Timeline for Analytical continuation of 2,3,4-point integrals
Current License: CC BY-SA 4.0
16 events
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| Jul 1, 2022 at 17:28 | history | edited | HirenPatel | CC BY-SA 4.0 |
Package-X no longer available
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| Jul 29, 2018 at 6:04 | vote | accept | CAF | ||
| Jul 29, 2018 at 6:03 | comment | added | CAF | Thanks! Sure, I think I have been in touch with you earlier on last year to request the PVReduce package so I will send a follow up email with some of the integrals I’ve been working with. I’m on holiday just now, however, so I’ll send it when I’m back if that’s ok (no access to mathematica) Thanks again! | |
| Jul 28, 2018 at 17:28 | comment | added | HirenPatel |
By the way, I would really appreciate it if you could send me more of those examples of $C_0$ functions that you expect to simplify under external Assumptions. Would you kindly send me an email containing some more examples? Getting my hands on examples like those would definitely guide me to improve that part of the code!
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| Jul 28, 2018 at 17:26 | comment | added | HirenPatel |
Using FullSimplify instead of Simplify combines the Conjugate[PolyLog[...]] with PolyLog[...] to get something in terms of Pi. To answer your last question LoopRefine is the only function in Package-X that does the $\epsilon$-expansion. To keep things compact, it abbreviates some $\mathcal{O}(\epsilon^0)$ functions in terms of DiscB, ScalarC0, etc.. So those abbreviations have no $\epsilon$ in them. Expanding functions like DiscExpand, C0Expand etc. only insert the analytic forms for the abbreviations; it does not do any further epsilon expansion.
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| Jul 28, 2018 at 17:18 | comment | added | HirenPatel |
@CAF Ah, that example very clearly demonstrates that I need to revisit and improve the code to produce better results! By playing around I found that you can get a nice result if you throw in KallenExpand, Simplify, etc as follows: Assuming[{xb > 1, x < -4*mc^2, mc > 0}, C0Expand[ScalarC0[mc^2, x/4 + mc^2/xb + x/(4 xb), 2 mc^2 + x/2 + (2 mc^2)/xb + x/(2 xb), mc, 0, mc]] // KallenExpand // Simplify // X`Utilities`SimplifyDiLog]. But you shouldn't have to do that! It should produce a nice answer on its own. (btw, Conjugate[PolyLog[]] can't be simplified).
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| Jul 26, 2018 at 10:14 | comment | added | CAF |
And as a final question if its ok, why does LoopRefine sometimes give me an epsilon expansion immediately but other times give me a ScalarC0 which then becomes an epsilon expansion only after it is within a C0Expand?
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| Jul 26, 2018 at 10:12 | comment | added | CAF |
In any case, it doesn't take long to analyse term by term to check whether the DiLog is a PolyLog or a Conjugate[PolyLog[]]. The PolyLog[2,x] is simply $\text{Li}_2(x)$. Is there a rewriting of Conjugate[PolyLog[2,x]]?
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| Jul 26, 2018 at 10:10 | comment | added | CAF |
Thanks for your reply. As an example, I tried the following Assuming[{xb > 1, x < -4*mc^2, mc > 0}, C0Expand[ScalarC0[mc^2, x/4 + mc^2/xb + x/(4 xb), 2 mc^2 + x/2 + (2 mc^2)/xb + x/(2 xb), mc, 0, mc]]]. There is no simplification made from DiLog to PolyLog that I can see. Does the Assuming only work for relatively simpler ScalarC0 functions?
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| Jul 25, 2018 at 4:46 | comment | added | HirenPatel |
...for example you can do something like Assuming[s<4m^2, ExpandC0[...]] and then it will try to come up with an explicit form of ScalarC0 with fewer Ln and DiLog. This part of Package-X definitely needs more work; in a future version, I'll provide the option Assumptions for LoopRefine, C0Expand, etc., and improve the code to produce better results in cases when some variables are restricted to a smaller domain.
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| Jul 25, 2018 at 4:40 | history | edited | HirenPatel | CC BY-SA 4.0 |
added 27 characters in body
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| Jul 25, 2018 at 4:39 | comment | added | HirenPatel |
@CAF Ah I missed your comment, sorry. The expression returned by Package-X is supposed to be valid for all real values of the scalar products and all positive values for the internal masses. Sometimes, results are expressed in terms of Ln and DiLog to accommodate the $+i\varepsilon$ prescription over the whole domain of reals. But, if you know that some quantities are restricted to a smaller domain, you can wrap Assuming around LoopRefine (or C0Expand, D0Expand, etc), and that will make it try to come up with something in terms of Mathematica's Log and PolyLog only...
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| Jul 22, 2018 at 9:21 | comment | added | CAF |
Thanks for your answer! In fact, I have already been using PackageX extensively. In, for example, LoopIntegrate[1,k,{k,0},{k+p1,mc},{k+p1+p2,mc}] together with scalar product relations, the package will sometimes just return an expression without me specifying the domain of parameters. In this case, would the expression be valid in the parameter region in which there are no imaginary parts? And how would one in general, through PackageX, specify the parameter region one wants the expression in? Thanks!
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| Jul 22, 2018 at 1:36 | history | edited | HirenPatel | CC BY-SA 4.0 |
edited body
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| Jul 22, 2018 at 1:31 | review | Late answers | |||
| Jul 22, 2018 at 1:33 | |||||
| Jul 22, 2018 at 1:15 | history | answered | HirenPatel | CC BY-SA 4.0 |