Higher-Twist

Overview

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Here we present some results for the "brick" problem from the Higher-Twist approach.

The basic calculation scheme followed here is outlined in the papers,

X.N. Wang and X.F. Guo, Nucl.Phys.A696:788-832,2001 [1]

A. Majumder, arXiv:0901.4516 nucl-th. [2]

A. Majumder, arXiv:0912.2987 nucl-th. [3]

Some details of the calculation, e.g., the details of the evolution in medium are not yet published and should appear soon.

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Details of the Calculations

The Higher-Twist scheme is all about evolving (changing) distributions of hadrons fragmenting from a jet due to the passage of the jet through a medium. One starts with a distribution of some detected hadrons (pions, kaons, protons etc.) and evolves it up in-vacuum or in-medium starting from some "final" lower virtuality up to some "initial" higher virtuality.

Evolution in-vacuum is performed using the standard DGLAP evolution equations. The evolution kernel in-medium is derived in Ref. [1] above and a simplified version appears in Ref. [2], which we use. Note: there is no on-shell quark or gluon in this formalism. The final states always have to be on-shell hadrons. The in-medium evolution kernel describes the scattering of a hard initial quark with energy $E$ and virtuality $Q^{2}$, off gluons in the medium and the induced emission of gluons with lower virtuality. The evolution charts the progress of the distributions from $Q^{2}$ down to some lower yet perturbatively large scale $\mu ^{2}$. Unless stated otherwise, $\mu ^{2}$ is chosen as 4 GeV$^{2}$.

The evolution kernel does not contain effects of scattering off quark states or soft-elastic energy loss. The results presented do not contain the absorption of collinear partons or energy gain in the medium. The number of scatterings per radiation is assumed to be between 1 and 2. Solving the evolution equation, in essence, resums the effect of infinite emissions.

The brick is usually held at a $T=400$ MeV which is considered equivalent to a ${\hat {q}}\simeq 2$ GeV$^{2}$/fm for a quark and ${\hat {q}}\simeq 4.5$ GeV$^{2}$/fm for a gluon. Results are usually presented for an initial quark entering the medium. The final distributions computed are usually for $\pi ^{0}$'s unless stated otherwise. The effect of different initial $Q^{2}$ and different initial energies are presented.

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The Plots

• Realistic plots: These included calculations which are closest in spirit to actual higher twist calculations and are focussed on dealing with the change in the fragmentation function of $\pi ^{0}$'s from a $u$-quark

• • Case for an initial quark energy of 20 GeV and two different lengths of medium 2 fm and 5 fm. HT_@_20_GeV

• • Case for an initial quark energy of 100 GeV and two different lengths of medium 2 fm and 5 fm. HT_@_100_GeV

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• Un-realistic plots: The calculations here have almost no relation to standard higher twist calculations and are simply presented for comparison with other formalisms such as AMY. We invent a fragmentation function for a quark with a virtuality of $\mu ^{2}=4$ GeV$^{2}$ to fragment into fictitious quarks and gluons with a virtuality of $m^{2}=0.25$ GeV$^{2}$ i.e., a mass of m = 0.5 GeV. This is similar to the masses of the partons in the AMY formalism. The initial distributions are given simply as

$D_{q}^{q}(z,\mu ^{2})={\frac {\alpha _{s}}{2\pi }}\log \left({\frac {\mu ^{2}}{m^{2}}}\right){\frac {1+z^{2}}{1-z}}$

$D_{q}^{g}(z,\mu ^{2})={\frac {\alpha _{s}}{2\pi }}\log \left({\frac {\mu ^{2}}{m^{2}}}\right){\frac {1+(1-z)^{2}}{z}}$

and so on. Appropriate cutoffs have been placed at ${\frac {1}{2}}-{\frac {1}{2}}{\sqrt {1-{\frac {m^{2}}{\mu ^{2}}}}}<z<{\frac {1}{2}}+{\frac {1}{2}}{\sqrt {1-{\frac {m^{2}}{\mu ^{2}}}}}$, and $\alpha _{s}=0.3$.

These distributions are then evolved in vacuum and in a brick at 400MeV of length 2 fm using the same evolution kernels as in the realistic calculations with pions in the section above. See the plots on the page for HT partonic distributions.

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Comparison with other approaches

In contrast to other formalisms of jet modification, the Higher Twist scheme starts with an input fragmentation function at a given virtuality scale $\mu ^{2}$ and uses an evolution kernel to change this distribution with the virtuality. The fragmentation function is present at all stages of the calculation and is not attached at the end of a purely partonic calculation as in other formalisms. This makes the comparison with other approaches not straightforward. However some schemes are described in the links below.

• comparison with AMY HT_to_AMY

• comparison with ASW/GLV HT_to_ASW

After considerable testing, it has been found that the best method is to simulate the incoming quark to quark fragmentation with a $\delta (1-z)$ and represent the $\delta$-function on the grid as $1/\Delta z$, where $\Delta z$ is the bin width.

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Testing the collinear approximation with a $\delta$-function input

Here we test the validity of the collinear approximation within the Higher twist approach.

We point out that we are using the ``standard`` version of the HT approach where a minimum hard scale of 1$GeV^{2}$ is used. No calculations may be performed below this scale. One may only introduce non-perturbative input at this scale.

We introduce the quark to quark fragmentation function as $D_{q\rightarrow q}(z)=\delta (1-z)$.

The quark to gluon fragmentation function is set to zero.

The gluon to quark-antiquark splitting function has been set to zero.

The plots always have a ${\hat {q}}_{G}=3GeV^{2}/fm$.

Calculations are carried out for two lengths of 2 fm and 5 fm.

The energy of the quark jet is always 20 GeV and the maximum virtuality is $Q^{2}=100GeV^{2}$.

The lowest virtuality on exit is given using a simple formation length argument

$E/Q_{0}^{2}>L$ i.e., radiations with these virtualities happen outside the medium.

This gives, $Q_{0}^{2}=E/L$. If this number fall below $1GeV^{2}$, we set it back to $1GeV^{2}$.

The HT already has an energy constraint in the calculation which is

${\frac {l_{\perp }^{2}}{2Ey(1-y)}}\leq M$ (BLACK lines)

where M is the maximum mass or energy that may be transferred from the medium to the jet in a single exchange. ] This is set to be 3T ~ 1 GeV in the plots.

The additional constraints that have been imposed are

$l_{\perp }\leq {\rm {Min}}[y,1-y]E$ (RED lines)

$45^{o}{\rm {constraint:}}l_{\perp }\leq 0.828{\rm {Min}}[y,1-y]E$ (GREEN lines)

Three plots are included.

1: L = 2 fm 2fm

2: L = 5 fm 5fm

3: L = 5 fm, with the vacuum contribution 5fm:b

Last edited by Amajum 14:35, 13 January 2010 (EDT)