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

Note: matching corrections beyond first order can only be applied for smoothly ordered showers. Matching corrections beyond first order will therefore be automatically switched off when strongly ordered showers are used, see the section on Evolution and Ordering.

Tree-Level Matching

mode  Vincia:matchingLO   (default = 3; 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.

One-Loop Matching

mode  Vincia:matchingNLO   (default = 2; minimum = 0; maximum = 2)
Selects the order of one-loop matrix-element corrections, also called Next-to-Leading-Order matching. The value should 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 one-loop matching up to and including X+1 partons (one emission plus one loop gives two powers of alphaS), to the extent the relevant matrix elements are available in the code, see the list below. In other words, to include one-loop matching for X+n partons, this switch must be set to n+1. The value 0 is equivalent to switching matching off. Note: values larger than the leading-order value above will be ignored. Thus, to switch off all matrix-element corrections, it is sufficient to switch off the LO ones.

The choice of functional form of the renormalization scale used for the alphaS power associated with the one-loop correction amounts to an NNLO effect and is hence formally beyond the explicit control of the NLO matching. It is controlled by the following parameter:

mode  Vincia:alphaSmodeNLO   (default = 0; minimum = 0; maximum = 1)

option 0 : The invariant mass of the parton system, m(ijk).
option 1 : Transverse momentum, defined as in ARIADNE, pT = m(ij)*m(jk)/m(ijk).
Note 1: for a generic multileg topology, the effective renormalization scale is computed as the geometric mean of such scales, taken over all ordered three-parton clusterings in the event.
Note 2: the default value, 0, has been chosen to limit the absolute size of the NLO corrections, especially for soft branchings. For hard corrections, there should not be much difference between the two choices (though one can of course always argue about factors of 2). For soft corrections, differences appear starting from order aS2*Log(s/pT2). Thus, changing to 1 increases the absolute size of the NLO corrections for soft branchings. Ideally, the shower and hadronization parameters should then be retuned.
Note 3: In the current formulation of the VINCIA NLO matching formalism, option 1 is intended mostly for theoretical reference. The matching expression used in the code is of the form (1 + V), which implicitly assumes that the correction, V, is small. For option 1, however, V becomes large for soft branchings. In this case, a resummed form of the matching expression would have to be used instead, but such an expression has so far not been derived.

Subleading Color

By default, tree-level matching is performed using full-color matrix elements, while the one-loop corrections are so far performed at leading color. 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 diagonalizes the full color matrix and guarantees positive-weight corrections.

List of Matrix-Element-Corrected Processes

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

BasicLOHelicity-NLO
Process Born * αsn Dependent Born * αsn

H0 → gg 1, 2, 3 y -
H0/H+/- → qq (massless) 1, 2, 3, 4 y -
H0 → qq (massive) 1, 2, 3, 4 n -
H+/- → qq (massive) 1, 2(g) n -
Z/W → qq (massless) 1, 2, 3, 4 y 1, 2
W → qq (massive) 1, 2(g) n -
Z → qq (massive) 1, 2, 3, 4 n -

Note: for helicity-dependent matrix elements, the symbols Z, W, H0, H+, are used for generic spin-charge states and thus apply also to, e.g., Z', W', etc.

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 choices described here affect how the scales Q and Qmatch in the regulators described above are computed.

mode  Vincia:MatchingRegType   (default = 1; minimum = 1; maximum = 2)
When Vincia:matchingRegOrder >= 1, choose argument of the regulator function (i.e., Q in the equations listed under Vincia:matchingRegShape).
option 1 : Impose matching scale in the type 1 evolution variable, Qmatch = pT (with unit normalization). The smallest pT scale of the current branching and the color neighbor on either side (if any) is used.
option 2 : Impose matching scale in the type 2 evolution variable, Qmatch = min(sij,sjk) (with unit normalization).

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 hadronization 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 normalization of the evolution variable will affect how this translates to invariants.

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