Matrix-Element Corrections (Matching)

  1. Matching Level
  2. List of ME-corrected Processes
  3. Matching Regulator
  4. IR Cutoff

Note: for how to include new (LO) matrix elements, see the section on the MADGRAPH Interface.

Matching Level

Tree-Level Matching

mode  Vincia:matchingLO   (default = 2; minimum = 0; maximum = 4)
Selects the order of tree-level matrix-element corrections, also called Leading-Order matching. The value should be interpreted as counting the total number of powers of alphaS beyond the Born that are matched to matrix elements. I.e., for the basic process X, setting this switch to 2 would invoke tree-level matching up to and including X+2 partons, to the extent the relevant matrix elements are available in the code, see the list below. The value 0 is equivalent to switching matching off.

Subleading Color

By default, tree-level matching is performed using full-color matrix elements. If desired, the subleading-color parts of the tree-level matching can be switched on and off using the following switch:

flag  Vincia:matchingFullColor   (default = true)

option off : Leading Color. Only include matching to leading-color matrix elements.
option on : Full Color. Include the full color structure of the matched matrix elements, absorbing the subleading-color pieces into each leading-color one in proportion to the relative sizes of the leading-color pieces. This procedure effectively diagonalises the full color matrix and guarantees positive-weight corrections.

Interference for Higgs processes

By default, in the matrix element correction procedure the Higgs is allowed to couple to gluons (via heft) and all quark flavours, taking all interferences into account. Say the user wants to include only Higgs to gluon couplings, the following switch can be used:

flag  Vincia:Hinterference   (default = on)

option off : On an event by event basis, depending on hard process at hand, either the Higgs to gluon coupling or all Yukawa couplings are set to zero.
option on : Full interference, no dependence on what the hard process is chosen to be.

List of Matrix-Element-Corrected Processes

Matrix-element corrections have been implemented for the following types of processes:

Basic Process LO (Born * αsn)

Z → jj (massless) 1, 2, 3, 4
W → jj (massless) 1, 2, 3, 4
H0 → jj (massless), heft to g and yukawa to quarks 1, 2, 3, 4

pp → Z (massless) 1, 2, 3
pp → W (massless) 1, 2, 3
pp → H0 (massless), heft to g and yukawa to quarks 1, 2, 3
pp → jj (massless), pure QCD 1, 2, 3

Note: In pure QCD processes the 3rd power of αs can only be calculated for the MHV case, and that the 7-gluon QCD amplitude can only be calculated at leading color order. The shower will skip the correction process if it doesn't have the required amplitude. Further, for W production, the following flavour combination can not be reached by a QCD shower off pp → W events and is therefore not included in the event generation:

q qbar → W q' qbar''

Matching Regulator

We use the term matching regulator to refer to a generic sharp or smooth dampening of the ME corrections as one crosses into a specified region of phase space. The purpose of this is to restrict the matching to regions of phase space that are free from subleading logarithmic divergences in the matrix elements. This is familiar from the CKKW and MLM approaches, where the matching scale is imposed as a step function in pT, with full ME corrections above that scale and no ME corrections below it. We explore a few alternatives to this approach.

mode  Vincia:matchingRegOrder   (default = 3; minimum = 0; maximum = 5)
Choose starting order from which matrix element corrections are regulated.
option 0 : Off. Matrix element corrections are not regulated at all. Not advised for production runs, but can be useful for theory studies.
option 1 : On, starting from 1st order in QCD. This would normally be overkill since the LL shower exactly reproduces the 1st order matrix-element singularities - the first-order correction should therefore normally be free of divergencies and should not need to be regulated.
option 2 : On, starting from 2nd order in QCD. The 2nd-order matrix element correction generally contains subleading logarithmic divergences which do not correspond exactly to those generated by the pure shower. Nonetheless, due to the unitary properties of VINCIA's matching formalism and the close approximation of its shower expansions to 2nd order matrix elements, however, 2nd order corrections can typically be applied over all of phase space, without ill effects.
option 3 : On, starting from 3rd order in QCD. This is the recommended option for the multiplicative matching strategy. Since the matrix-element corrections are exponentiated, the subleading divergencies in the higher-order corrections are effectively resummed. However, due to the LL nature of the underlying shower, it appears from empirical studies that a matching scale is still needed starting from 3rd order even in the multiplicative case.
option 4 : On, starting from 4th order in QCD. Not recommended for production runs, but can be useful for theory studies.
option 5 : On, starting from 5th order in QCD. Not recommended for production runs, but can be useful for theory studies.

mode  Vincia:matchingRegShape   (default = 1; minimum = 0; maximum = 1)
When Vincia:matchingRegOrder >= 1, choose the functional form of the regulator. (See below for how to modify the choice of Q and Qmatch.)
option 0 : Step function at Q=Qmatch, i.e.,

option 1 : Suppress the shower-subtracted ME corrections by a function that is unity above Q2 = 2*Q2match, zero below Q2 = Q2match/2, with a simple interpolation (logarithmic in Q2) between those scales, i.e.,

Matching Scale

The scale associated with the matrix element correction, Q, is the minimum of all branching scales in the given configuration. The following choices affect how the reference / matching scale, Qmatch, in the regulators described above is computed.

flag  Vincia:matchingRegScaleIsAbsolute   (default = false)
Selects whether the user wants to input the value of Qmatch either by giving an an absolute number in GeV or by giving a ratio with respect to the hard scale.
option false : Relative. The matching scale is determined automatically in relation to the hard scale in the process (e.g., the Z mass) by the factor Vincia:matchingRegScaleRatio below. This is the default option and the one recommended for non-experts. It should allow a wide range of processes to be considered without having to manually adjust the matching scale.
option true : Absolute. The matching scale is set by the value Vincia:matchingRegScale (in GeV). Care must then be taken to select a matching scale appropriate to the specific process and hard scales under consideration. For non-experts, the relative method above is recommended instead.

parm  Vincia:matchingRegScaleRatio   (default = 0.05; minimum = 0.0; maximum = 1.0)
When Vincia:matchingRegScaleIsAbsolute == false (default), this sets the ratio of the matching scale to the process-dependent hard scale; inactive otherwise. Since the unresummed logarithms depend on ratios of scales, it is more natural to express the matching scale in this way than as an absolute number in GeV. Note that this parameter should normally not be varied by more than a factor of 2 in either direction. The default value has been chosen so as to allow one order of magnitude between the hard scale and the matching scale. Setting it too close to unity will effectively switch off the matching, even at high scales. Settings around 0.01 and below risk re-introducing large unresummed logarithms in the matching coefficients.

parm  Vincia:matchingRegScale   (default = 20.0; minimum = 0.0)
When Vincia:matchingRegScaleIsAbsolute == true, this sets the absolute value of the matching scale, in GeV; inactive otherwise. Care must be taken to select a matching scale appropriate to the specific process and hard scales under consideration.

IR cutoff

Due to the freezing of alphaS in the infrared, it is possible to run VINCIA with very low hadronisation cutoffs. Though this formally continues the perturbative treatment into the infrared, allowing the emission of gluons with very soft momenta, it is doubtful whether matching corrections would be of any value in that region.

Our intuition is that, at best, continuing such corrections into the region below ~ 1 GeV would merely slow down the code. At worst they could generate unphysically large corrections (e.g., the scale-dependent terms in the NLO corrections are unphysical at scales near ΛQCD).

The parameter below sets an absolute lower scale for the evolution variable, in GeV, below which matrix-element corrections are not applied. Note that the normalisation of the evolution variable will affect how this translates to invariants.

parm  Vincia:matchingIRcutoff   (default = 2.0; minimum = 0.0; maximum = 100.0)