The minimal order differential equation for the all equal mass multiloop sunset integrals in general dimensions¶

\begin{equation} \Omega^{(n-1)}_\circleddash( t,\epsilon)={\Omega^{(n)}_0\over \textbf{F}_{n-1}(t)}\left(\textbf{U}_{n-1}^n\over \textbf{F}_{n-1}(t)^{n-1}\right)^\epsilon \end{equation}

with \begin{equation} \textbf{ U}_{n-1}= x_1\cdots x_n\sum_{i=1}^n {1\over x_i},\qquad \textbf{ F}_{n-1}(t)= \textbf{U}_{n-1} \left(\sum_{i=1}^n x_i\right)-t x_1\cdots x_n \end{equation}

The all equal mass case up to 20 loops

The Picard-Fuchs operator at loop L is in PF[L] $$\mathscr{L}^{(l),\epsilon} I^{(l)}(t,\epsilon)=-(l+1)! {\Gamma(1+\epsilon)^l\over \Gamma(1+l\epsilon)}$$ with $$ \mathscr{L}^{(l),\epsilon}=\sum_{r=0}^l \mathscr{L}^{(l),r} \epsilon^r$$ where $ \mathscr{L}^{(l),r}$ is differential operator in $t$ of order $l-r$.

In [1]:

epsilon=var('epsilon')

In [2]:

from ore_algebra import *
OA, t, Dt = DifferentialOperators(QQ, 't')
OAepsilon, t, Dt = DifferentialOperators(QQ[epsilon], 't')

---------------------------------------------------------------------------
ModuleNotFoundError                       Traceback (most recent call last)
Cell In[2], line 1
----> 1 from ore_algebra import *
      2 OA, t, Dt = DifferentialOperators(QQ, 't')
      3 OAepsilon, t, Dt = DifferentialOperators(QQ[epsilon], 't')

ModuleNotFoundError: No module named 'ore_algebra'

In [3]:

# PFEqualMass[l] where l is the loop order  l=1,..,20
PFEqualMass=dict()
with open("PFSunsetEqualMass-1to20.txt") as f:
    for e in f.read().replace("\n", "").split(";"):
            if e:
                name, expr = e.split(":=")
                num=int(name.strip("PFEqualMass"))
                print("loading result at loop order ",num)
                PFEqualMass[num] =  SR(expr)

loading result at loop order  1
loading result at loop order  2
loading result at loop order  3
loading result at loop order  4
loading result at loop order  5
loading result at loop order  6
loading result at loop order  7
loading result at loop order  8
loading result at loop order  9
loading result at loop order  10
loading result at loop order  11
loading result at loop order  12
loading result at loop order  13
loading result at loop order  14
loading result at loop order  15
loading result at loop order  16
loading result at loop order  17
loading result at loop order  18
loading result at loop order  19
loading result at loop order  20

In [6]:

PFEqualMass[1] 

Out[6]:

-(t^2 - 4*t)*Dt - epsilon*t - t + 2

In [7]:

PFEqualMass[2] 

Out[7]:

(t^3 - 10*t^2 + 9*t)*Dt^2 + 2*epsilon^2*t + (3*epsilon*t^2 - 10*epsilon*t + 3*t^2 - 9*epsilon - 20*t + 9)*Dt + 2*epsilon^2 + 3*epsilon*t - 5*epsilon + t - 3

In [4]:

PFEqualMass[3] 

Out[4]:

-(t^4 - 20*t^3 + 64*t^2)*Dt^3 - 6*epsilon^3*t - 6*(epsilon*t^3 - 10*epsilon*t^2 + t^3 - 15*t^2 + 32*t)*Dt^2 - 12*epsilon^3 - 11*epsilon^2*t - (11*epsilon^2*t^2 - 28*epsilon^2*t + 18*epsilon*t^2 - 64*epsilon^2 - 120*epsilon*t + 7*t^2 - 68*t + 64)*Dt + 14*epsilon^2 - 6*epsilon*t + 18*epsilon - t + 4

In [ ]: