Why do NLO cross sections sometimes decrease relative to LO ones?

[ChrisVer]

I was wondering, can the calculation of the NLO cross section decrease the cross section (relative to the LO one)? or not?
I am asking because I read here
http://www.pa.msu.edu/~huston/Kfactor/Kfactor.pdf
the following

While explicit loop corrections often do increase the NLO cross sections with respect to the LO ones, it can be observed from the table above that the K-factors do not have to be greater than unity.

Which refers to Table 1 and cases where the K-factor appears to be less than "1". Do those cases belong to the not-oftenly increased cross section? Or is there something else that decreases the K-factor from >1 yet the NLO is relatively larger than LO $\sigma$s?

 

[mfb]

NLO cross sections can be smaller. Think of both LO and NLO as complex amplitudes with random relative phase. An increase is a more likely result, but you can also decrease the cross section.

"Experimentalists typically deal with leading order (LO) calculations, especially in the context of parton shower Monte Carlos."
That seems to be a bit outdated. LO rarely has the precision you need, at least for standard model processes (if you search for a new particle, 20% more or less doesn't matter much). Some measurements have to move to NNLO already.

 

[ChrisVer]

LO rarely has the precision you need

So you need to apply 2 "K"-factors?
I don't know if it's outdated, maybe it has to be (its references span the period of 2008-2009, so I'd put it at least 2010)... But, don't the MC simultions produce LO results?

I have some more questions on this 'article'...
Trying to explain the slope as a function of the # of jets in the W+n-jets K-factors, the author says:

At LO, one parton equals one jet. By choosing a jet algorithm with size parameter D, we are requiring any two partons to be a distance D or greater apart. The matrix elements have 1/∆R poles, so a larger value of D means smaller cross sections. At NLO, there can be two partons in a jet, and jets for the first time can have some structure. No ∆R cut is needed since the virtual corrections cancel the collinear singularity from the gluon emission (but there are residual logs that can become important if the value of D is too small). Increasing the size parameter D increases the phase space for including an extra gluon in the jet, and thus increases the cross section at NLO (in most cases).

(Ch 3.2- W+3jets)

In fact this paragraph contains so much information that becomes mind-blowing. It would not be an exaggeration to say that I didn't understand a single phrase here (Except for the last one)...
Firstly, what does one parton=one jet mean? and what does two partons=one jet mean?
Second, how can I see the $1/\Delta R$ poles in the matrix elements? or is this information irrelevant?
Third, what does it mean for a jet to have a structure? Is it something similar to the cluster shape?
Next, who ever applied a $\Delta R$ cut? OK for the jet algorithm, the $\Delta R$ cut is applied with $D$ but I think this is always the case for jets...I mean you have to set a cone in which you define your jet to be...
Finally what is a virtual correction and a collinear singularity?
Thanks.

 

[mfb]

So you need to apply 2 "K"-factors?

No, you should apply NLO calculations. Many MC generators can do that for many processes out of the box. But what you actually want are dedicated predictions from theory groups. Those are usually not done with the full hadronization and detector simulation, so you still need your MC to estimate the hard process from the observations.

Your hard process produces some number of partons. Every quark and gluon in that process will then form one jet.

Second, how can I see the 1/ΔR1/\Delta R poles in the matrix elements?

They should be there somehow but don't ask me for details.

Third, what does it mean for a jet to have a structure? Is it something similar to the cluster shape?

Sometimes jets look like two narrow jets that overlap, for example.

ΔR cuts come from the jet algorithm, but there are also analyses requiring larger separation between their objects for various reasons.

The virtual loops in NLO can cancel the problems of collinear gluons (gluons very close to quarks) somehow, but I don't know details here either.

 

[RGevo]

I can add to mfb's already nice answer.

So for the matrix element, you are literally calculating the process ij -> W + k l m (where i,j,k,l,m are quarks or gluons - this would contribute to part of the W+3jet process if you applied the jet algorithm to it).

At the level of the matrix element, it has poles/Infrared divergences when the k l m partons are emitted soft or collinear with respect to one of the other partons in the process. The easiest way to see this would be to calculate the matrix element, which is a function of invariants like p_i . p_j, p_k . p_l ... etc. (in principle there will be a minimal subset of invariants, for 2->2 it is the usual S,T,U parameters, for 2->3 there are 5 minimal invariants etc. so on).

The function for this matrix element will involve combinations like:
p_i . p_k / (p_l . p_m)... etc.

And the divergences are when the stuff in the denominator goes to zero. To see these as divergences in Delta R, (eta,phi), I think you should write the distance between the four-vectors in eta and phi.

If for this process we are considering ( 2 -> W +3 ), we require that these 3 final state partons are well separated in Delta R all possible combinations of these invariants A / (p_l . p_m) ... will kinematically not contain any divergences. As such, you can compute the process and have no problems.

If instead, two of your 3 final state partons were very close. In terms of jets, the two partons would be reconstructed as one. The final state would then be W+2 jets. The IR divergences of this 3 partons final state (2 jet final state) are then canceled by a virtual correction to the 2 parton final state (where these two partons are well separated and form two jets).

Let me summarise the main points for W+n jets. Firstly, the W+1 parton final state is part of the NLO QCD correction to W+0 parton production. If we require this 1 parton to be well separated from other partons (that is to say the incoming partons) it counts as a jet. If the parton were collinear/soft with respect to the incoming partons then I would need the IR divergences from the virtual correction to cancel it. This argument then holds for the calculation of W+2 partons... bla bla bla.

The cancelation of these divergences which is performed is a consequence of KLN-theorem. Notice here we are only talking about Infrared divergences (not UV divergences).

Does this help, or is something not so clear?

%%%%%
There are only a very few number of processes which have been computed at NNLO in such a way that they can be interfaced with a parton shower (NNLO+PS see for example http://arxiv.org/abs/1309.0017 and subsequent publications). However, NLO+PS has been automated now. This is one of the most important experimental tools, since it combines NLO accuracy for some observables (total cross section etc.), but has the benefit that it includes parton shower effects (so lots of gluon/quark splittings resummed to all orders at collinear leading log accuracy), as well as many non-perturbative effects which have been modeled/fit to data.

Of course, as mfb says, sometimes it is appropriate to compare to a calculation which is truly fixed-order (no parton shower). For example, comparisons of top quark differential measurements at NNLO fixed order accuracy (which are available at NLO+PS production and decay accuracy).

####
Edit: finally, as an after thought. It's probably better to look at these divergences at the level of the squared matrix element. Collinear divergences factorise at the level of the matrix element squared (not at the amplitude level like soft divergences). So I think it's better to analyse them not at the level of a matrix element. Not totally sure here though.

 

[ChrisVer]

So the main problem of divergences comes from "soft" jets? and they are soft because they are collinear to the initial state partons? I think this is pretty clear in your post.
I have a small problem in grasping the virtual correction idea however. You say that if I reconstruct the 3 partons as 3 jets , the IR-divs cancel out. If however I reconstruct the 3 partons as 2 jets (3-->2), the IR-divs are canceled by a virtual correction to a 2 parton to 2 jets (2-->2)?

_{(I guess I will need to check for the KLN theorem tomorrow, it's late here.)}

about the parton showers; is there any difference between the NLO $\it{W} + \text{1 jet}$ and the LO $\it{W} + \text{0 jet}$ + PS?

 

[RGevo]

Good god, physics forums deleted by lengthy response when I logged in :-/ This attempt will be less detailed - sorry.

Partons which are soft or collinear. The jets are (by definition of the algorithm) meant to be insensitive to soft and collinear partons, and in fact will contain both soft and collinear partons inside them.

Lets take W+0 jet production as an example. We can do the QCD correction to this process (both the real emission and the virtual) to the NLO correction.

At the LHC, this would be a 2->1 process (q qbar > W) and the W boson must have 0 transverse momentum to conserve momentum (which is only in the Z-axis in the initial state). Good.

I do the virtual correction, and this applies to the 0 pT configuration (since I still have a 2>1).

Now I do the real emission diagram (which can spit off a gluon from one of the quarks, or i can have a gluon in the initial state). This will have a kinematic contribution like:
p_i . p_j / (p_i p_k) / (p_j . p_k) where p_k is the four momentum of the emitted parton, i/j are the incoming parton labels. If these are massless partons we can write these invariants as

p_i. p_k = E_i E_k (1 - cos(ik) ), where ik is the angle between the partons. This invariant tends to zero if E_k -> 0 (soft), or cos(ik) -> 1 which occurs when the angle between the two tends to zero (collinear). So in other words, the emitted parton has very low energy or the parton goes shooting along the z-direction (with small angle compared to the incoming partons). In other words, it gives an increasingly large contribution as the pT of the W goes towards zero. This positive divergence is ultimately canceled versus the virtual divergence which lives in the 0 pT bin... but the region at low pT is not well modeled by an NLO calculation.

On the other hand, the LO+PS description will include many parton splittings. Specifically, the splittings are produced according to an all-order solution to parton splittings in the collinear limit (DGLAP evolution). Consequently you get a very good (or at least much better) description of the low pT region. On the other hand, the parton splittings are correct only in the collinear limit. In the normal limit (or hard limit away from collinear configurations) the parton shower does not know the full answer for the emission of a parton, and consequently it does not describe well this region.

So the NLO calculation would do well at high pT values of the W. The LO+PS would do better at describing the shape at low pT. This is one of the main differences.

You see, that there are benefits of both. This is why now people developed NLO+PS (for over 10 years now) and also the few NNLOPS calculations which are available. Like all physics, its important to know when to apply which theory description.