Basics of dilepton decay distributions. Examples: quarkonium and vector bosons A general demonstration of an old and surprising “ perturbative QCD” relation, using only rotation invariance Modelindependent spin characterization of the Higgslike di photon resonance.
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Basics of dilepton decay distributions. Examples: quarkonium and vector bosons
A general demonstration of an old and surprising “perturbativeQCD” relation,using only rotation invariance
Modelindependent spin characterization of the Higgslike diphoton resonance
Angular momentum and decay distributionsin high energy physics: an introduction and use cases for the LHCPietroFaccioli, CERN, April 23th, 2013
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Example: formationofψ andcoloursinglet state
J = 1
red
antired
purely perturbative
+ analogous colour combinations
red
antired
J = 1
colouroctet state
J = 0, 1, 2, …
green
antiblue
Transition to the
observable state.
Quantum numbers change!
J can change! polarization!
perturbative
nonperturbative
J = 1 → three Jzeigenstates1, +1 ,1, 0 ,1, 1 wrt a certain z
Measure polarization = measure (average) angular momentum composition
Method: study the angular distribution of the particle decay in its rest frame
The decay into a fermionantifermion pair is an especially clean case to be studied
The shape of the observable angular distribution is determined by a few basic principles:
1) “helicity conservation”
=
2) rotational covarianceof angular momentum eigenstates
f
* , Z, g, ...
NO
YES
z'
1, +1
3) parity properties
1
1
1
1, +1 + 1, 1 1, 0
2
2
√2
z
?
EW and strong forces preserve the chirality (L/R) of fermions.
In the relativistic (massless) limit, chirality = helicity = spinmomentum alignment
→ the fermion spin never flips in the coupling to gauge bosons:
f
* , Z, g, ...
YES
NO
YES
NO
z'
1, M’ℓ+ℓ
(–1/2)
+1/2
c
( )
ℓ+
ℓ+
( )
J/ψ rest frame:
c
*
z
J/ψ
1, MJ/ψ
( )
(–1/2)
ℓ
( )
+1/2
ℓ
J/ψangular momentum component alongthepolarizationaxisz:
MJ/ψ = 1, 0, +1
(determinedbyproductionmechanism)
Thetwoleptonscan onlyhave total angular momentumcomponent
M’ℓ+ℓ = +1 or 1along their common direction z’
0 is forbidden
z'
change of quantization frame:
R(θ,φ): z → z’
y → y’
x → x’
Jz’ eigenstates
J, M’
θ,φ
z
+ J
J, M
Σ
J, M’ =DMM’(θ,φ)J, M
J
M =  J
Jz eigenstates
Wigner Dmatrices
z'
1, +1
Example:
90°
z
1
1
1
1, +1 + 1, 1 1, 0
Classically, we wouldexpect1, 0
2
2
√2
z'
J/ψ (MJ/ψ= 0) →ℓ+ℓ(M’ℓ+ℓ = +1)
ℓ+
1, +1
J/ψ rest frame
θ
z
1,0
ℓ
→ the Jz’eigenstate1, +1 “contains” the Jzeigenstate1, 0
with component amplitude D0,+1(θ,φ)
1
→ the decay distribution is
1
D0,+1(θ,φ)2 = ( 1 cos2θ)
1, +1 O1, 02
1*
1, +1 =D1,+1(θ,φ)1, 1 + D0,+1(θ,φ) 1, 0 + D+1,+1(θ,φ) 1, +1
1
1
1
2
z
ℓ+ℓ← J/ψ
ℓ+
ℓ+
3: parityθ
θ
z
z
1,+1
1,1
1,1 and 1,+1 distributions
are mirror reflections
of one another
ℓ
ℓ
z
z
1*
1*
Decay distribution of 1,0 stateis always paritysymmetric:
Are they equally probable?
z
z
z
dN
dN
dN
dN
z
dΩ
dΩ
dΩ
dΩ
P (1) > P(+1)
P (1) = P(+1)
P (1) < P(+1)
1*
z
“Transverse” polarization,
like for real photons.
The word refers to the
alignment of the field vector,
not to the spin alignment!
J/ψ = 1,+1
or 1,1
1 + cos2θ
x
(parityconserving case)
y
z
“Longitudinal” polarization
J/ψ = 1,0
1 – cos2θ
x
dN
dN
y
dΩ
dΩ
We can apply helicity conservation at the production vertex to predict that
all vector states produced in fermionantifermion annihilations (qq or e+e–) at Born level have transverse polarization
qq rest frame
= V rest frame
q
( )
(–1/2)
+1/2
( )
( )
q
z
V
V = *, Z, W
q
q
( )
The “natural” polarization axis in this case is
the relative direction of the colliding fermions
(CollinsSoper axis)
1
.
5
DrellYan
1
.
0
DrellYan is a paradigmatic case
But not the only one
(2S+3S)
0
.
5
E866, CollinsSoper frame
dN
0
.
0
dΩ
1 +λcos2θ

0
.
5
0
1
2
pT [GeV/c]
z
z
chosenpolarization axis
θ
ℓ +
production plane
φ
y
x
particle
rest frame
average
polar anisotropy
average
azimuthal anisotropy
correlation
polar  azimuthal
parity violating
zHX
zHX
zHX
zGJ
zGJ
Helicity axis (HX): quarkonium momentum direction
GottfriedJacksonaxis (GJ):direction of one or the other beam
zCS
CollinsSoperaxis (CS): average of the two beam directions
h1
h2
Perpendicular helicity axis (PX): perpendicular to CS
hadron
collision
centre
of mass
frame
h2
h2
h2
h1
h1
h1
production plane
p
zPX
particle
rest
frame
particle
rest
frame
particle
rest
frame
For pL << pT , the CS and HX frames differ by a rotation of 90º
z
z′
helicity
CollinsSoper
90º
x
x′
y
y′
longitudinal
“transverse”
(pure state)
(mixedstate)
What do different detectors measure with arbitrary frame choices?
(HX in this case)
+1/3
λθ = +0.65
λθ = 0.10
1/3
artificial (experimentdependent!)
kinematic behaviour
measure in more than one frame!
V = *, Z, W
_
Due to helicity conservation at the qqV(qq*V)vertex,
Jz = ± 1 along the qq (qq*) scattering direction z
_
z
_
_
q
z = relative dir. of incoming q and qbar
( CollinsSoper frame)
q
q
V
important only up to pT = O(partonkT)
V
q
z = dir. of one incoming quark
( GottfriedJackson frame)
q
V
q*
q*
g
QCD
corrections
q
V
z = dir. of outgoing q
(= partoncmshelicity labcmshelicity)
q*
g
q
Different subprocesses have different “natural” quantization axes
q
For schannel processes the natural axis is
the direction of the outgoing quark
(= direction of dilepton momentum)
V
q*
g
q
example: Z
y = +0.5
HX
CS
PX
GJ1
GJ2
(negative beam)
(positive beam)
1/3
Different subprocesses have different “natural” quantization axes
V
q
q
V
For t and uchannel processes the natural axis is
the direction of either one or the other incoming parton ( “GottfriedJackson” axes)
q*
q*
g
_
q
example: Z
y = +0.5
HX
CS
PX
GJ1 = GJ2
MZ
1/3
The shape of the distribution is (obviously) frameinvariant (= invariant by rotation)
→ it can be characterized by frameindependent parameters:
z
λθ = –1
λφ = 0
λθ = +1
λφ = 0
λθ = –1/3
λφ = –1/3
λθ = +1/5
λφ = +1/5
λθ = +1
λφ = –1
λθ = –1/3
λφ = +1/3
rotations in the production plane
Gedankenscenario: vector state produced in this subprocess admixture:
60%processes with natural transverse polarization in the CS frame
40%processes with natural transverse polarization in the HX frame
assumed indep.
of kinematics,
for simplicity
CS
HX
M = 10 GeV/c2
polar
azimuthal
rotation
invariant
V = *, Z, W
_
Due to helicity conservation at the qqV(qq*V)vertex,
Jz = ± 1 along the qq (qq*) scattering direction z
_
z
q
_
_
“natural” z = relative dir. of q and qbar
λθ(“CS”)= +1
q
q
V
~
~
~
λ = +1
wrtanyaxis:λ = +1
λ = +1
~
V
q
z = dir. of one incoming quark
λθ(“GJ”)= +1
q
V
λ = +1
q*
q*
any frame
g
(LO) QCD
corrections
z = dir. of outgoing q
λθ(“HX”)= +1
q
V
q*
~
N.B.: λ = +1 in both
ppHX and qgHX frames!
g
q
In all these cases the qqV lines are in the production plane (planar processes);
The CS, GJ, ppHX and qgHX axes only differ by a rotation in the production plane
Example: Z/*/W polarization (CS frame) as a function of contribution of LO QCD corrections:
Case 2: dominating qg QCD corrections
Case 1: dominating qqbar QCD corrections
pT = 50 GeV/c
y = 2
pT = 50 GeV/c
pT = 50 GeV/c
pT = 50 GeV/c
(indep. of y)
pT = 200 GeV/c
y = 2
pT = 200 GeV/c
1
pT = 200 GeV/c
pT = 200 GeV/c
(indep. of y)
y = 0
y = 0
y = 2
y = 2
0
.
5
y = 0
y = 0
~
~
~
~
~
M = 150 GeV/c2
M = 80 GeV/c2
M = 150 GeV/c2
M = 80 GeV/c2
λ = +1
λ = +1
λ = +1
λ = +1
λ = +1 !
mass dependent!
0
fQCD
fQCD
fQCD
fQCD
0
1
0
2
0
3
0
4
0
5
0
6
0
7
0
8
0
λθCS
W by CDF&D0
λθ
“unpolarized”?
No,
~
λ is constant, maximal and
independent of process admixture
pT [GeV/c]
Example: Z/*/W polarization (CS frame) as a function of contribution of LO QCD corrections:
Case 2: dominating qg QCD corrections
Case 1: dominating qqbar QCD corrections
pT = 50 GeV/c
y = 2
pT = 50 GeV/c
pT = 50 GeV/c
pT = 50 GeV/c
(indep. of y)
pT = 200 GeV/c
y = 2
pT = 200 GeV/c
pT = 200 GeV/c
pT = 200 GeV/c
(indep. of y)
y = 0
y = 0
y = 2
y = 2
y = 0
y = 0
~
~
~
~
M = 150 GeV/c2
M = 80 GeV/c2
M = 150 GeV/c2
M = 80 GeV/c2
λ = +1
λ = +1
λ = +1
λ = +1
mass dependent!
fQCD
fQCD
fQCD
fQCD
~
On the other hand, λ forgets about the direction of the quantization axis.
This information is crucial if we want to disentangle the qg contribution,
the only one resulting in a rapiditydependentλθ
Measuring λθ(CS) as a function of rapidity gives information on the gluon content
of the proton
A fundamental result of the theory of vectorboson polarizations (DrellYan, directly produced Z and W) is that, at leading order in perturbative QCD,
independently of the polarization frame
LamTung relation, PRD 18, 2447 (1978)
This identity was considered as a surprising result of cancellations in the calculations
Today we know that it is only a special case of general frameindependent polarization relations, corresponding to a transverse intrinsic polarization:
It is, therefore, simply a consequence of
1) rotational invariance
2) properties of the quarkphoton/Z/W coupling
Experimental tests of the LT relation are not tests of QCD!
Even when the LamTung relation is violated,
can always be defined and is always frameindependent
→ LamTung. New interpretation: only vector boson – quark – quarkcouplings (in planar processes) automatically verified in DY at QED & LO QCD levels and in several higherorder QCD contributions
→ vectorboson – quark – quarkcouplings in
nonplanar processes (higherorder contributions)
→ contribution of different/new couplings or processes
(e.g.: Z from Higgs, W from top, triple ZZ coupling, highertwist effects in DY production, etc…)
z'
±1
g
θ
T restframe
T
z
g
±1
Usual approach to “determine” the J of T:
comparison between J=0 hypothesis and ONE alternative hypothesis. Example:
graviton with minimalcouplings to SM bosons ( “boson helicity conservation”)
SM Higgs boson
J=2
J=0
+
+
OR
OR
OR
OR
Mgg = 0
Mgg = 0
Mgg = 0, ±1, ±2
Mgg = 0, ±1, ±2
Decay distribution calculated casebycase
z'
±1
g
θ
T restframe
T
z
g
±1
Usual approach to “determine” the J of T:
comparison between J=0 hypothesis and ONE alternative hypothesis. Example:
graviton with minimalcouplings to SM bosons ( “boson helicity conservation”)
SM Higgs boson
J=2
J=0
Mgg = ±2
Mgg = ±2
L decay angular distribution
L [B] / L [A]
MPC = Minimal Physical Constraints
dN
1 + λ2 cos2θ + λ4 cos4θ + λ6 cos6θ + … + λNcosNθ
dΩ
[J=1 hypothesis forbidden by LandauYang theorem]
The general physical parameter domains of the J=2, 3 and 4 cases are mutually exclusive!
And do not include the origin (J=0)!
MPC = Minimal Physical Constraints
J = 3
J = 4
J = 0
J = 2
dN
1 + λ2 cos2θ + λ4 cos4θ + λ6 cos6θ + … + λNcosNθ
dΩ
The cosθ distribution discriminates the spin univocally:
MPC = Minimal Physical Constraints
dN
1 + λ2 cos2θ + λ4 cos4θ + λ6 cos6θ + … + λNcosNθ
dΩ
The binary strategy of the LR approach aims at discriminating between two hypotheses:
From this point of view,
this measurementwould correspond to a J=0 characterization
99% C.L.
J = 2 minimallycoupling graviton
J=2
In the MPC approach
this measurementwould represent an unequivocal spin0 characterization
In the MPC approach it would exclude all models lying outside the ellipse, but it would not exclude J=2, nor J=3!
99% C.L.
J=3
J = 0
(SM Higgs)