Jacobian factors

[enocera]

As reported in https://github.com/scarlehoff/Hawaiian_vrap/issues/6, cross sections computed with Vrap+PineAPPL differ from the available FK tables computed with APFEL+APFELComb. Here is a summary of my understanding.

1. The baseline computation performed by APFEL corresponds to Eq. (7) in [arXiv:1002.4407], that is (at LO, for simplicity): $$\frac{d^2\sigma}{dM^2dy}(M^2,y)=\frac{4\pi\alpha^2}{9M^2s}\sum_i\left[q_i(x_1,M^2)\bar{q}_i(x_2,M^2) + \bar{q}_i(x_1,M^2)q_i(x_2,M^2)\right].$$
2. The data for DYE605 is in the form $$s\frac{d^2\sigma}{d\sqrt{\tau}dy},$$ with $\tau=M^2/s$. One can change variables and write $$s\frac{d^2\sigma}{d\sqrt{\tau}dy}=2Ms^{3/2}\frac{d^2\sigma}{dM^2dy},$$ with the Jacobian $J=2Ms^{3/2}$.
3. The data for DYE866P is instead in the form $$M^3\frac{d^2\sigma}{dMdy}.$$ One can change variables and write $$M^3\frac{d^2\sigma}{dMdy}=2M^4\frac{d^2\sigma}{dM^2dy},$$ with the Jacobian $J=2M^4$.
4. By comparing the Jacobian factors in 2. and 3., one can see that the cross sections for DYE605 and DYE866P differ for a factor $$\frac{2Ms^{3/2}}{2M^4}=\left(\frac{\sqrt{s}}{M}\right)^3.$$ This explains the factor observed by @scarlehoff in #6. Note that the cross section computed by APFEL, and therefore encapsulated in the FK tables, is always consistent with the format of the data. The Jacobian factors are indeed included as multiplicative factors on top of the baseline computation performed by APFEL: for DYE605 here; for DYE866P here.
5. I understand that Vrap computes the cross section in the form $$\frac{d^2\sigma}{dMdy}.$$ This is what I grasp from the documentation and from here and here. This fact makes me think that, once the $\left(\frac{\sqrt{s}}{M}\right)^3$ factor is fixed in the DYE886P data, the DYE605 and the DYE866P predictions obtained with Vrap+PineAPPL differ from those obtained with APFEL+APFELComb by the same factor. This is consistent with what is reported by @scarlehoff in #6 . The natural thing to believe is that this factor is $2M$. However, this does not explain the factor ~54 observed by @scarlehoff , which is independent from $M$. Maybe Vrap computes something else? I'll investigate it further tomorrow.

[enocera]

Let me try to go back to this.

[enocera]

@scarlehoff I think that I now understand the origin of the ~54 factor. We must rescale the Vrap result by $\sqrt{2s}$. With $\sqrt{s}=38.76$ GeV being the c.m.e. of both DYE605 and DYE866, one has $\sqrt{2s}=54.81$ GeV. I'll write the explanation of this in detail later, for the time being please rescale the Vrap result by a factor $\sqrt{2s}$ by default (this also applies to the ratio observables, though of course there it is immaterial because of the num/den simplification).

[scarlehoff]

The factor works perfectly, although with this factor the agreement is a bit worse (now the agreement between apfel and vrap is at worst 1.2% instead of being below 1% all across -with 54.35-).

[enocera]

Which, I think, is a mater of life. Note that there may be other sources for this "discrepancy", such as different values of the physical parameters (such as MZ), as @cschwan noted yesterday, or in the pbarn conversion factor, as noted in #9 .

[cschwan]

Are all the input parameters the same between APFEL and Vrap?

[scarlehoff]

The pbarn conversion is fixed. The factors hardcoded in vrap now are all using the latest edition of the PDG (but this might not be the case for apfel!) But yes, I think we can live with 1.2% :P

[enocera]

OK, but are these values consistent between APFEL and Vrap? I'd say no (so this may explain the difference). Question: are you taking the value of $\sqrt{s}$ from the commondata? I've taken $\sqrt{s}=38.76$ from the paper, BUT I haven't checked that this is the SAME as int he commondata. Actually, the commondata implementation of the old FT DY data (everything but DYE906) is a mess...

[enocera]

Sorry - I see that the values are not as in APFEL, you've said this explicitly!

[scarlehoff]

Actually, the commondata implementation of the old FT DY data (everything but DYE906) is a mess...

Well, about DYE906, turns out that the fktable expect a jacobian factor computed with $\sqrt{s} = 38.76$. Of course, it "doesn't matter" because we always use it as a ratio...