Four-loop banana at the mass threshold

The four-loop banana with squared masses (1,1,1,1,16), tuned so that the pseudo-threshold falls at zero momentum, where the maximal cut is a Calabi–Yau threefold period with monodromy of order four. Continued to the pseudo-threshold, the period acquires two components with quarter-integer powers of the squared momentum; their coefficients are new, with closed forms in $\pi$ and the lemniscate constant $\Gamma(\tfrac14)$.

The integral

The four-loop banana is five massive propagators joining the same two vertices, with one external leg at each vertex:

$$I(p^2)\;=\;\int d^d l_1\, d^d l_2\, d^d l_3\, d^d l_4\;\frac{1}{\big(l_1^2-M_1\big)\big(l_2^2-M_2\big)\big(l_3^2-M_3\big)\big(l_4^2-M_4\big)\,\big((l_1+l_2+l_3+l_4-p)^2-M_5\big)}\,.$$

It depends on one kinematic variable, $t=p^2$, the squared external momentum, and on the five internal squared masses $(M_1,\ldots,M_5)$, here fixed at $(1,1,1,1,16)$ so that $\sqrt{M_5}=4=1{+}1{+}1{+}1$. The computation is at $d=2$, where the integral is finite, and concerns its maximal cutthe integral with every internal line put on shell, which solves the homogeneous part of the diagram's differential equation and carries the diagram's geometry, which is a period of a one-parameter family of Calabi–Yau threefolds. The holomorphic period $\varpi_0(t)$ is the power-series solution of the diagram's order-six Picard–Fuchs operatorthe linear differential operator in $t$ that annihilates the maximal-cut period, normalized to $1$ at the MUM pointthe point of maximal unipotent monodromy, the standard expansion point for Calabi–Yau periods, here at $t\to\infty$. At this tuning the pseudo-threshold falls at $t=0$, and there the operator has a pair of quarter-integer roots in its indicial polynomialthe polynomial whose roots $\rho$ are the allowed leading powers $t^\rho$ of solutions at the singular point $t=0$:

$$3057647616\,(\rho-2)^2(\rho-1)^2(4\rho-7)(4\rho-5)\;\Longrightarrow\;\{\,1,1,2,2,\,\tfrac54,\,\tfrac74\,\}.$$

The exponents $\tfrac54$ and $\tfrac74$ give monodromythe linear map on the solution space induced by carrying the integration contour once around a singular point and back eigenvalues $e^{5\pi i/2}=+i$ and $e^{7\pi i/2}=-i$, a semisimple block of order four: one circuit of $t=0$ rotates two of the six period solutions by $\pm i$, and four circuits bring them back. The quantities computed are the two connection coefficients $c_{5/4}$ and $c_{7/4}$ with which $\varpi_0$, once continued to the pseudo-threshold $t=0$, projects onto the fractional branches $\Phi_{5/4}=t^{5/4}(1+O(t))$ and $\Phi_{7/4}=t^{7/4}(1+O(t))$. They are pure numbers fixed by the mass configuration.

Banana integrals are the standard examples of Calabi–Yau geometry in Feynman integrals. Klemm, Nega and Safari (2019) and Bönisch, Fischbach, Klemm, Nega and Safari (2020) expand the periods at every loop order around the MUM point, where the monodromy is unipotent by definition: a circuit of that point shifts the period only by logarithms. Bönisch, Duhr, Fischbach, Klemm and Nega (2021) treat the degenerations of these families. When the heaviest mass equals the sum of the others the monodromy at the pseudo-threshold $t=0$ acquires a part of finite order: two for the three-loop configuration $(1,1,1,9)$, whose maximal cut is a K3 period, and four here. Because the exponents at $t=0$ differ by multiples of $\tfrac14$, the connection coefficients at that point are built from $\Gamma(\tfrac14)$, $\Gamma(\tfrac34)$ and $\Gamma(\tfrac12)=\sqrt\pi$, and $\Gamma(\tfrac14)$ is a new transcendental, not expressible in $\pi$ and square roots.

The branches $\Phi_{5/4}$ and $\Phi_{7/4}$ are not combinations of products of holomorphic periods expanded about the MUM point, so an ansatz in those products cannot produce the two coefficients, and to our knowledge they have not been computed before. The closed forms follow from the Bessel representation of the period, and transportnumerical solution of the differential equation along a path in $t$, starting from a point where the period is known exactly of $\varpi_0$ from the MUM point to $t=0$ followed by an exact projection onto the fractional branches gives their numerical values. The integration-by-parts reduction of the family at symbolic dimension gives, for the maximal cut of $I$ at $d=2$, the order-six Picard–Fuchs operator of $\varpi_0$ up to a unit shift of the Euler operator $\theta_t=t\,d/dt$, so the fractional exponents are properties of the Feynman family's own differential equation. At the nearby configuration $(1,1,1,1,9)$ the local exponents are $\{1,1,2,2,3,3\}$ and the monodromy is unipotent, so the order-four rotation belongs to the threshold tuning alone. As the loop order increases the finite-order part of the pseudo-threshold monodromy alternates: order four for the two-loop sunrise $(1,1,4)$, two at three loops, four here, and two again for a two-mass slice of the five-loop banana, whose order-eight indicial polynomial has a single half-integer root $\tfrac32$. The alternation follows from the exact indicial polynomials through five loops and is conjectural beyond.

Closed form

Near $t=0$ the six-dimensional solution space is spanned by four Frobenius solutionsthe canonical local solutions at a regular singular point: one power series per indicial root, with logarithms where roots repeat $\Phi^{(j)}$ with integer exponents $\{1,1,2,2\}$, which carry the logarithms, and the two quarter-integer branches. Transported from the MUM point, $\varpi_0$ decomposes on this basis as

$$\varpi_0(t) \;=\; \sum_{j=0}^{3} A_j\,\Phi^{(j)}(t) \;+\; c_{5/4}\,\Phi_{5/4}(t) \;+\; c_{7/4}\,\Phi_{7/4}(t), \qquad \Phi_\alpha(t)=t^{\alpha}\bigl(1+O(t)\bigr),$$

with $A_j$ the coefficients on the integer-exponent solutions and $c_{5/4}$, $c_{7/4}$ the two connection coefficients onto the fractional branches. Explicitly,

$$c_{5/4}\;=\;-\frac{1}{\sqrt{2\pi}\,\Gamma(\tfrac14)^2}\;=\;-\frac{\Gamma(\tfrac34)^2}{2\sqrt2\,\pi^{5/2}},\qquad c_{7/4}\;=\;-\frac{5\,\Gamma(\tfrac14)^2}{384\sqrt2\,\pi^{5/2}},$$

$$c_{5/4}\,c_{7/4}\;=\;\frac{5}{768\,\pi^3}\,.$$

The two forms of $c_{5/4}$ are related by the reflection identity $\Gamma(\tfrac14)\Gamma(\tfrac34)=\pi\sqrt2$, and the same identity makes the product $c_{5/4}c_{7/4}$ a rational multiple of $\pi^{-3}$. The product expresses the Wronskian normalization of the conjugate pair of branches with eigenvalues $+i$ and $-i$; the factor $64$ in $768=12\cdot64$ is the normal threshold $t=(\sum_i\sqrt{M_i})^2=64$ of the family, where all five lines go on shell. Under the order-four monodromy each $c_\alpha$ picks up a factor of $\pm i$ per circuit of $t=0$, so only $|c_\alpha|$ is free of convention; the signs above hold on the branch $\arg t=-\pi$, the one reached from the Euclidean axis.

Beyond the normal threshold the holomorphic period has the position-space Bessel representation of Groote, Körner and Pivovarov (1999), an integral of a Hankel-function kernel against the product $J_0(y)^4J_0(4y)$ of Bessel functions, one factor $J_0(\sqrt{M_i}\,y)$ per line, obtained from the position-space form by rotating the radial variable to the imaginary axis. Fractional powers of $t$ at $t\to0$ come only from the non-oscillatory tail of that product at large $y$, the cross term of the five large-argument asymptotics whose phases cancel because $4=1+1+1+1$. Integrating the two leading tail terms against that kernel, term by term with the Mellin transform of the Hankel function, gives the closed forms for $c_{5/4}$ and $c_{7/4}$ displayed above; the higher tail terms match the Frobenius expansions of $\Phi_{5/4}$ and $\Phi_{7/4}$. The same derivation yields Bessel-moment identities for the coefficients $A_j$ of the integer-exponent solutions, among them $\int_0^\infty y\,J_0(y)^4 J_0(4y)\,dy=0$. In this derivation quarter-integer exponents, and with them $\Gamma(\tfrac14)^2$ and $\Gamma(\tfrac34)^2$, occur exactly when the number of lines is odd; for an even number the exponents are half-integers and the squared gamma factors are rational multiples of $\pi$, as in the three-loop coefficient $c_{3/2}=-\sqrt3/(36\pi)$; this parity rule holds through six lines and is expected to hold at every loop order.

Special values found previously in the banana family involve constants other than $\Gamma(\tfrac14)$: $\Gamma(\tfrac13)$ in the Bessel moments of the equal-mass sunrise (Bailey, Borwein, Broadhurst and Glasser, 2008), products of $\Gamma(a/15)$ values in the critical $L$-values of the modular form attached to the equal-mass three-loop banana, and critical $L$-values of Hecke eigenforms at the attractor points of the equal-mass four-loop banana family of threefolds (Candelas, de la Ossa, Elmi and van Straten, 2019). The same tail derivation applied to the two-loop sunrise $(1,1,4)$, whose period has local exponents $\tfrac34$ and $\tfrac54$ at its own pseudo-threshold, gives connection coefficients in $\Gamma(\tfrac14)$ as well.

In the transported values of $\varpi_0$ and its first five derivatives $\theta_t^k\varpi_0$ at small $t$, the logarithmic solutions exceed the fractional admixture by roughly $10^4$, so the two coefficients are extracted with a spectral projector. Let $M_0$ be the $6\times6$ matrix by which one circuit of $t=0$ acts on the solution space and $\mathbb{1}$ the identity. With eigenvalues $\{1,1,1,1,+i,-i\}$ the rank-four unipotent block is annihilated by $(M_0-\mathbb{1})^4$, and each of the two semisimple branches is isolated by one more linear factor:

$$P_{+i}=\tfrac{i}{8}(M_0-\mathbb{1})^4(M_0+i\mathbb{1}),\qquad P_{-i}=-\tfrac{i}{8}(M_0-\mathbb{1})^4(M_0-i\mathbb{1}),$$

so that $P_{\pm i}\cdot\varpi_0=c_{5/4,\,7/4}\,\Phi_{5/4,\,7/4}$, and each of the six functions $\theta_t^k\varpi_0$ gives the same coefficient. In the full, uncut integral $I$ the $t^{5/4}$ and $t^{7/4}$ branches are absent to the accuracy of the transport and no logarithm appears at $t=0$, so the non-unipotent monodromy is confined to the maximal cut.

The complete expression, with the indicial data, the projector and the two coefficients to full precision, is in the expression file inside the bundle under Evaluator.

Checks

Evaluator

Same family

The three-loop banana at the mass threshold, squared masses $(1,1,1,9)$, is the K3 case: one half-integer exponent, monodromy of order two and a single coefficient $c_{3/2}$ in $\sqrt3/\pi$. The four-loop equal-mass banana has the same five-line graph with all masses equal; the geometry is again a Calabi–Yau threefold, but the monodromy at every singular point is unipotent and the quantity of interest is the $\varepsilon$ expansion in $d=2-2\varepsilon$ about the MUM point. The two-loop member of the threshold sequence, the sunrise with squared masses $(1,1,4)$, is on the unequal-mass sunrise page.

The papers

• Polylogarithmic, elliptic, and Calabi–Yau Feynman integrals from a hybrid bootstrap (PDF) — Matthew D. Schwartz.
• Bootstrapping elliptic and Calabi–Yau Feynman integrals (PDF) — Matthew D. Schwartz. A preliminary version of the paper.

References

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