# 7. A posteriori scale variations¶

There are several ways to compute the effects of changing the scales and PDFs of any event produced by Sherpa. They can computed explicitly, cf. Explicit scale variations, on-the-fly, cf. On-the-fly event weight variations (restricted to multiplicative factors), or reconstructed a posteriori. The latter method needs plenty of additional information in the event record and is (depending on the actual calculation) available in two formats:

## 7.1. A posteriori scale and PDF variations using the HepMC GenEvent Output¶

Events generated in a LO, LOPS, NLO, NLOPS, MEPS@LO, MEPS@NLO or
MENLOPS calculation can be written out in the HepMC format including
all infomation to carry out arbitrary scale variations a
posteriori. For this feature HepMC of at least version 2.06 is
necessary and both `HEPMC_USE_NAMED_WEIGHTS: true`

and
`HEPMC_EXTENDED_WEIGHTS: true`

have to enabled. Detailed
instructions on how to use this information to construct the new event
weight can be found here
https://sherpa.hepforge.org/doc/ScaleVariations-Sherpa-2.2.0.pdf.

## 7.2. A posteriori scale and PDF variations using the ROOT NTuple Output¶

Events generated at fixed-order LO and NLO can be stored in ROOT NTuples that allow arbitrary a posteriori scale and PDF variations, see Event output formats. An example for writing and reading in such ROOT NTuples can be found here: Production of NTuples. The internal ROOT Tree has the following Branches:

`id`

Event ID to identify correlated real sub-events.

`nparticle`

Number of outgoing partons.

`E/px/py/pz`

Momentum components of the partons.

`kf`

Parton PDG code.

`weight`

Event weight, if sub-event is treated independently.

`weight2`

Event weight, if correlated sub-events are treated as single event.

`me_wgt`

ME weight (w/o PDF), corresponds to ‘weight’.

`me_wgt2`

ME weight (w/o PDF), corresponds to ‘weight2’.

`id1`

PDG code of incoming parton 1.

`id2`

PDG code of incoming parton 2.

`fac_scale`

Factorisation scale.

`ren_scale`

Renormalisation scale.

`x1`

Bjorken-x of incoming parton 1.

`x2`

Bjorken-x of incoming parton 2.

`x1p`

x’ for I-piece of incoming parton 1.

`x2p`

x’ for I-piece of incoming parton 2.

`nuwgt`

Number of additional ME weights for loops and integrated subtraction terms.

`usr_wgt[nuwgt]`

Additional ME weights for loops and integrated subtraction terms.

### 7.2.1. Computing (differential) cross sections of real correction events with statistical errors¶

Real correction events and their counter-events from subtraction terms are highly correlated and exhibit large cancellations. Although a treatment of sub-events as independent events leads to the correct cross section the statistical error would be greatly overestimated. In order to get a realistic statistical error sub-events belonging to the same event must be combined before added to the total cross section or a histogram bin of a differential cross section. Since in general each sub-event comes with it’s own set of four momenta the following treatment becomes necessary:

An event here refers to a full real correction event that may contain several sub-events. All entries with the same id belong to the same event. Step 2 has to be repeated for each event.

Each sub-event must be checked separately whether it passes possible phase space cuts. Then for each observable add up

`weight2`

of all sub-events that go into the same histogram bin. These sums \(x_{id}\) are the quantities to enter the actual histogram.To compute statistical errors each bin must store the sum over all \(x_{id}\) and the sum over all \(x_{id}^2\). The cross section in the bin is given by \(\langle x\rangle = \frac{1}{N} \cdot \sum x_{id}\), where \(N\) is the number of events (not sub-events). The \(1-\sigma\) statistical error for the bin is \(\sqrt{ (\langle x^2\rangle-\langle x\rangle^2)/(N-1) }\)

Note: The main difference between `weight`

and `weight2`

is that they
refer to a different counting of events. While `weight`

corresponds to
each event entry (sub-event) counted separately, `weight2`

counts events
as defined in step 1 of the above procedure. For NLO pieces other than the real
correction `weight`

and `weight2`

are identical.

### 7.2.2. Computation of cross sections with new PDF’s¶

#### 7.2.2.1. Born and real pieces¶

Notation:

```
f_a(x_a) = PDF 1 applied on parton a, F_b(x_b) = PDF 2 applied on
parton b.
```

The total cross section weight is given by:

```
weight = me_wgt f_a(x_a)F_b(x_b)
```

#### 7.2.2.2. Loop piece and integrated subtraction terms¶

The weights here have an explicit dependence on the renormalization and factorization scales.

To take care of the renormalization scale dependence (other than via
`alpha_S`

) the weight `w_0`

is defined as

```
w_0 = me_wgt + usr_wgts[0] log((\mu_R^new)^2/(\mu_R^old)^2) +
usr_wgts[1] 1/2 [log((\mu_R^new)^2/(\mu_R^old)^2)]^2
```

To address the factorization scale dependence the weights `w_1,...,w_8`

are given by

```
w_i = usr_wgts[i+1] + usr_wgts[i+9] log((\mu_F^new)^2/(\mu_F^old)^2)
```

The full cross section weight can be calculated as

```
weight = w_0 f_a(x_a)F_b(x_b)
+ (f_a^1 w_1 + f_a^2 w_2 + f_a^3 w_3 + f_a^4 w_4) F_b(x_b)
+ (F_b^1 w_5 + F_b^2 w_6 + F_b^3 w_7 + F_b^4 w_8) f_a(x_a)
```

where

```
f_a^1 = f_a(x_a) (a=quark), \sum_q f_q(x_a) (a=gluon),
f_a^2 = f_a(x_a/x'_a)/x'_a (a=quark), \sum_q f_q(x_a/x'_a)x'_a (a=gluon),
f_a^3 = f_g(x_a),
f_a^4 = f_g(x_a/x'_a)/x'_a
```

The scale dependence coefficients `usr_wgts[0]`

and `usr_wgts[1]`

are normally obtained from the finite part of the virtual correction
by removing renormalization terms and universal terms from dipole
subtraction. This may be undesirable, especially when the loop
provider splits up the calculation of the virtual correction into
several pieces, like leading and sub-leading color. In this case the
loop provider should control the scale dependence coefficients, which
can be enforced with option `USR_WGT_MODE: false`

.

Warning

The loop provider must support this option or the scale dependence coefficients will be invalid!