Collins and IFF Angles at STAR/COMPASS/HERMES
This is the extended version of the message that I posted to the spin hypernews
Hi All,
Sorry for digging out the old thread. But I've spent the past few days looking
at the definitions of Collins angles at STAR/Hermes/Compass and I seem to
have found a way relating one to another. We can also resolve the sign issue between
our Collins and IFF asymmetries.
Here is the short conclusion.
a). In our Collins analysis phi_s is different from Trento/Hermes convention while phi_h is
the same. As a result, our Collins angle phi_collins(star) is shifted w.r.t Trento as follows,
phi_collins(trento) = pi - phi_collins(star) = pi - (phi_s(star) - phi_h(star))
with phi_s(star) = pi - phi_s(trento)
phi_h(star) = phi_h(trento)
But in terms of the sine modulation our definition gives the same moments <sin(phi_collins))
as Hermes' results.
b). In our IFF analysis phi_s and phi_R is defined in the same way as phi_s(star) and phi_h(star).
But that doesn't mean the sign of A_{UT}^{sin(phi_RS)} from IFF is opposite to A_{N}^{sin(phi_c)}
from Collins once we corrected the spin patterns. The correction depends on how you calculated the
symmetries with the wrong spin states.
c). The spin rotation around y axis mentioned by Anselm exists in both pp and
SIDIS in the same way. The reason that Compass has negative Collins asymmetry
is simply because of a different choice in defining the Collins angle. The physics
remains the same --when the spin of the 'outgoing' quark is pointing up, pi+ will be
produced preferentially to the left if looking down to this quark's momentum.
----
Argument for a).
In our Collins analysis phi_s and phi_h are defined in the same way(please confirm?)
as D'Alesio's paper Phys.Rev.D 83.034021(2011) where you firstly build a hadronic c.m.
frame on the scattering plane in such a way that (p_jet)_x > 0, then the helicity frame
of the fragmenting parton follows from rotating the hadronic c.m. frame around its
y-axis so that the z-axis is in the same direction as the outgoing parton. In this
way phi_s(star) is simply the azimuthal angle of proton's spin in hadronic c.m. frame,
phi_h(star) is the azimuthal angle of hadron's momentum in the helicity frame of the
fragmenting parton.
The above scheme is different from Trento convention, but they are connected by
phi_s(star/initial) = pi - phi_s(trento/initial)
phi_h(star) = phi_h(trento)
Of course phi_s(trento/initial) is in terms of the spin of initial quark (target). You can make sense
out of these relations by transforming fig.1 of D'Alesio's paper into a frame where incoming and outgoing
partons are on the same z-axis, therefore the picture of the new frame is similar to fig.1 of Trento paper
(arxiv: hep/ph0410050v2).
However, in going from the initial to final hard scattered parton,
phi_s(star/final) = phi_s(star/initial)
phi_s(trento/final) = pi - phi_s(trento/initial)
because the spin transfer from inital to final generates a spin rotation around
y-axis (by pi) due to the opposite momentum directions of incoming/outgoing quarks
in gamma-nucleon(as in sidis) or partonic helicity(as in pp) frames. However
in STAR/D'Alesio's scheme this process becomes hidden because the x-axis of the
partonic helicity frame is flipped instead of changing phi_s. Therefore,
phi_s(star/final) = phi_s(trento/final)
and the Collins angle,
phi_collins(trento) = phi_s(trento/initial) + phi_h(trento)
= pi - phi_s(trento/final) + phi_h(trento)
= pi - phi_s(star/final) + phi_h(star)
= pi - [phi_s(star/initial) - phi_h(star)]
= pi - phi_collins(star)
So in terms of sine modulation sin(phi_collins(trento)) = sin(phi_collins(star)), and
our collins asymmetry should have the same sign as Hermes' result.
Argument for b).
The way to figure out how phi_s(iff) is related to Trento is similar as above, by
Transforming fig.1 of Bacchetta & Radici's paper (arxiv: hep/ph0409174v2) into a
partonic c.m. frame where the new P_B and P_C are lying on the new z-axis, in order
to mimic the gamma-nucleon frame as outlined by fig.1 of the Trento paper. Notice that
you can say P_C -> q only in that frame.
Still we are looking at the moments of sin(phi_s(star) - phi_R(star)) as our Collins analysis
with phi_h(star) replaced by phi_R(star). There is no problem with phi_R(star) and phi_h(star),
they are the same as phi_h(trento).
Here is my opinion regarding how we should correct the sign of A_{UT}^{sin(phi_RS)}.
b1.) Correct phi_s(star) to account for the wrong spin pattern, where
phi_s(star/right) = pi + phi_s(star/wrong).
So what's previously been plotted as a function of phi_RS(wrong) ( = phi_s(star/wrong) - phi_R(star) )
should now be changed to pi + phi_RS(wrong) = phi_RS(right) = pi + phi_s(star/wrong) - phi_R(star).
For example, when extracting A_UT from raw yield asymmetries vs phi_RS Anselm plotted
asy.(wrong) = N(up/wrong)-N(down/wrong) / [N(up/wrong)+N(down/wrong)]
vs phi_RS(wrong) and the amplitude of sin(phi_RS(wrong)) modulation is A_UT (positive).
Now with the right spin states the x-axis should be shifted by pi, which will give you negative
A_UT.
b2). Correct the sign of raw yield asymmetry.
The asy. plotted on the y-axis should also have its sign flipped in order to have
asy.(right) = N(up/right)-N(down/right) / [N(up/right)+N(down/right)] = -asy.(wrong)
This will flip the sign of A_UT again.
Therefore the sign of A_{UT}^{sin(phi_RS)} will be flipped twice to account for the corrections to the spin
pattern. This still produces a positive asymmetry for IFF as what it is now.
As a comparsion, since in Collins analysis one does not have to calculated asy. of the raw yield between spin states
and everything can be done with a single spin state,in principle we only need to flip the sign of A_{N}^{sin(phi_s - phi_h)}
once. But as the original amplitude was plotted in terms of sin(phi_h - phi_s), you need a second flip from there.
Argument for c).
No arugment for now, but fig.1 of arxiv: hep/ex-1111.0869v1 is a nice plot showing the definition of anlges at
COMPASS. Both of phi_s and phi_h follow Trento convention, except that they have chosen a different collins
angle phi_c = phi_s - (pi - phi_h)
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