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A second-order differential equation for the two-loop sunrise graph with arbitrary masses. (English) Zbl 1275.81069

The two-loop integral corresponding to the sunrise graph with arbitrary masses take the form \(S(D, t)= S(D, t, m^2_1,m^2_2,m^2_3,\mu^2)\); \[ S(D,t)= \Gamma(3- D)(\mu^2)^{3-D} \int_\sigma {{\mathcal U}^{3-{3\over 2}D}\over{\mathcal F}^{3-D}}\,\omega. \] Here \(\sigma= \{[x_1,x_2,x_3]\in \mathbb{P}^2\mid x_i\geq 0\), \(i= 1,2,3\}\), \[ {\mathcal U}= x_1x_2+ x_2x_3+ x_3x_1, \]
\[ {\mathcal F}=- x_1x_2x_3 t+ (x_1 m^2_1+ x_2m^2_2+ x_3m^2_3){\mathcal U}, \] \(\omega\) is the volume form of the sphere, and \(\mu\) is the scaling parameter. It is known \(S\) satisfies a system of four coupled first-order equations [F. V. Tkachov, “A theorem on analytical calculability of four loop renormalization group functions”, Phys. Lett. B 100, No. 1, 65–68 (1981)].
If \(D= 2\), \(S(2,t)\) takes the simpler form \(\mu^2\int_\sigma{\omega\over{\mathcal F}}\). In this case, if \(m_1= m_2= m_3\), it was shown the system of four coupled equations reduces to a single second-order equation [S. Laporta and E. Remiddi, Nucl. Phys., B 704, No. 1–2, 349–386 (2005; Zbl 1119.81356)]. In this paper, regarding \(S(2,t)\) to be a period of some mixed Hodge structure, this result is extended for the arbitrary masses case, as follows:
Theorem. \(S(2,t)\) satisfies the second-order differential equation \[ \Biggl[{d^2\over dt^2}+ {p_1(t)\over p_0(t)} {d\over dt}+ {p_2(t)\over p_0(t)}\Biggr]\, S(2,t)= \mu^2 {p_3(t)\over p_0(t)}. \] Here \(p_i(t)\), \(0\leq i\leq 3\) are polynomials (§4). Their explicit forms are also given ((3.16) and (3.18) for \(1\leq i\leq 3\), and (3.25) and (3.26) for \(i=0\)). For to prove the theorem, the set \(X\) of the points in \(\mathbb{P}2^2\times\Delta^*\) on which \({\mathcal F}\) does not vanish, where \(\Delta^*\) is an open set of \(\mathbb{C}\) is used.
Let \(X_t\) be the fiber over \(t\). Then if \(t\) is not a real number larger than \((m_1+ m_2+ m_3)^2\), it is shown a intersects \(X_t\) precisely three points [1:0:0], [0:1:1] and [0:0:1] (Lemma 3.1). Let \(\pi:P\to \mathbb{P}^2\) be the blow up of \(\mathbb{P}^2\) in these three points, and let \(Y_t\) the transform of \(X_t\), \(B\) the total transform of \(B_0= \{x_1x_2x_3= 0\}\subset\mathbb{P}^2\). Then \(H_t= H^2(P\setminus Y_t, B\setminus B\cap Y_t)\) is the right mixed Hodge structure and \(S(2,t)\) is a period of \(H_t\). Hence the theorem is obtained computing its Picard-Fuchs equation [cf. S. Bloch, H. Esnault and D. Kreimer, Commun. Math. Phys. 267, No. 1, 181–225 (2006; Zbl 1109.81059)]. Concrete calculations of coefficients are divided into the homogeneous part (§3.2) and the inhomogeneous part (§3.3). In the calculation of the inhomogeneous part the authors adopt the ansatz in [P. A. Griffiths, Ann. Math. (2) 90, 460–495, 496–541 (1969; Zbl 0215.08103)] (cf. (3.20)).
In appendix, relations between \(S(2, t)\) and \(S(4- 2\varepsilon, t)\) is discussed.

MSC:

81T18 Feynman diagrams
13F50 Rings with straightening laws, Hodge algebras
35B44 Blow-up in context of PDEs
14D07 Variation of Hodge structures (algebro-geometric aspects)
32G20 Period matrices, variation of Hodge structure; degenerations

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