Equal-mass sunrise

The two-loop sunrise with three internal lines of the same mass, the simplest Feynman integral whose answer is elliptic. Laporta and Remiddi (2005) computed its finite part analytically and Adams and Weinzierl (2017) wrote it as an iterated integral of modular forms. The integral is recomputed here in that form, with its expansion through fourth order in the dimensional regulator.

The integral

The sunrise is the minimal two-loop diagram: two vertices joined by three massive propagators, with one external momentum $p$ flowing through. The family is

$$J_{111}(d,\,p^2) \;=\; \int d^d l_1\, d^d l_2\; \frac{1}{\bigl(l_1^2-m_1^2\bigr)\,\bigl(l_2^2-m_2^2\bigr)\,\bigl((l_1+l_2-p)^2-m_3^2\bigr)}\,, \qquad d = 2-2\varepsilon,$$

where $l_1$ and $l_2$ are the loop momenta and the subscript denotes one power of each propagator. The conventions are those of Adams and Weinzierl (2017): a factor $1/(i\pi^{d/2})$ in each loop measure, $m^2=\mu^2=1$ with $\mu$ the scale of dimensional regularization, no prefactor $e^{\gamma\varepsilon}$ ($\gamma$ is the Euler–Mascheroni constant), and $J_{111}=-S_{111}$ in their notation. In the equal-mass case $m_1=m_2=m_3=m$, so the integral depends on one dimensionless variable, $t = p^2/m^2$, the squared external momentum in units of the squared internal mass. The quantity computed is the order-$\varepsilon^0$ coefficient $J^{(0)}(t)$ of the top-sector masterthe master integral with every propagator of the diagram present once $J_{111}$ as a function of $t$, together with the higher orders through $\varepsilon^4$.

On the maximal cut one integration is left, over a Baikov variable $y$ (a squared momentum invariant in units of $m^2$), with integrand $dy/\sqrt{y(y-4)\bigl(y^2-2y(t+1)+(t-1)^2\bigr)}$. The four branch points of the square root make the remaining integral a periodthe integral of the holomorphic one-form $dx/y$ around a cycle of the elliptic curve; the basic elliptic transcendental, playing the part $2\pi i$ plays for logarithms of the elliptic curve defined by the quartic. The two master integrals of the top sector (master integrals being the finite set of integrals to which every integral of the family reduces by integration by parts) obey a second-order Picard–Fuchs equation, the differential equation in $t$ satisfied by the periods of the curve. The integral can be singular only at $t=0$, $1$, $9$ or $\infty$, and those four points are exactly the four cuspsthe points at the boundary of a modular curve where the elliptic fiber pinches to a nodal rational curve; the natural boundary data of a modular problem of the modular curve of the congruence subgroup $\Gamma_1(6)$ of $SL(2,\mathbb{Z})$. Over those four points the curve degenerates into Kodaira fibers of types $I_1$, $I_2$, $I_3$ and $I_6$ (cycles of one, two, three and six rational curves), and in Beauville's classification of rational elliptic surfaces that set of fibers picks out the $\Gamma_1(6)$ curve uniquely. The natural functions for the answer are therefore modular formsholomorphic functions on the upper half-plane that transform with a fixed weight under a congruence subgroup; at each weight they span a finite-dimensional vector space for $\Gamma_1(6)$ and their iterated integrals, the Eichler integralsiterated integrals of modular forms along a path in the upper half-plane, the elliptic generalization of the polylogarithms.

Laporta and Remiddi (2005) gave the equal-mass graph a full analytic treatment. Bloch and Vanhove (2015) recast the answer as an elliptic dilogarithm and showed that the family is modular for $\Gamma_1(6)$, and Adams, Bogner and Weinzierl (2014) extended the elliptic-dilogarithm form to arbitrary internal masses in two dimensions. Adams and Weinzierl (2017) assembled the all-orders representation as iterated integrals of modular forms for $\Gamma_1(6)$, which is the form used here: the function space is taken from that work and the coefficients are determined again without using the published expression.

The three finite singular points are the soft (light-like) point $t=0$ and the two thresholds $t=1$ and $t=9$; they supply the polylogarithmic letters. The elliptic part of the function space needs only two objects: the holomorphic period of the curve, as an overall factor, and one Eichler integral, $I(1,f_3)$, the weight-three modular form $f_3$ of $\Gamma_1(6)$ integrated twice. Only one transcendental constant can appear at order $\varepsilon^0$, the Dirichlet L-valuethe value at an integer point of an $L$-series built from a periodic arithmetic character; the number-theoretic constants that replace $\zeta(2)$ and $\zeta(3)$ on an elliptic curve $L(\chi_{-3},2)$, where $\chi_{-3}$ is the quadratic Dirichlet character of conductor 3. If $J^{(0)}$ is written as a general linear combination of these functions with unknown coefficients, two conditions set most of the coefficients to zero: integrability of the differential equation, and the first-entry condition that only the soft point and the physical thresholds may open a branch cut. The coefficients that remain are rational numbers, fixed by integer-relation (PSLQ) fitsPSLQ takes high-precision decimals and returns the small-integer linear combination of them that vanishes, if one exists to high-precision values of the integral at Euclidean points.

Closed form

At order $\varepsilon^0$ the sunrise is the period of its curve times the sum of a boundary constant and an Eichler integral on $\Gamma_1(6)$:

$$ J^{(0)}(t) \;=\; -\,\frac{\psi_1(t)}{\pi}\,\Bigl(\,\tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},2) \;+\; I(1,f_3;\,q_C)\Bigr)\,, $$

where

$$ \frac{\psi_1}{\pi} \,=\, \frac{4}{\pi}\,\bigl[(t-1)^3(t-9)\bigr]^{-1/4}\,K(k^2)\,, \qquad q_C \,=\, -\,e^{\,i\pi\,\psi_2/\psi_1}\,, \qquad I(1,f_3;\,q_C) \,=\, \sum_{n\ge 1}\frac{a_n}{n^{2}}\,q_C^{\,n}\,. $$

Here $\psi_1$ and $\psi_2$ are the two periods of the sunrise elliptic curve, $\psi_1$ the one holomorphic at $t=0$; $K$ is the complete elliptic integral of the first kind, and its modulus $k^2(t)$ is a ratio of differences of the three roots of the cubic that puts the curve in Weierstrass form. The ratio of the two periods fixes the nome $q_C$ of $\Gamma_1(6)$, abbreviated $q$. In practice $q$ is obtained from $t$ by inverting the hauptmodulthe one function of the nome that generates all functions on a genus-zero modular curve; here it expresses the kinematic variable $t$ in terms of $q$ of $\Gamma_1(6)$, $t(q) = 9q\prod_{n\ge1}(1-q^{6n})^8(1-q^n)^4(1-q^{2n})^{-8}(1-q^{3n})^{-4}$, and $q$ is negative real for Euclidean $t\lt0$. The explicit form of the period used here is its series in the nome, $\psi_1/\pi = 2\sqrt3\,\bigl(e_1(q)+e_2(q)\bigr)$, with $e_1$ and $e_2$ given below. The $a_n$ are the Fourier coefficients of the weight-three modular form

$$ f_3 \;=\; 36\sqrt{3}\,\bigl(e_1^3 - e_1^2 e_2 - 4\,e_1 e_2^2 + 4\,e_2^3\bigr)\,, \qquad e_1(q) \;=\; \tfrac{1}{6} + \sum_{m\ge 1}\Bigl(\sum_{d\mid m}\chi_{-3}(d)\Bigr) q^m\,, \qquad e_2(q) \;=\; e_1(q^2)\,. $$

In these formulas $e_1$ and $e_2$ are Eisenstein series of weight one and $\chi_{-3}$ is the character introduced above. The boundary constant is $\tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},2) = 2.029883212\ldots$. The period multiplying the bracket is the holomorphic period of the cut curve in the normalization of Adams and Weinzierl (the Feynman-curve period, holomorphic at $t=0$); only in that normalization is $-\pi J^{(0)}/\psi_1$ a constant plus the single Eichler integral $I(1,f_3;q)$. Every $q$-series above is an expansion about $q=0$, that is, about $t=0$. For physical $t\gt9$ the nome is continued along $t+i0$ and $J^{(0)}$ acquires an imaginary part.

Higher orders in $\varepsilon$

Iterated integrals of modular forms for $\Gamma_1(6)$ also suffice for the expansion through order $\varepsilon^4$, that is, through transcendental weight six:

$$ e^{2\gamma\varepsilon} J_{111}(2-2\varepsilon,\, t) \;=\; \tfrac{2}{\sqrt3}\; u(q)\; e^{-\varepsilon \tilde w(q)} \sum_{j\ge0} \varepsilon^j H_j(q)\,, $$

where the factor $e^{2\gamma\varepsilon}$, one $e^{\gamma\varepsilon}$ per loop, removes the Euler–Mascheroni constant from the coefficients $H_j$. The prefactor $u = (\sqrt3/2)\,\psi_1/\pi$ is the holomorphic period normalized to $u(0)=1$. The exponent $\tilde w(q)$ is fixed by the Picard–Fuchs equation so that the differential equation obeyed by the sum $\sum_j \varepsilon^j H_j$ contains $\varepsilon$ only as an overall multiplier ($\varepsilon$-factorized form). Each $H_j$ is a combination, with coefficients in $\mathbb{Q}(\sqrt3)$, of iterated integrals in $q$ of modular forms of $\Gamma_1(6)$ of weight at most four: the constant, a form of weight two, $f_3$ (normalized as $-f_3/(3\sqrt3)$) and an Eisenstein series of weight four; the weight-four cusp form of $\Gamma_1(6)$ is admitted as a fifth letter and its coefficient comes out exactly zero. The term of each $H_j$ with no iterated integral is a constant $B_j$; the first three have closed forms, with $r_3 = e^{2\pi i/3}$:

$$ B_0 = -\tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},2)\,, \qquad B_1 = -6\,\operatorname{Im}\operatorname{Li}_3(1-r_3) \;-\; \tfrac{1}{2}\,\pi\log^2 3 \;-\; \tfrac{5}{54}\,\pi^3\,, $$

$$ B_2 = 12\,\operatorname{Im}\operatorname{Li}_4(1-r_3) \;+\; \tfrac{3}{2}\sqrt{3}\,L(\chi_{-3},4) \;-\; \tfrac{1}{4}\sqrt{3}\,\pi^2 L(\chi_{-3},2) \;+\; \tfrac{5}{54}\,\pi^3\log 3 \;+\; \tfrac{1}{6}\,\pi\log^3 3\,. $$

In this normalization $B_0$ is the constant of the order-$\varepsilon^0$ result with the opposite sign. The constants $B_3$ and $B_4$ are known numerically, to any precision, because every $B_j$ is a coefficient in the $\varepsilon$ expansion of the equal-mass vacuum integral, the sunrise at $p=0$,

$$ B_j \;=\; -\tfrac{\sqrt{3}}{2}\,\bigl[\varepsilon^j\bigr]\Bigl(e^{2\gamma\varepsilon}\, S_{\rm vac}(\varepsilon)\Bigr)\,, \qquad S_{\rm vac}(\varepsilon) \;=\; \frac{2^{\,2-\varepsilon}}{\Gamma(1-\varepsilon)}\int_0^\infty x^{1+\varepsilon}\,K_\varepsilon(x)^3\,dx\,, $$

with $K_\varepsilon$ the modified Bessel function of the second kind of order $\varepsilon$; an integer-relation search finds neither $B_3$ nor $B_4$ among weight-five and weight-six combinations of $\operatorname{Im}\operatorname{Li}_k(1-r_3)$, the values $L(\chi_{-3},k)$, powers of $\pi$ and $\log 3$, odd zeta values and generalized log-sine values at $2\pi/3$.

The complete expression, with the conventions and the expansion through order $\varepsilon^4$, is in sunrise-expression.md.

Checks

Evaluator

Same family

• The unequal-mass sunrise: squared masses $(1,1,2)$, $(1,2,3)$ and $(1,1,4)$ on the three lines; the curve is no longer modular for $\Gamma_1(6)$, and the answer through order $\varepsilon^1$ is written in elliptic polylogarithms at three marked points of the torus.
• The equal-mass kite: the same three massive lines with two massless lines added, so that this sunrise is a subgraph of the kite and both live on the same curve.
• The three-loop equal-mass banana: a fourth massive line between the two vertices; the geometry becomes a K3 surface whose Picard–Fuchs operator is the symmetric square of the sunrise's.

The paper

• Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF) — Matthew D. Schwartz.

References

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