Unequal-mass sunrise

The two-loop sunrise with squared internal masses (1,1,2), (1,2,3) or (1,1,4), whose maximal cut is a non-modular elliptic curve. Adams, Bogner and Weinzierl (2014) gave its finite part as elliptic dilogarithms and Bogner, Müller-Stach and Weinzierl (2019) every order in the dimensional regulator. The function space comes from those works; every coefficient is determined here independently of the published expressions.

The integral

The sunrise is the two-loop self-energy in which three massive propagators join two vertices and a single external momentum $p$ flows through. With masses $m_1$, $m_2$, $m_3$ on the three lines the integral 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 each subscript is the power of one propagator. Each loop integration is divided by $i\pi^{d/2}$, and the sign and period normalization follow Adams and Weinzierl (2017), in whose notation $J_{111}=-S_{111}$. The lightest mass is the unit, $m_1^2=1$, which is also the scale $\mu^2$ of dimensional regularization, and no factor $e^{\gamma\varepsilon}$, with $\gamma$ the Euler–Mascheroni constant, is included. The kinematic variable is $t=p^2$ in these units, and the three mass assignments are $(m_1^2,m_2^2,m_3^2)=(1,1,2)$, $(1,2,3)$ and $(1,1,4)$. In the expansion $J_{111}=\sum_{k\ge0}\varepsilon^k J^{(k)}(t)$, the quantities computed here are the finite part $J^{(0)}(t)$ of this top-sector masterthe master integral with every propagator of the diagram present once for each assignment and the next coefficient $J^{(1)}(t)$, for real $t\lt0$; the $(1,1,2)$ result is also continued above its threshold.

The sunrise with three massive lines is the simplest Feynman integral whose maximal cut is an elliptic curve. For equal masses the curve is modular for $\Gamma_1(6)$ and the integral is an iterated integral of modular formsfunctions of the period ratio $\tau$ that transform covariantly under a finite-index subgroup of SL(2,$\mathbb Z$), here $\Gamma_1(6)$; a family of elliptic curves is called modular when its base is the quotient of the upper half-plane by such a group; for unequal masses the curve depends on the assignment and is not modular. Bloch and Vanhove (2015) expressed the equal-mass graph in two dimensions as an elliptic dilogarithm and Adams, Bogner and Weinzierl (2014) extended that form to arbitrary masses; Bogner, Müller-Stach and Weinzierl (2019) then gave the arbitrary-mass integral to all orders in $\varepsilon$ as iterated integrals on the moduli space $\overline{\mathcal M}_{1,3}$ of a torus with three marked points. The elliptic dilogarithms and the iterated integrals on $\overline{\mathcal M}_{1,3}$ are both elliptic multiple polylogarithms in the sense of Broedel, Duhr, Dulat and Tancredi (2017), so the function space of every quantity computed here is known in advance.

On the maximal cut at $(1,1,2)$ the curve degenerates at $t=0$, $2$, $6\pm4\sqrt2$ and $\infty$, into cycles of one, four, two and three rational curves respectively (Kodaira types $I_1$, $I_4$, $I_2$ and $I_3$), and the family of cut curves over the $t$-line is a rational elliptic surface that, unlike the equal-mass family, is not the modular surface of $\Gamma_1(6)$. The polylogarithmic letters are $\{t,\,t-2,\,t^2-12t+4\}$, the last one an irreducible quadratic with roots $6\pm4\sqrt2$. The arguments of the elliptic polylogarithms are the three marked pointsthe images on the torus, under the Abel–Jacobi map, of the three points of the cut where a propagator goes on shell; their positions move with $t$ and with the masses $z_1,z_2,z_3$ of the propagators, which sum to zero modulo the period lattice; they are distinct for $(1,2,3)$, and $z_1=z_2$ for $(1,1,2)$ and $(1,1,4)$. Two conditions restrict which iterated integrals (words) can appear at each order in $\varepsilon$: integrability of the differential equation that the master integralsa finite set of integrals to which every integral of the family reduces by integration-by-parts identities satisfy in $t$, and the first-entry condition that only the soft point $t=0$ and the physical thresholds open a branch cut. The only admissible constants are then the ones generated at these degeneration points: the value $L(\chi_{-3},2)$ of the Dirichlet $L$-function of the quadratic character modulo $3$ at order $\varepsilon^0$ and $\zeta(3)$ at order $\varepsilon^1$. The remaining rational coefficients are fixed by integer-relation (PSLQ) fits to high-precision values of the master integrals at Euclidean points.

Closed form

The results are stated for the ratio $E^{(k)}(t)=J^{(k)}(t)/\hat\psi_1(t)$, where $\psi_1(t)$ is the holomorphic periodthe integral of the holomorphic one-form $dx/y$ around a cycle of the elliptic curve; the basic elliptic transcendental, playing the role $2\pi i$ plays for logarithms of the cut curve in the normalization of Adams and Weinzierl and $\hat\psi_1=\psi_1/\pi$. Besides the period, the closed forms involve two functions on the torus, the first of which is the depth-two word built from the third Kronecker–Eisenstein formone of the generating family of doubly periodic kernels $g^{(n)}(z,\tau)$ on a torus of modulus $\tau$, from which elliptic polylogarithms are built as iterated integrals $g^{(3)}$ at a marked point,

$$W(z,N)=I\!\big(1,g^{(3)}(z,N\tau_C);q_C\big)=\sum_{n\ge1}b_n(z,N)\,\frac{q_C^{\,n}}{n^2}\,, \qquad N\in\{1,2\},$$

where $\tau_C$ is the modulus of the cut torus and $q_C=e^{2\pi i\tau_C}$ its nomethe variable $q=e^{2\pi i\tau}$ in which functions on the torus of modulus $\tau$ have convergent power series; in terms of the period ratio $\tau_F$ of Adams and Weinzierl, $q_C=-e^{i\pi\tau_F}$, which is real and negative for $t\lt0$. The symbol $I(1,g;q_C)$ denotes the twice-iterated integral of $g$ along $d\log q_C$ (the entry $1$ is the trivial kernel of the outer integration), and $b_n(z,N)$ are the $q_C$-expansion coefficients of $g^{(3)}(z,N\tau_C)$, so that the two integrations produce the factor $1/n^2$. The second function on the torus is the elliptic dilogarithm of Bloch and Vanhove and of Adams, Bogner and Weinzierl, $C_{4,2}(t)=\sum_{j}\tfrac1{2i}\big[\mathrm{Li}_2(\bar w_j)-\mathrm{Li}_2(1/\bar w_j)\big]$ with $\bar w_j=e^{2\pi i z_j}$, summed over the three marked points; in the equal-mass limit it reduces to the constant $\tfrac32\sqrt3\,L(\chi_{-3},2)$.

Squared masses (1,1,2)

With $z_1=z_2$ the finite part is

$$E^{(0)}(t)=-\,C_{4,2}(t)-\frac{6}{(2\pi)^3}\,\frac{\big(2W(z_1,1)+W(z_3,1)\big)-8\big(2W(z_1,2)+W(z_3,2)\big)}{3}.$$

Three rational numbers in this formula are fixed by the integer-relation fit: the coefficient $-1$ of $C_{4,2}$, and the coefficients $-\tfrac13$ of the $W(z,1)$ terms and $+\tfrac83$ of the $W(z,2)$ terms multiplying the common factor $6/(2\pi)^3$. The formula holds as written for real $t\lt0$, where $q_C$ and the marked points are real. The same expression, continued to $t+i0$ along a path in the upper half of the complex $t$-plane, gives the integral above its threshold $t=(2+\sqrt2)^2=6+4\sqrt2$, where it acquires an imaginary part. For the continuation the depth-two words are resummed into dilogarithms, and the periods and marked points are continued analytically along the path so that each stays on the branch reached from $t\lt0$.

Squared masses (1,2,3)

With three distinct marked points the result has one term per marked point, with the same rational coefficients,

$$E^{(0)}(t)=-\,C_{4,2}(t)-\frac{6}{(2\pi)^3}\,\frac13\sum_{j=1}^{3}\big(W(z_j,1)-8\,W(z_j,2)\big),$$

and the $(1,1,2)$ expression above is the $z_1=z_2$ case of this formula. The masses enter only through the curve, its period and its marked points, so the formula can be evaluated at any squared masses, for example $(1,1,3)$.

Squared masses (1,1,4)

At $(1,1,4)$ the two light lines again give $z_1=z_2$, and the finite part is the $(1,1,2)$ expression evaluated with the curve, period and marked points of the $(1,1,4)$ assignment, with the same three rational coefficients.

Order $\varepsilon^1$

For all three assignments the next coefficient is a fixed combination of two functions of Bogner, Müller-Stach and Weinzierl:

$$E^{(1)}(t)=-\,J^{(3)}_4(t)+\big(2\gamma+2\log m_3^2\big)\,J^{(2)}_4(t).$$

Both are written in their notation and normalization; the superscript is the polylogarithmic weight, not the order in $\varepsilon$. The first, $J^{(2)}_4=-E^{(0)}$, is the order-$\varepsilon^0$ function above, and $J^{(3)}_4$ is their combination of 101 words of depth at most three in an alphabet of 14 Kronecker–Eisenstein forms at the three marked points; the alphabet is taken from their paper. The constant $\zeta(3)$ enters through the boundary term $C_{4,3}$ of $J^{(3)}_4$, a Nielsen polylogarithma two-index generalization $S_{n,p}(x)$ of the classical polylogarithm; $S_{1,2}$ is the simplest one beyond $\mathrm{Li}_3$ $S_{1,2}$ evaluated on the unit circle. Neither the coefficient $-1$ of $J^{(3)}_4$ nor the mixing constant $b=2\gamma+2\log m_3^2$ ($b=2.5407256\ldots$ for $(1,1,2)$) is fitted: $b$ accounts for the difference between the normalization of $J_{111}$ above, with $m_1^2=1$ and no $e^{\gamma\varepsilon}$ prefactor, and the one in which $J^{(2)}_4$ and $J^{(3)}_4$ are defined. The formula is evaluated with the published word coefficients. As a test of those coefficients, all 101 of them and the two normalization terms $\gamma\,J^{(2)}_4$ and $\log m_3^2\,J^{(2)}_4$ are left free in an integer-relation fit to numerical values at fifteen Euclidean points. The fit returns the value of $b$ and, for every word but one, exactly the published coefficient. The exception is a word that vanishes identically in the regularization used for the iterated integrals: no fit can determine its coefficient, which is fixed instead by regularity at the soft point $t=0$.

Checks

Evaluator

Same family

The equal-mass sunrise is the same graph with three equal masses: there the curve is modular for $\Gamma_1(6)$, the three marked points coincide at a torsion point and the finite part becomes an iterated integral of a modular form for that group. The unequal-mass kite, with internal masses $1$, $1$ and $\sqrt2$, contains the $(1,1,2)$ sunrise as a subdiagram and inherits its curve. The ice-cream cone with generic masses is a two-loop three-point graph whose elliptic curve is likewise not modular. With two lines at the top-quark mass $m_t$ and one at an electroweak-boson mass $M$, the same sunrise is the elliptic subgraph of the two-loop mixed QCD–electroweak corrections to $gg\to H$.

The paper

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

References

← back to Feynman diagrams