FRGRealTime

Coeffgamm2(k, T, mfun::Spline1D)

The coefficient before the two point function flow.

Coeffgamm2Simple(k, T, m)

The coefficient before the two point function flow.

Epi(k,m)

compute $\sqrt{(x^2+m)}$

F1All(p0, ps, k, m, T)

$F_1$ is an odd function about $p_0$

when p0>0,F1All=loopfunpp

when p0<0,dkF1All=-loopfunpp(-p0)

F1All doesn't contains the type-2 delta function

F2All(p0, ps, k, m, T)

$F_2$ is an odd function about $p_0$

when p0>0,F2All=loopfunpm

when p0<0,F2All=-loopfunpm(-p0)

PvdkF1Tildeps(p0, ps, qsmax, k, m, T; kwargs...)

Piecewise hints:

We only integrate the p0' domain which the flowpp_intcostheqs is non zero.

VImSimple(p0, ps, q0, k, m, T, Npi, lam4pik)

compute $\mathrm{Im}V_k(q_0)$, the k dependence of $\lambda_{4\pi}$ and $m$ are neglected.

VImSimple only contains $V(q_0)$, no $V(-q_0)$

VImSimple contains type-1 delta function

VImSimple doesn't contains type-2 delta function

VImintqs(p0, ps, k, T, Npi,UVScale,mfun::Function,lamfun::Function)

compute $\int_0^{k}dq_s qs^2\int_{-1}^{1}d\cos\theta \mathrm{Im}V(q_0,k)$. In our code, we perform integration over kprim, q0 & qs does not involved, so qs=k, q0=Epi(k, mfun(k)).

VImintqs don't have any delta function contribution, we include the type-1 delta function in VImintqs_delta1 separately.

Arguments

• mfun::Function: $m^2(k)$, input from zero momentum result
• lampifun::Function: $\lambda_{4\pi}(k)$, input from zero momentum result.

VImintqsSimple(p0, ps, k, T, Npi, m, lamda,UVScale)

compute $\int_0^kq_s^2dqs\int_{-1}^{1}d\cos\theta\mathrm{Im}V_k(q_0)$, the k dependence of $\lambda_{4\pi}$ and $m$ are neglected.

VImintqsSimple only contains $V(q_0)$, no $V(-q_0)$

VImintqsSimple contains type-1 delta function

VImintqsSimple doesn't contains type-2 delta function

VImintqs_delta1(p0, ps, k, T, Npi,IRScale,UVScale, mfun::Spline1D, lamfun::Spline1D)

compute type-1 delta function contribution in $\mathrm{Im}V_k$, we do the triple integral analytically.

VReintqs(p0, ps, k, T, Npi,IRScale,UVScale, mfun, lamfun)

compute $\int_0^{k}dq_s qs^2\int_{-1}^{1}d\cos\theta \mathrm{Re}V(q_0,k)$. In our code, we perform integration over kprim, q0 & qs does not involved, so qs=k, q0=Epi(k, mfun(k)).

VReintqs contains type-1 and type-2 delta function.

Arguments

• mfun::Function: $m^2(k)$, input from zero momentum result
• lampifun::Function: $\lambda_{4\pi}(k)$, input from zero momentum result.

dkF1All(p0, ps, k, m, T)

$F_1$ is an odd function about $p_0$

when p0>0,dkF1All=flowpp

when p0<0,dkF1All=-flowpp(-p0)

dkF1All doesn't contains the type-2 delta function

dkF1Allintqs(p0, ps, qsmax, k, m, T)

$F_1$ is an odd function about $p_0$

when p0>0,dkF1Allintqs=flowpp_intcostheqs

when p0<0,dkF1Allintqs=-flowpp_intcostheqs(-p0)

dkF1Allintqs doesn't contains the type-1 delta function

dkF1Allintqs doesn't contains the type-2 delta function

dkF1TildeintqsAll(p0, qsmax, k, m, T; kwargs...)

costh is not integrated we need an extra 2 at somewhere

dkF2All(p0, ps, k, m, T)

$F_2$ is an odd function about $p_0$

when p0>0,dkF2All=flowpm

when p0<0,dkF2All=-flowpm(-p0)

flowpm(p0, ps, k, m, T)

$F_2$ is an odd function about $p_0$

when p0>0,dkF2Allintqs=flowpm_intcostheqs

when p0<0,dkF2Allintqs=-flowpm_intcostheqs(-p0)

dkF2TildeintqsAll(p0, qsmax, k, m, T; kwargs...)

costh is not integrated we need an extra 2 at somewhere

dkVIm(p0, ps, q0, k, m, T, Npi, lam4pik)

compute $\tilde{\partial_k}\mathrm{Im}V_k(q_0)$.

dkVIm only contains $V(q_0)$, no $V(-q_0)$

dkVIm contains type-1 delta function

dkVIm doesn't contains type-2 delta function

dkVImintqs(p0, ps, q0, qsmax, k, m, T, Npi, lam4pik)

compute $\int_0^{qsmax}dq_s qs^2\int_{-1}^{1}d\cos\theta \tilde{\partial_k}\mathrm{Im}V(q_0)$.

dkVImintqs only contains $V(q_0)$, for $-q_0$, we have $\int d\cos\theta V(q_0)=\int d\cos\theta V(-q_0)$, so we need an extra $2$ at somewhere.

dkVImintqs doesn't have any delta function contribution, we include the type-1 delta function in VImintqs_delta1 separately.

Arguments

• qsmax: we integrate $q_s$ from $0$ to $k$, qsmax will set to k when we do the integration $dk'$, it should be distinguished from $k'$
• m: mass square, it will be $m(k')$ when we do the integration $dk'$.
• lam4pik: $\lambda_{4\pi}$, it will be $\lambda_{4\pi}(k')$ when we do the integration $dk'$ .

dkVReintqs(p0, ps, q0, qsmax, k, m, T, Npi, lam4pik)

compute $\int_0^{qsmax}dq_s qs^2\int_{-1}^{1}d\cos\theta \tilde{\partial_k}\mathrm{Re}V(q_0)$.

dkVImintqs only contains $V(q_0)$, for $-q_0$, we have $\int d\cos\theta V(q_0)=\int d\cos\theta V(-q_0)$, so we need an extra $2$ at somewhere.

dkVReintqs contains type-1 and type-2 delta function

Arguments

• qsmax: we integrate $q_s$ from $0$ to $k$, qsmax will set to k when we do the integration $dk'$, it should be distinguished from $k'$
• m: mass square, it will be $m(k')$ when we do the integration $dk'$.
• lam4pik: $\lambda_{4\pi}$, it will be $\lambda_{4\pi}(k')$ when we do the integration $dk'$ .

flowpm(p0, ps, k, m, T)

compute $-\frac{1}{\pi}\tilde{\partial_k}\Im I_{2, k}(p)$

flowpm_intcostheqs(p0, ps, qsmax, k, m, T)

compute

\[\begin{aligned} &\int_0^{qsmax}\!\!dq_s q_s^2\int_{-1}^{1}\!\!d\cos\theta \tilde{\partial_k}F_2\left(\sqrt{p_s^2+q_s^2+2p_sq_s\cos\theta}\right)\\ &=\int_0^{qsmax}\!\!dq_s q_s^2\int_{-1}^{1}\!\!d\cos\theta \tilde{\partial_k}F_2\left(\sqrt{p_s^2+q_s^2-2*p_s*q_s\cos\theta}\right) \end{aligned}\]

flowpp(p0, ps, k, m, T,δ=0.02)

compute $-\frac{1}{\pi}\tilde{\partial_k}\Im I_{1, k}(p)$

To be noticed that, flowpp doesn't contains $\tilde{\partial_k}\mathcal{F}_4$, we will consider it separately.

At $p_0=2E_{\pi,k}$, flowpp has a $\delta$ function contribution, we use a rectangle function with width $\delta$ to approximate.

flowpp_intcostheqs(p0, ps, qsmax, k, m, T)

compute

\[\begin{aligned} &\int_0^{qsmax}\!\!dq_s q_s^2\int_{-1}^{1}\!\!d\cos\theta \tilde{\partial_k}F_1\left(\sqrt{p_s^2+q_s^2+2*p_s*q_s\cos\theta}\right)\\ &=\int_0^{qsmax}\!\!dq_s q_s^2\int_{-1}^{1}\!\!d\cos\theta \tilde{\partial_k}F_1\left(\sqrt{p_s^2+q_s^2-2*p_s*q_s\cos\theta}\right) \end{aligned}\]

loopfunpm(p0, ps, k, m, T)

compute $-\frac{1}{\pi}\Im I_{2, k}(p)$

loopfunpp(p0, ps, k, m, T)

compute $-\frac{1}{\pi}\Im I_{2, k}(p)$, but without $\mathcal{F}_4$ contribution

loopfunppfix(p0, ps, k, m, T)

compute $-\frac{1}{\pi}\Im I_{1, k}(p)$

propImSimple(p0, ps, T,IRScale,UVScale, Npi, m, lamda)

x[1] is qs, x[2] is costh, x[3] is k

Arguments

• m: mass square, it's a constant number.
• lamda: $\lambda_{4\pi}$, it's a constant number.

propImintqs_delta1_dq0_delta(p0, ps, T, IRScale, UVScale, Npi, mfun, lamfun)

Compute the delta function part appears

star1fun(qp, qm, ps, m, T)

compute $-\frac{1}{\pi}\mathcal{F}_{1}\left(q_{+}, q_{-}, p, \bar{m}_{\pi, k}^{2}\right)$

star1funpm(qp, qm, ps, m, T)

compute $-\frac{1}{\pi}\mathcal{F}_{1}^{\prime}\left(q_{+}, q_{-}, p, \bar{m}_{\pi, k}^{2}\right)$

Compared with $\mathcal{F}$, the $\mathcal{F'}$ has subtracted vacuum contributions. it will be used in $\mathrm{Im} I_2$

star2fun(qp, qm, ps, k, m, T)

compute $-\frac{1}{\pi}\mathcal{F}_{2}\left(q_{+}, q_{-}, k, p, \bar{m}_{\pi, k}^{2}\right)$

star2funpm(qp, qm, ps, k, m, T)

compute $-\frac{1}{\pi}\mathcal{F}^{\prime}_{2}\left(q_{+}, q_{-}, k, p, \bar{m}_{\pi, k}^{2}\right)$

star3fun(qp, ps, k, m, T)

compute $-\frac{1}{\pi}\mathcal{F}_{3}\left(q_{+},k, p, \bar{m}_{\pi, k}^{2}\right)$

star3funpm(qp, ps, k, m, T)

compute $-\frac{1}{\pi}\mathcal{F}^{\prime}_{3}\left(q_{+}, k, p, \bar{m}_{\pi, k}^{2}\right)$

star4fun(p0, p, k, m, T)

compute $-\frac{1}{\pi}\mathcal{F}_{4}\left(p_{0}, k, p, \bar{m}_{\pi, k}^{2}\right)$