ramanujan_qft.one_loop#

Scalar one-loop integrals in LoopTools conventions, precision-checked at the requested number of digits. User guide: One-loop integrals; conventions: Conventions.

ramanujan_qft.one_loop.B0(p2, m1sq, m2sq, mu2=1, delta=0, dps=None)[source]#

Scalar two-point function \(B_0(p^2; m_1^2, m_2^2)\), precision-checked to dps digits.

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

in the LoopTools normalisation (measure \(\mu^{4-D}\,d^Dq/(i\pi^{D/2} r_\Gamma)\), propagators \(1/(q^2 - m^2 + i0)\), \(D = 4 - 2\epsilon\)). The Feynman-parameter integral is done in closed form.

Parameters:

  • p2 (number or str) – External momentum squared (Minkowski metric, p2 > 0 is time-like).

  • m1sq (number or str) – Internal squared masses, \(\ge 0\).

  • m2sq (number or str) – Internal squared masses, \(\ge 0\).

  • mu2 (number or str, default 1) – Renormalisation scale squared, \(\mu^2 > 0\).

  • delta (number or str, default 0) – Value assigned to the UV divergence \(\Delta = 1/\epsilon - \gamma_E + \log 4\pi\).

  • dps (int, optional) – Decimal digits requested (default: mpmath.mp.dps).

Return type:

mpmath.mpc

Raises:

  • ValueError – For a negative squared mass, a non-finite input, mu2 <= 0, or the scaleless \(B_0(0; 0, 0)\).

  • ArithmeticError – If the precision check fails.

See also

B0_mob

the equal-mass Method-of-Brackets series.

Examples

>>> B0(5, 1, 1)          # LoopTools 2.16: 1.56959105903599583 + 1.40496294620814521 i
mpc(real='1.5695910590359961', imag='1.4049629462081452')
>>> import mpmath
>>> mpmath.nstr(B0(10, 1, 2, dps=30), 30)
'(0.6225920730905419577001843621 + 2.01160080643417843946062487128j)'
ramanujan_qft.one_loop.B0_mob(p2, msq, mu2=1, delta=0, dps=None)[source]#

Equal-mass \(B_0(p^2; m^2, m^2)\) from its Method-of-Brackets series, precision-checked.

The Method of Brackets gives \(B_0(\epsilon) = \Gamma(\epsilon)(m^2)^{-\epsilon}\, {}_2F_1(\epsilon, 1; 3/2; w)\). Its \(\epsilon\) expansion is the exact series

\[B_0 = \Delta - \log\frac{m^2}{\mu^2} + \frac{2w}{3}\,{}_2F_1(1, 1; 5/2; w), \qquad w = \frac{p^2}{4m^2},\]

continued across the threshold \(w = 1\) with \(w \to w + i0\).

Parameters:

  • p2 (number or str) – External momentum squared.

  • msq (number or str) – The common squared mass, \(> 0\).

  • mu2 – As in B0().

  • delta – As in B0().

  • dps – As in B0().

Return type:

mpmath.mpc

Raises:

ValueError – For msq <= 0, mu2 <= 0 or a non-finite input.

Examples

>>> B0_mob(8, 1) == B0(8, 1, 1)
True
ramanujan_qft.one_loop.C0(s, m1sq, m2sq, m3sq, dps=None)[source]#

Scalar three-point function \(C_0(0, 0, s; m_1^2, m_2^2, m_3^2)\), precision-checked.

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

with \(+i0\) on each propagator. This is LoopTools’ C0(0, 0, s, m1sq, m2sq, m3sq). Equal masses use the closed form of C0_mob(); unequal masses the ‘t Hooft-Veltman dilogarithm reduction (real part) and the exact cut formula (imaginary part); \(s = 0\) the divided-difference closed form.

Parameters:

  • s (number or str) – The squared momentum of the massive leg.

  • m1sq (number or str) – Internal squared masses, \(\ge 0\). A massless line is allowed when the integral is infrared finite.

  • m2sq (number or str) – Internal squared masses, \(\ge 0\). A massless line is allowed when the integral is infrared finite.

  • m3sq (number or str) – Internal squared masses, \(\ge 0\). A massless line is allowed when the integral is infrared finite.

  • dps (int, optional) – Decimal digits requested (default: mpmath.mp.dps).

Return type:

mpmath.mpc

Raises:

  • ValueError – For the collinear-divergent cases \(m_1 = m_2 = 0\) or \(m_2 = m_3 = 0\) (and, at \(s = 0\), two vanishing masses), a negative squared mass, or a non-finite input.

  • ArithmeticError – If the precision check fails.

Examples

>>> C0(10, 1, 2, 3)      # LoopTools 2.16: -0.421932531674735434 - 0.336732268033680282 i
mpc(real='-0.42193253167473543', imag='-0.33673226803368017')
ramanujan_qft.one_loop.C0_mob(s, msq, dps=None)[source]#

Equal-mass \(C_0(0, 0, s; m^2, m^2, m^2)\) in closed form, precision-checked.

The Method of Brackets gives \(C_0 = -\frac{1}{2m^2}\,[{}_2F_1(\tfrac12, \tfrac12; \tfrac32; \tfrac{s}{4m^2})]^2 = -\frac{2}{s}\arcsin^2\sqrt{s/4m^2}\), continued to all real \(s\) as

\[C_0 = \frac{1}{2s}\log^2\frac{\beta - 1}{\beta + 1}, \qquad \beta = \sqrt{1 - \frac{4m^2}{s + i0}}.\]
Parameters:

  • s (number or str) – The squared momentum of the massive leg.

  • msq (number or str) – The common squared mass, \(> 0\).

  • dps (int, optional) – Decimal digits requested (default: mpmath.mp.dps).

Return type:

mpmath.mpc

Examples

>>> C0_mob(2, 1)
mpc(real='-0.61685027506808487', imag='0.0')
ramanujan_qft.one_loop.bubble_3d(p2, m1sq, m2sq, dps=None)[source]#

The finite two-point function in \(D = 3\), precision-checked.

\[\Gamma(\tfrac12)\int_0^1 dx\,\big(x m_1^2 + (1-x)m_2^2 - x(1-x)p^2 - i0\big)^{-1/2},\]

with the measure \(d^3q/(i\pi^{3/2})\) and propagators \(1/(q^2 - m^2 + i0)\). With \(M = m_1 + m_2\): \(\sqrt\pi\,(2/Q)\arctan(Q/M)\) for \(p^2 = -Q^2 < 0\); \(\sqrt\pi\,(2/k)\operatorname{artanh}(k/M)\) for \(0 < p^2 = k^2 < M^2\); \(\sqrt\pi\,(2/k)[\tfrac12\log\frac{k+M}{k-M} + i\pi/2]\) above threshold.

Parameters:

  • p2 (number or str) – External momentum squared.

  • m1sq (number or str) – Internal squared masses, \(\ge 0\).

  • m2sq (number or str) – Internal squared masses, \(\ge 0\).

  • dps (int, optional) – Decimal digits requested (default: mpmath.mp.dps).

Return type:

mpmath.mpc

Raises:

ValueError – Exactly at threshold \(p^2 = (m_1 + m_2)^2\), for the scaleless case \(p^2 = 0\) with massless lines, or for invalid inputs.

Examples

>>> bubble_3d(10, 1, 2)
mpc(real='1.1259454203297292', imag='1.7608599228871051')