Conventions#

The one_loop and sunset functions follow the conventions of LoopTools, so that values can be compared one to one with LoopTools, FF and pySecDec output.

The Massive Euclidean multiloop graphs and Multivariate Ramanujan series for the D=2 sunset graph APIs instead use positive Euclidean propagators and the measure \(d^Dk/\pi^{D/2}\); their conventions are given in the linked guides. The conventions below apply to one_loop and sunset.

Kinematics#

  • Squared masses. Every mass argument (m1sq, m2sq, m3sq, msq) is a squared mass \(m^2 \ge 0\). Thresholds are written with the masses themselves, for example \((m_1+m_2)^2\).

  • Metric. Minkowski, \(p^2 = p_0^2 - \vec p^{\,2}\). \(p^2 > 0\) is time-like, and physical thresholds lie at positive \(p^2\) or \(s\). Euclidean (space-like) momenta are \(p^2 < 0\).

  • Feynman prescription. Every propagator is \(1/(q^2 - m^2 + i0)\). The \(+i0\) is implemented exactly: imaginary parts come from the analytic structure, not from a small numerical width. The resulting signs are listed below.

Loop measure and dimension#

Each loop is integrated with

\[ \frac{\mu^{4-D}}{r_\Gamma}\int\frac{d^Dq}{i\pi^{D/2}}, \qquad D = 4-2\epsilon, \qquad r_\Gamma = \frac{\Gamma^2(1-\epsilon)\,\Gamma(1+\epsilon)}{\Gamma(1-2\epsilon)} . \]

The factor \(1/(i\pi^{D/2})\) makes one-loop integrals real and of order one below threshold. For UV-finite integrals (\(C_0\), the \(D=3\) bubble, the sunset combinations) the factor \(r_\Gamma\) and the scale \(\mu\) drop out, and the measure is \(d^4q/(i\pi^2)\) (or \(d^3q/(i\pi^{3/2})\) for bubble_3d).

The UV pole of B0#

\(B_0\) is UV divergent. As in LoopTools, the divergence is returned through the constant

\[ \Delta = \frac{1}{\epsilon} - \gamma_E + \log 4\pi , \]

so that

\[ B_0(p^2;m_1^2,m_2^2) = \Delta - \int_0^1 dx\,\log\frac{x m_1^2 + (1-x) m_2^2 - x(1-x)p^2 - i0}{\mu^2} + O(\epsilon). \]

The parameter delta is the value assigned to \(\Delta\), and mu2 is \(\mu^2\). The defaults delta=0, mu2=1 reproduce LoopTools’ default output, which is the \(\overline{\rm MS}\) finite part at \(\mu^2 = 1\). To obtain the finite part at another scale, pass mu2; to keep track of the pole, pass a symbolic stand-in value for delta, for example delta=1, and subtract.

>>> import mpmath as mp
>>> from ramanujan_qft import B0
>>> B0(10, 1, 2)
mpc(real='0.62259207309054199', imag='2.0116008064341786')
>>> mp.nstr(B0(10, 1, 2, mu2=4) - B0(10, 1, 2), 12), mp.nstr(mp.log(4), 12)
('(1.38629436112 + 0.0j)', '1.38629436112')
>>> mp.nstr(B0(10, 1, 2, delta=1) - B0(10, 1, 2), 12)
'(1.0 + 0.0j)'

The three-point function#

C0(s, m1sq, m2sq, m3sq) is \(C_0(0,0,s;m_1^2,m_2^2,m_3^2)\), LoopTools’ C0(0, 0, s, m1sq, m2sq, m3sq):

\[ C_0 = \int\frac{d^4q}{i\pi^2}\, \frac{1}{(q^2-m_1^2)\,((q+k_1)^2-m_2^2)\,((q+k_2)^2-m_3^2)}, \qquad k_1^2 = (k_2-k_1)^2 = 0,\quad k_2^2 = s . \]

\(m_2\) is the line between the two light-like legs, so \(m_1 = m_2 = 0\) or \(m_2 = m_3 = 0\) gives a collinear divergence. \(C_0\) is symmetric under \(m_1\leftrightarrow m_3\), and its \(s\)-channel cut opens at \(s = (m_1+m_3)^2\).

The two-loop sunset#

\[ S(s) = \int\frac{d^Dk}{i\pi^{D/2}}\,\frac{d^Dl}{i\pi^{D/2}}\; \frac{1}{(k^2-m_1^2)\,(l^2-m_2^2)\,((p-k-l)^2-m_3^2)}, \qquad s = p^2 , \]

with the same per-loop measure. The package provides only UV-finite combinations (\(\operatorname{Im}S\), \(S - S(0) - sS'(0)\), \(S''\)), which do not depend on the scheme or on \(\mu\). See Two-loop sunset.

Signs of imaginary parts#

With the \(+i0\) prescription and the measure above, imaginary parts above threshold have these signs:

function

sign of the imaginary part above threshold

B0

positive, \(\operatorname{Im}B_0 = \pi\,\lambda^{1/2}(p^2,m_1^2,m_2^2)/p^2\)

C0

from the cut formula on the One-loop integrals page; negative in all its examples

bubble_3d

positive, \(\pi^{3/2}/\sqrt{p^2}\)

sunset_im

negative

 
>>> p2, m1, m2 = 10, 1, 2
>>> mp.pi * mp.sqrt(p2**2 + m1**2 + m2**2 - 2*p2*m1 - 2*p2*m2 - 2*m1*m2) / p2
mpf('2.0116008064341786')

Converting to other normalisations#

  • A measure \(\int d^Dq/(2\pi)^D\) instead of \(\int d^Dq/(i\pi^{D/2})\) multiplies each one-loop integral by \(i/(16\pi^2)\) in \(D = 4\) (and each loop of the sunset by the same factor), up to the \(\epsilon\)-dependent factors that only matter for the pole terms.

  • Euclidean conventions (\(p_E^2 = -p^2\), propagators \(1/(q_E^2 + m^2)\)): evaluate at \(p^2 = -p_E^2\).

Units and numbers#

  • All quantities are in arbitrary but consistent units: only ratios such as \(p^2/m^2\) matter, apart from the \(\log\mu^2\) of B0. Results scale exactly under \(p^2, m^2, \mu^2 \to \lambda p^2, \lambda m^2, \lambda\mu^2\) (\(B_0\) is invariant, \(C_0\) scales as \(1/\lambda\), \(\operatorname{Im}S\) as \(\lambda\)).

  • Inputs may be ints, floats, strings or mpmath numbers. Floats are exact binary numbers; pass strings for exact decimals (Precision and numerical checks).

  • dps is the number of decimal digits requested. Results are mpmath.mpc for the one-loop functions and sunset_subtracted, and mpmath.mpf for sunset_im, sunset_d2 and sunset_threshold.