displaystyle{ 0 =\zeta (1)\,\zeta (n) =\zeta (1,n) +\zeta (n,1) +\zeta (n + 1)\,. }$$

(68)

In particular, for n = 2, we find

$$\displaystyle{ \zeta (2,1) = -\zeta (3) -\zeta (1,2) = -2\,\zeta (3)\,. }$$

(69)

From Euler’s formula

$$\displaystyle{ \zeta (2) = \frac{\pi ^{2}} {6} }$$

(70)

(a special case of (28)) and from the shuffle and stuffle relations one deduces that all zeta values of weight four are rational multiples of π ⁴ (in accord with the Zagier conjecture (66)). In particular, the relations for , $10 \times 10$ and (67) for n = 4,

$$\displaystyle{\zeta (2)^{2} =\zeta _{ 10\times 10} = 2\,\zeta (2,2) +\zeta (4)\,;\quad \zeta (1,3) +\zeta (2,2) =\zeta (4)\,,}$$

allow to express all weight four words of length not exceeding two as integer multiples of $\zeta (1,3)$:

$$\displaystyle{\zeta (4) = 4\,\zeta (1,3)(=\zeta (1,1,2))\,,\ \zeta (2,2) = 3\,\zeta (1,3)\,,\ \zeta (2)^{2} = 10\,\zeta (1,3)}$$

$$\displaystyle{ \Rightarrow \zeta (1,3) = \frac{\pi ^{4}} {360}\,. }$$

(71)

Proceeding in a similar fashion with the two products of the words 10 and 100 we find

$$\displaystyle{\zeta (2)\,\zeta (3) = 3\,\zeta _{10100} + 6\,\zeta _{11000} +\zeta _{10010} = 3\,\zeta (2,3) + 6\,\zeta (1,4) +\zeta (3,2)\,,}$$

$$\displaystyle{\zeta (2)\,\zeta (3) =\zeta (2,3) +\zeta (3,2) +\zeta (5)\,;\quad \zeta (1,4) +\zeta (2,3) +\zeta (3,2) =\zeta (5)\,.}$$

These three equations determine a two-dimensional space of zeta values of weight five (in accord with (66)). Selecting as a basis ζ(1, 4) and ζ(2, 3) we express the remaining convergent ζ-values of weight 5 in terms of this basis with positive integer coefficients

$$\displaystyle{\zeta (1,1,3) =\zeta (1,4)\,,\ \zeta (1,2,2) =\zeta (2,3)\,,}$$

$$\displaystyle{\zeta (5) = 2\,\zeta (2,3) + 6\,\zeta (1,4)\,,\ \zeta (3,2) =\zeta (2,1,2) =\zeta (2,3) + 5\,\zeta (1,4)\,,}$$

$$\displaystyle{ \zeta (2)\,\zeta (3) = 4\,\zeta (2,3) + 11\,\zeta (1,4) }$$

(72)

(while $\zeta (4,1) = -\zeta (1,4) -\zeta (5) = -7\,\zeta (1,4) - 2\,\zeta (2,3)$).

For the study of single valued MZV it is more natural to use the basis $(\zeta (5),\zeta (2)\,\zeta (3))$ instead. Then we find

$$\displaystyle{(\zeta (1,1,3) =)\,\zeta (1,4) = 2\,\zeta (5) -\zeta (2)\,\zeta (3)\,,}$$

$$\displaystyle{(\zeta (1,2,2) =)\,\zeta (2,3) = 3\,\zeta (2)\,\zeta (3) -\frac{11} {2} \,\zeta (5)}$$

$$\displaystyle{ (\zeta (2,1,2) =)\,\zeta (3,2) = \frac{9} {2}\,\zeta (5) - 2\,\zeta (2)\,\zeta (3)\,;\ \zeta (4,1) =\zeta (2)\,\zeta (3) - 3\,\zeta (5)\,. }$$

(73)

Brown [6] has demonstrated that a basis for “motivic” MZV for all weights is given by $\zeta (s_{1},\ldots,s_{k})$, with s _i  ∈ { 2, 3}.

From the iterated integral representation of MZV it follows that the generating function (29) satisfies:

$$\displaystyle{ Z_{e_{0}e_{1}}^{-1} = Z_{ e_{1}e_{0}} =\tilde{ Z}_{-e_{0},-e_{1}}\,. }$$

(74)

(The first equation incorporates, in particular, (67a).)

Appendix 3: Monodromy at z = 1: Single Valued MZV

The representation (33) can be obtained from (32) by noticing that the substitution z → 1 − z corresponds to the exchange $e_{0} \leftrightarrow e_{1}$ and that the path from 0 to z can be viewed as a composition of two paths: from 0 to 1 and from 1 to z. For 0 < z < 1 one should just set $h_{1}(z) = h_{0}(1 - z)$. Equation (30) follows from (32) (33) and the relations

$$\displaystyle{ \mathcal{M}_{0}\ln z =\ln z + 2\pi i\,,\quad \mathcal{M}_{1}\ln (1 - z) =\ln (1 - z) + 2\pi i\,. }$$

(75)

Applying (30) one should take into account the relation (74)

$$\displaystyle{ Z_{e_{0},e_{1}}^{-1} = Z_{ e_{1},e_{0}} =\tilde{ Z}_{-e_{0},-e_{1}} }$$

(76)

where the tilde indicates that each word is replaced by its opposite. We leave it to the reader to verify that the first few terms in the expansion of (30) reproduce (75) and give

$$\displaystyle{\mathcal{M}_{1}\,L_{01}(z) = L_{01}(z)(=\ln z\ln (1 - z) + Li_{2}(z))}$$

$$\displaystyle{ \mathcal{M}_{1}\,L_{10}(z)(= \mathcal{M}_{1}(-Li_{2}(z))) = L_{10}(z) + 2\pi i\ln z\,. }$$

(77)

We now proceed to the evaluation of the element e ₁ ^′ defined by Eq. (47). To this end we introduce the Lie algebra valued function

$$\displaystyle{ F(e_{0},e_{1}) = Z_{e_{0}e_{1}}e_{1}Z_{e_{0}e_{1}}^{-1} - e_{ 1} =\zeta (2)[[e_{0},e_{1}],e_{1}] +\zeta (3)[[[e_{0},e_{1}],e_{1}],e_{0} + e_{1}]+\ldots }$$

(78)

Equation (47) can then be solved recursively, writing $e_{1}^{{\prime}} =\mathop{\lim }\limits_{ k \rightarrow \infty }\,e_{1}^{(k)}$ with

$$\displaystyle{ e_{1}^{(0)} = e_{ 1}\,,\ e_{1}^{(k+1)} = e_{ 1} + F(e_{0},e_{1}) + F_{0}(-e_{0},-e_{1}^{(k)})\,. }$$

(79)

The weight three term with ζ(2) cancels out and one finds

$$\displaystyle{ e_{1}^{{\prime}} = e_{ 1} + 2\,\zeta (3)\,[[[e_{0},e_{1}],e_{1}],e_{0} + e_{1}] +\zeta (5)(\ldots )+\ldots }$$

(80)

where, according to Schnetz [30], the ζ(5) contribution consists of eight bracket words of weight six. (The ζ(3) contribution will be sufficient to the application that follows.)

The SVMPs in the right hand side of (51) are obtained from those in g ₃(z) by adding a letter 0 in front and at the end of each labeling word. Evaluating the regularized limit at z = 1 (and noting that for L ₁₁(z) it is zero) while $\bar{L}_{01}(1) = -\bar{L}_{10}(1) =\zeta (2)$ we find that for each (5-letter) word-label w in (51) we obtain the following counterpart of (50)

$$\displaystyle{P_{w}(1) = P_{w}^{0}(1) + 2\,\zeta (2)\,\zeta (3)\,\langle w,w_{ 23}\rangle }$$

$$\displaystyle{ w_{23}:= \left [e_{0},[[[e_{0},e_{1}],e_{1}],e_{0} + e_{1}]\right ]\,. }$$

(81)

We shall see that the role of the second term in the right hand side of (81) is to cancel the product ζ(2) ζ(3) in P _w ⁰(1), in accord with the observation that ζ ^SV(2) = 0.

Indeed the depth one contributions are proportional to ζ(5):

$$\displaystyle{P_{0^{3}10}(1)(= P_{0^{3}10}^{0}(1)) = L_{ 0^{3}10}(1) + L_{010^{3}}(1) = 8\,\zeta (5)\,,}$$

$$\displaystyle{P_{0^{2}10^{2}}(1) = 2\,L_{0^{2}10^{2}}(1) = -12\,\zeta (5)}$$

and their difference reproduces (52). For depth two we find (after cancelling the products ζ(2) ζ(3)) a negative multiple of ζ(5):

$$\displaystyle\begin{array}{rcl} P_{0^{2}101}^{0}(1)& =& \zeta _{ 0^{2}101} +\zeta _{1010^{2}} +\zeta _{100}\,\zeta _{01} {}\\ & =& 3\,\zeta (4,1) + 2\,\zeta (3,2) + 2\,\zeta (2,3) -\zeta (2)\,\zeta (3) = 4\,\zeta (2)\,\zeta (3) - 11\,\zeta (5)\,, {}\\ \end{array}$$

where in the last step we used (73), $\langle 0^{2}101,w_{23}\rangle = -2$ so that $P_{0^{2}101}(1) = P_{0^{2}101}^{0} + 2\,\zeta (2)\,\zeta (3)\langle 0^{2}101,w_{23}\rangle = -11\,\zeta (5)$; similarly $P_{01010}(1) = 4\,\zeta (5) = P_{0^{3}1^{2}}(1)$, $P_{01^{2}0^{2}}(1) = -\zeta (5)$, so that

$$\displaystyle{ P_{0^{2}101}(1) - P_{01010}(1) + P_{01^{2}0^{2}}(1) - P_{0^{3}1^{2}}(1) = -20\,\zeta (5)\,. }$$

(82)

Finally, the depth three contribution is equal to that of depth one. Indeed we find, using [7],

$$\displaystyle{P_{0101^{2}}(1) =\zeta _{0101^{2}} +\zeta _{1^{2}010} +\zeta _{10}\,\zeta _{01^{2}} + 6\,\zeta (2)\,\zeta (3) = 11\,\zeta (5)\,,P_{01^{2}01}(1) = -9\,\zeta (5)}$$

$$\displaystyle{ \Rightarrow P_{0101^{2}}(1) - P_{01^{2}01}(1) = 20\,\zeta (5)\,. }$$

(83)

This completes the proof of (53). (The expressions (82) and (83) can be also extracted from the polylog- and polyzeta-procedures of [29].)