Some questions about Mincer exact

[vsht]

I apologize if my questions are too naive, but I'm not a Mincer expert (when I started using IBPs FIRE was already there :) ) and I'd really like to understand certain things in a better way.

1. Where can I see which master integrals are implemented to which order in ep? For example, it seems that the product of 3 1-loop bubbles is available only up to ep^4, even if one asks for more orders. Of course, it might be that I'm using the code in a wrong way:

off stat;
#include minceex.h
.global
#define MSBAR "1"

* massless bubble ^3 , LA topology, fig. 7 in the
* mincer-form paper, remove p7,p8
L myInt = 1/p1.p1/p2.p2/p3.p3/p4.p4/p5.p5/p6.p6;

#call integral(la,0)
#call subvalues
* expand up to ep^6
#call expansion(8)
id Q.Q^s? = 1;
Print +f;
.sort
Format mathematica;
#write  <Bubble3L.m> "(%E)",myInt

.end 

2. Could some kindly point me to a reference that gives the explicit normalization factor for Mincer integrals? Empirically, I found that if one wants to convert Mincer output (in the MSbar scheme), to the "standard" multiloop normalization (Exp[ep EulerGamma]/(I Pi^(d/2)))^L, I need to multiply it by 1 - (Pi^2 ep^2)/4 + (Pi^4 ep^4)/32 - (Pi^6 ep^6)/384 + ... (not 100% sure about the ep^6-term though). Is this correct?

Cheers, Vladyslav

[vermaseren]

In Mincer the master integrals have been provided only to a limited depth. When Mincer was made, people did not consider yet more than 3 loops. Of course one may want more by now. One way would be to highjack the appropriate master integrals from the Forcer program. Those again have been taken from Lee+Smirnov+Smirnov and go to weight 12. Because of a slightly different convention as to what constitutes a master integral there were some ’transformations’, but mincer and forcer use the same definitions.

On 30 Apr 2021, at 19:31, Vladyslav Shtabovenko @.***> wrote:

I apologize if my questions are too naive, but I'm not a Mincer expert (when I started using IBPs FIRE was already there :) ) and I'd really like to understand certain things in a better way.

Where can I see which master integrals are implemented to which order in ep? For example, it seems that the product of 3 1-loop bubbles is available only up to ep^4, even if one asks for more orders. Of course, it might be that I'm using the code in a wrong way: off stat;

include minceex.h

.global

define MSBAR "1"

• massless bubble ^3 , LA topology, fig. 7 in the
• mincer-form paper, remove p7,p8 L myInt = 1/p1.p1/p2.p2/p3.p3/p4.p4/p5.p5/p6.p6;

call integral(la,0)

call subvalues

• expand up to ep^6

  call expansion(8)

  id Q.Q^s? = 1; Print +f; .sort Format mathematica;

  write "(%E)",myInt

.end Could some kindly point me to a reference that gives the explicit normalization factor for Mincer integrals? Empirically, I found that if one wants to convert Mincer output (in the MSbar scheme), to the "standard" multiloop normalization (Exp[ep EulerGamma]/(I Pi^(d/2)))^L, I need to multiply it by 1 - (Pi^2 ep^2)/4 + (Pi^4 ep^4)/32 - (Pi^6 ep^6)/384 + ... (not 100% sure about the ep^6-term though). Is this correct? Cheers, Vladyslav

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[apik]

You can use MINCER for reduction only, and in the end substitute master integrals in any chosen normalization. Here is an example for chain of self-energies:

-
#include minceex.h
* master integrals
S [l1(1,1)];
S [t1(1,1,1,1,0)],[t1(1,0,1,0,1)];
S [no(1,1,1,1,1,1,1,1)],[la(1,1,1,1,1,1,0,0)],[la(1,0,1,0,1,0,1,1)],
[la(1,0,1,1,1,0,1,0)],[la(1,0,1,1,0,1,1,1)],[la(0,0,1,0,0,1,1,1)];

L se3 = 1/<p1.p1>/.../<p6.p6>;

#define TOP "la"
#call momsubs(`TOP')
#call integral(`TOP',1)

* 3-loop
#if(`TOP'==la)||(`TOP'==be)||(`TOP'==no)
id BNO = [no(1,1,1,1,1,1,1,1)]*Q.Q^2*rat(1, 1);
id GC0*BT1 = rat(3*ep-1,-2*ep^3)*[la(1,0,1,1,0,1,1,1)] - [la(0,0,1,0,0,1,1,1)]*rat( - 60*ep^3 + 97*ep^2 - 47*ep + 6,ep^5)/Q.Q^2;
id GC0^3 = [la(1,1,1,1,1,1,0,0)]*rat(1, 1);
id GC0^2*GC1 = ([la(1,0,1,1,1,0,1,0)]*rat(2*(-2 + 3*ep), ep))/Q.Q;
id GC0^2*GC2 = ([la(1,0,1,0,1,0,1,1)]*rat(3*(-1 + 2*ep), ep))/Q.Q;
id GC0*GC1*GC2 = ([la(0,0,1,0,0,1,1,1)]*rat(6*(-2 + 3*ep)*(-3 + 4*ep), ep^2))/Q.Q^2;
* 2-loop
#elseif(`TOP'==t1)
id GC0*GC1 = [t1(1,0,1,0,1)]*rat(6*ep - 4,ep)/Q.Q;
id GC0^2 = [t1(1,1,1,1,0)];
* 1-loop
#elseif(`TOP'==l1)
id GC0 = [l1(1,1)];
#else
#message Unknown topology `TOP'
#endif

Print+s;
.end

[vsht]

Thanks to the two of you for the explanations!

@apik Actually, I wanted to do the opposite: Upon finishing the reduction with FIRE I noticed that some of my masters where "Mincer" integrals, so it seemed a natural thing to get the analytic results form Mincer instead of looking them up in the literature or recalculating with HyperInt.

The normalization was not so clear to me, but then I simply fixed it by comparing the 1L bubble cubed with the known literature result. However, when checking the so-obtained results numerically (for real 3L integrals), I noticed that for some integrals O(ep^3) pieces were not correct, while O(ep^2) pieces matched perfectly. I guess this is because such high orders are simply not implemented, as explained by Jos. On the other hand, in all cases Mincer gave me analytic results up to O(ep^4), which was the reason for my question regarding the maximal O(ep)-order implemented for each master available in Mincer.

I'll then switch to FORCER for higher orders in the ep-expansion.