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Theoretical physics for experimentalists:
Branching ratio and helicity amplitudes
for Lb  L(pK) g decays (L spin = 3/2)
Combined work of:
Gudrun Hiller (Dortmund UNI), the Bearer of the Light
Thomas Schietinger (PSI), the Scholar
Mathias Knecht and Federica Legger (EPFL), the water Carriers
Outline

Once upon a time:




the electromagnetic penguin bsg
the photon polarization (theory and
experiment)
my thesis results & open questions
The mighty quest for L spin = 3/2:
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Branching ratio for Lb  L(pK) g
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
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the tools: Mathematica
Helicity amplitudes
Sensitivity to photon polarization
Summary and outlook
2
Motivations

Standard Model (SM): best description of known
elementary particles and their interactions:




e m t
ne nm nt
passed all experimental tests up to now;
still one missing particle, the Higgs boson.
However...

leptons
19 (!!!) free parameters;
gravity is not included.
quarks
Quest for new physics in the quark sector:

CKM picture is very successful

but we still know little about b  s, d transitions !
u
d
c
s
t
b
3
The electromagnetic penguin bsg
u,c,t
g
b
W

s
New physics in the decay rate :


are there any contribution from supersymmetric particles?
the measured bsg branching fraction is compatible with SM prediction
 Theory:
BF(bsg) [10-6]= 357 ± 30
Gambino, Misiak, NPB 611 (2001) 338
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Experiment: BF(bsg) [10-6]= 355 ± 24 +9-10 ± 3
http://www.slac.stanford.edu/xorg/hfag/rare


from HFAG (combined measurements by Belle, BaBar, CLEO)
Need other observables to test the SM...
4
The electromagnetic penguin bsg
u,c,t
g
b


W
s
The W boson only couples
to a left-handed s quark
Left-handed photon (to
conserve ang. momentum)
Photon polarization:


pure 2-body decay: right-handed components of the
order of r = ms/mb
when considering bsg + gluons

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right-handed components may be up to 10-15%
explicit calculations only for BK*g, Brg
“Naïve” SM
Atwood, Gronau, Soni,
PRL 79, 185 (1997)
SM + QCD
Grinstein, Grossman,
Ligeti, Pirjol, PRD 71,
011504 (2005)
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Photon polarization measurements
LHCb
B factories
Exp. status
Theor. Refs.
e+e- conversion
First measurements of K*
polarization in B->K*l+lby Belle/Babar
Grossman, Pirjol, JHEP06, 029 (2000)
Melikov, Nikitin, Simula, PLB 442,
381 (1998)
B-B interference
Latest world average
sin2b = 0.0 ± 0.3
Atwood, Gronau, Soni, PRL 79,
185 (1997)
Higher K*
resonances
Difficult to disentangle
resonance structure
(Babar, hep/0507031)
Gronau, Pirjol, PRD 66, 054008 (2002)
Gronau, Grossman, Pirjol, PRL 88,
051802 (2002)
Charmonium
res. interference
No results so far...
b-baryons
Exploit ang. correlations between
polarized initial state and final
state. Under study at LHCb
(F.Legger, M. Knecht)
Knecht, Schietinger, PLB 634,
403 (2006)
Mannel, Recksiegel, JPG:
NPP 24, 979 (1998)
Hiller, Kagan, PRD 65,
074038 (2002)
Legger, Schietinger,
PLB 644 (2007) xxx
6
Polarized b baryons decays

If initial state is polarized:
 exploit angular correlations between initial and final states
 only possible with b baryons
 feasible at hadron colliders
Case study: Lb  (L(1115)  pp) g
g
Lb
b
s
d
d
u
u
L
Long distance contributions
from internal W exchange,
or vector meson cc
contributions are expected to
be small
Mannel, Recksiegel, JPG: NPP 24, 979 (1998)
Hiller, Kagan, PRD 65, 074038 (2002)
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Polarized Lb  L(1115) g decays

Angular distributions:
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PB = Lb polarization
ap = weak decay
parameter
depend on photon polarization ag
Lb  L(1115) g
Lb  L(1115) g
Evtgen
ag = 1
PB = 1
ag (fit) = 1.036
ag (theory) = 1
ap (fit) = 0.679
ap (theory) = 0.642
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However...


From the experimental point of view the decay Lb 
L(1115) g is quite hard to observe (ct = 7.89 cm)
Can we probe the photon polarization in heavier L
resonance decays?
g
 Lb  (L(X)  pK) g
b
s
Lb
d
u

u
u
d
u
K
p
what do we need?


Branching ratios for Lb  L(X) g
Angular distributions for L spin = 1/2, 3/2

spin > 3/2: helicity states > observables
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L(X) resonance spectrum
1520
L spin = 3/2
L spin = 1/2
1670 1690
PDG 2004
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Invariant pK mass spectrum obtained with:

BR(Lb  L(X) g ), calculated rescaling BR(Lb  L(1115) g ) with a
kinematical factor, assuming the same form factors and no spin
dependence for all L(X) resonances.
Legger, Schietinger, PLB 644 (2007) xxx
10
Helicity formalism for Lb  L(pK) g
JL = 1/2
JL = 3/2

Photon helicity = ±1,
L helicity = ±1/2
2 helicity amplitudes
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Photon angular distribution
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Proton angular distribution flat because of P conservation
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Photon helicity = ±1,
L helicity = ±1/2, ±3/2
4 helicity amplitudes
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Photon angular distribution
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Legger, Schietinger, PLB 644 (2007) xxx
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Lb  L(pK) g decays
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(JL = 3/2)
ag,3/2 depends on the asymmetry of Lb spin with respect
to photon momentum
and can be factorized into the photon helicity parameter
ag and the strong parameter 
 can be extracted from the proton angular distribution
Legger, Schietinger,
PLB 644 (2007) xxx
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Open questions

The photon helicity can be probed in decays
involving L resonances of spin 3/2 by measuring
ag,3/2 and 
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Can we get a better estimate of the BR ?
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Include at least the spin dependence
Form factors will have to be measured
Can we get an estimation of ?
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Decay amplitude for Lb  L(1520) g
The effective hamiltonian:
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Electromagnetic dipole
operators:
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long distance effects
non perturbative approach
(HQET)
Wilson coefficients: C7, C7’
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short distance
Fermi theory (point-like
interactions)
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Decay amplitude for Lb  L(1520) g
The effective hamiltonian:
The matrix element:
g (q, e)
L(1520)
(p´,s´)
Lb (p, s)
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Decay amplitude for Lb  L(1520) g
The effective hamiltonian:
u(p,s) = Dirac spinor to
describe the Lb (spin 1/2)
Rarita-Schwinger (RS)
spinor to describe the L
(spin 3/2) Rarita, Schwinger,
The matrix element:
Phys Rev 60(1941) 61
Dirac
spinor
Find Gan and G(5)an!!
Polarization
vector
1/2  1 = 3/2
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Conditions
Equations of motion
(EOM)
On-shell photon
RS spinors
Gauge invariance
Main actors:
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Gan and G(5)an
We define the tensor Gamn(antisymmetric in m and n):
Ansatz:
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Gan and G(5)an
We define the tensor Gamn(antisymmetric in m and n):
Ansatz:
On-shell photon!
Reabsorbed in B
and C using EOM
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Gan and G(5)an
We define the tensor Gamn(antisymmetric in m and n):
Ansatz:
On-shell photon!
Reabsorbed in B
and C using EOM
Contracting with qm
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Gan and G(5)an
Form factors
G(5)an is related to G(5)an through the identity:
it is straightforward to obtain (ask Mathias) :
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Spin averaged matrix element
To evaluate the BR we need:
Writing explicitely
the spinor indices!
where
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Spin averaged matrix element
Sum over spins:
Aliev, Ozpineci, hep-ph/0406331
We finally obtain:
To calculate the trace we use:
with the TRACER package
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Trace evaluation
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Trace evaluation
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Branching Ratio
In the limit
BR (Lb  L0g ) ~ 7·10-5
f2
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HFAG ICHEP 2006
K*(892) = vector
K1(1270) = axial vector
K1(1400) = axial vector
K2*(1430) = tensor
From B+ and B0 radiative decays, and
dedicated form factors studies,
BR should have the same order of
magnitude
S. Veseli, M.G. Olsson, Z. Phys. C 71 (1996) 287
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Helicity amplitudes
We use the Lb rest frame:
g
q=(E,0,0,-E)
Lb
L(1520)
z
p´=(E´,0,0,E)
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Helicity amplitudes
The amplitudes A3/2 (A1/2 ) result from a Lb-baryon with h = 1/2
(h = +1/2) and a photon with Jz = +1
RS spinor
Dirac spinor
Photon polarization vectors:
Jz
in Lb rest frame:
L polarization vectors:
L helicity
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Helicity amplitudes: results
Right-handed
photon
In the limit
and f1~f2
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Helicity amplitudes: naïve picture
b
Quark level:
g
Left-handed photon = SM
Spin flip b vs s
s
b
Lb
Spin flip Lb vs L
s L(1520)
g
b
Opposed b and Lb spin
-> suppressed ~ O(1/mb)
Lb
g
s
L(1520)
M. Suzuki, J. Phys. G: Nucl. Part. Phys. 31 (2005) 755
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Sensitivity to the photon polarization
Lb Polarization = 20%
10k L(1520) events
(~3 yrs LHCb running)
3s significance
Photon polarization:
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Conclusions and outlook
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The BR(Lb  L(1520) g) has been calculated in
the framework of HQET
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form factors will need to be measured
Helicity amplitudes for the decay Lb  L(1520) g
have been evaluated
 straightforward extension to decay involving
JP = 3/2+ resonances, by replacing C ’7-> -C ’7

Still to do: work out a better estimate of the Lb
polarization
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(Some) theoretical models and calculations are
(also) accessible to experimentalists!
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Backup slides
Lb production at LHC:
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bb cross section in pp collision = 500 mb
10% of produced bb hadronize in baryons
Lb dominates (90%)
Lb produced with transversal polarization
n
p1

Lb
p2
Expectations are PB ~ 20%
Ajaltouni, Conte, Leitner, PLB, 614 (2005) 165

ATLAS plans to measure it with a statistical precision
better than 1%
Feasibility of Beauty Baryon Polarization Measurement in Lb 
J/Y L decay channel by ATLAS – Atlas note 94-036 PHYS
35
Photon polarization
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Lb  L(1670) g selected evts. (transversally
polarized Lb)
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efficiency corrected (from unpolarized decays)
from data, the correction can be obtained from
B  K*g decays
ag
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Sensitivity on |r| measurement
1 year, 3s
SM
naive
SM + QCD


5 years, 3s
SM
naive
SM + QCD
Values of |r| that can be probed from
single measurements
Getting close to the SM expected range,
becomes interesting if NP!
Lb Polarization = 20%
37
Combined measurements
1 year, 3s
SM + QCD
SM
naive


5 years, 3s
SM +
QCD
SM
naive
Combining measurement increases
range by a few percent at most
L(X) measurements have good
sensitivity (in case L(1115) turns out
to be difficult)
Lb Polarization = 20%
38
Dependence on Lb polarization
Lb  L(X) g

Lb  L(1115) g
1 year
If only the photon asymmetry is measured, a polarization of
at least 20% is needed to have good sensitivity
39