 Dalitz decay:
From theory…
…to HADES experimental spectra
B. Ramstein, IPN Orsay
In collaboration with J. Van de Wiele
GSI, HADES Collaboration Meeting , 05/07/08
 Dalitz decay in transport codes:
C+C 2 GeV
No medium effects
HSD
IQMD
 Dalitz
pn
Me+e-(GeV/c2)
Thomère,Phys ReV C 75,064902 (2007)
E. Bratkovskaya nucl-th 07120635
Important issue for understanding intermediate mass dilepton yield
 Dalitz decay :
intrinsic interest of the measurement
 Dalitz decay
p
+
*
e+
Branching ratio not measured
experimental challenge
q2 = M2inv(e+e-) = M*2 > 0
e-
Time Like  - N transition:
Complementary probes of electromagnetic
structure of N -  transition
Space Like N -  transition :
ee-
*
p
q2 = M* = - Q2 < 0
+ →Np
Pion electro/photo-production
extraction of electromagnetic
form factors
GE(q2), GM(q2), GC(q2)
Lots of data, Mainz, Jlab
N- Dalitz decay dilepton yield:
ingredients of the calculation
Strong interaction model
1) N N N  :
• Mass dependent width Breit-Wigner, with possible cut-offs
• model for t dependence or angular distribution
QED
N
2 )   N e- e +
• exact field theory calculation
• 3 independent amplitudes:
e.g. Electric, Magnetic and Coulomb
N
p
N
+
*
e+
q2 =
0
M*
2>
 Dalitz decay
e--
QCD
3) electromagnetic form factors
GM(q2),GE(q2),GC(q2)
N- em transition : what do we know?
• at q2=0, mainly M1+ (magnetic) transition
N
« Photon point » : q2=0
GM(0)=3, GE(0)~0
• At finite q2, many recent data points from Mainz, Jlab:
multipole analysis of p° or p+ electroproduction
(%)
GM(q2)
REM 
Im( E1 )
Im( M 1 )
related to GE(q2 )
(%)
RSM 
Im( S1 )
Im( M 1 )
related to GC( q2 )
Many models: dynamical models (Sato,Lee), EFT (Pascalutsa and Vanderhaeghen),
Lattice QCD, two component model Q. Wan and F. Iachello
What about time-like region ?

N- transition em structure: what about time-like region?
Problems in Time-like region

No data

Electromagnetic form factors are complex
Space Like: q2<0
Analytic continuation :
real GSL(q2)
Models constrained by data
Time Like: q2>0
complex GTL(q2)
eg. GTL(q2) = GSL(-q2)
or GTL(q2) = GSL(-q2ei),…
But,…
  decay width doesn’t depend on phases of form factors
 q2 stays small in  Dalitz decay
at M =1232 MeV/c2 , q2 < 0.09 GeV/c2
2 options:
 take constant form factors
HSD, UrQMD, IQMD

use models for form factors GE(q2),GM(q2),GC(q2) :
VDM,eVDM, (RQMD) two component Iachello model
Sensitivity to Iachello form factor
two component model:
GM
(q2)
 Unified description of all baryonic transition form
factors
 Direct coupling to quarks + coupling mediated by 
0.6m2
M=1.1 GeV/c2
M=1.3 GeV/c2
__ pure QED
__ Iachello FF
Analytic formula
4 parameters fitted on
• elastic nucleon FF
(SL+TL)
• SL N- transition GM
M=1.5 GeV/c2
M=1.7 GeV/c2
PLUTO simulations: sensitivity to Iachello’s form
factor in pe+e- events from  Dalitz decay
E. Morinière, PHD thesis
pp @ 1.25 GeV
pe+e- events
Normalisation problem now solved →no sensitivity at E=1.25 GeV
N- Dalitz decay dilepton yield:
ingredients of the calculation
Strong interaction model
1) N N N  :
• Mass dependent width Breit-Wigner, with possible cut-offs
• model for t dependence or angular distribution
QED
N
2 )   N e- e +
• exact field theory calculation
• 3 independent amplitudes:
e.g. Electric, Magnetic and Coulomb
N
p
N
+
*
e+
q2 =
0
M*
2>
 Dalitz decay
e--
QCD
3) electromagnetic N- transition
form factors GM(q2),GE(q2),GC(q2)
 Dalitz decay in « reference » papers
HSD before 2007,
IQMD, UrQMD
Wolf, Nucl.Phys. A517 (1990) 615
GMWolf=2.7
GM=4.1
HSD after 2007
PLUTO
Ernst, Phys.Rev C 58, 447 (1998)
GMErnst=3
(HSD 2.7)
GM=4.5
(HSD=4.1)
(PLUTO= 3.2)
RQMD
Krivoruchenko Phys.Rev.D 65, 017502
e-VDM
GM(0)~3
Zetenyi and Wolf
Zetenyi and Wolf, nucl-th0202047
g1= 2
GM(0)~3
Jones and Scadron convention
 form factor conventions
(including or not isospin factor
differences
2
3
of the amplitude)
 choices of form factors
d( e  e  )
 analytic formula for
dq 2
See Krivoruchenko et al. Phys.Rev.D 65, 017502
« remarks on  radiative and Dalitz decays »
Comparing different  Dalitz decay
dilepton spectra:
• analytic formula for
d( e  e  )
dq 2
• and form factors values at q2=0 from 4 papers
 compare dilepton spectra for M=1232 MeV/c2
X4 (misprint)
 mass dependence
factor 1.5
factor 1.7
factor 2
Discrepancy increases with  mass
But also off-shell effects problem at high  mass
factor 2.2
Me+e-( GeV/c2)
Check: radiative decay width values
d(Δ  Ne  e- )


(Δ  N *)
2
2
dq
3π q
M=1232 MeV/c2
For M* =0
radiative decay width
Dalitz decay width
d(Δ  Ne  e- ) 2
(Δ  Ne e )  
dq
2
dq

(Δ  N )
-
Branching
Ratio
Expt
« Wolf »:
HSD before
2007, IQMD,
UrQMD
« Ernst »
HSD after
2007
« Krivoruchenko »
RQMD
Zetenyi
Radiative
decay (10-3)
5.6± 0.4
6.0
8.7
7.05 (HSD)
5.6
5.6
4.6
6.5
5.3 (HSD)
4.12 (const.GM)
4.25 (e-VDM)
4.12
Dalitz
decay (10-5)
?
Radiative decay width OK
PLUTO
4.4
Pretty
well!
Direct effect: different normalisation of  Dalitz decay
dilepton spectrum
Same « Ernst » formula
Pluto BR(+→pe+e-) = 4.4 10-5
HSD BR (+→pe+e-) =5.3 10-5
Field theory calculation:
From reference papers and Jacques Van de Wiele’s work
• Differential decay width:

e
• Amplitude
Mc

phase space
d 
1 1
2
 

M



Ne
e

2
s
s
s
s
dq d q d e 2M  4 mN ,m ,m  ,m 
5



e
Leptonic current


1
s 
L
s 
s 
J m , p N , mΔ , pΔ J m e , p e , m e , p e 2
q
H
isospin 
hadronic current
s
N
Same as for
→N
• Electromagnetic hadronic current: 2 sets of covariants can be used:
E,M,C : eg Krivoruchenko
« standard normal parity set »: eg Wolf
J H  G M (q 2 ) J M  G E (q 2 ) J E  GC (q 2 ) J C
J H  g1 (q 2 ) J1  g 2 (q 2 ) J 2  g3 (q 2 ) J3
• Spin ½ projector (Dirac spinors)
• spin 3/2 projector (Rarita-Schwinger spinors)
• Traces of products of  matrices
Calculation of
JH(..) JH ’*(..)* JL’(..) JL(…) *
 Dalitz decay width calculation:
results
 Jacques Van de Wiele’s calculation → same
analytical function as Krivoruchenko’s
2
2


2 1/ 2
2 3/ 2 
q
d(Δ  Ne  e- )  2  3 2  mΔ  mN  
2
2
2
2
2 



    q  mΔ  mN   q   GM  3 GE  2 GC 

 c

dq2
48π  2 isospin  2 3 2  mΔ mN
2 mΔ


q m Δ mN


 Can also be expressed in terms of g1,g2,g3:
 GM 
 g1 


 
2
G

M
(
q
)
 E
 g2
G 
 g3
 C
 
• Shyam and Mosel; Kaptari and Kämpfer:
g1=5.42, g2=6.61, g3=7 equivalent to GM=3.2 GE=0.04 GC~0.2
• Zetenyi and Wolf: g1=1.98, g2=0,g3=0
fitted to reproduce radiative decay width
→ same Dalitz decay width as Van de Wiele/Krivoruchenko
q2 dependence negligible
for  Dalitz decay
 Dalitz decay width calculation:
results and suggestions for new PLUTO inputs
 * angular distribution
p
+
d 3(   N *)
dq 2 d *
*
e+
q2 = M* 2
 Dalitz decay
 « helicity distribution »
e--
Ok with E. Bratkovskaya, Phys. Lett. B348 (1995) 283

isotropic
d 5 (   N *)
2
* 
~
1

cos

e
dq 2 d *d*e 
Krivoruchenko/Van de Wiele ( or « Zetenyi » ) expression for
d(Δ  Ne  e- )
dq2
 Electromagnetic N- transition form factors
 Branching ratio
d(Δ  Ne  e- )
 (Δ  Ne e )  
dq2
dq 2
M=1232 MeV/c2
N- Dalitz decay dilepton yield:
ingredients of the calculation
Strong interaction model
1) N N N  :
• Mass dependent width Breit-Wigner, with possible cut-offs
• model for t dependence or production angular distribution
QED
N
2 )   N e- e +
• exact field theory calculation
• 3 independent amplitudes:
e.g. Electric, Magnetic and Coulomb
N
p
N
+
*
e+
q2 =
0
M*
2>
 Dalitz decay
e--
QCD
3) electromagnetic form factors
GM(Q2),GE(q2),GC(Q2)
N N N  model:  polarisation effects
 Dalitz decay
N
N

q
N
 polarization 4x4 density matrix
ms= -3/2,-1/2,1/2,3/2
+
*
m
p
Anisotropy of * angular distribution


s m 's
Same as in p
photoproduction
Spin-isospin excitation
1p exchange
p +  exchange
Effective interaction,…
Long. polarization :  
S .q
(pure 1p exch.)
1/2 1/2 =  -1/2 -1/2 =1/2
others ij=0
Transv. polarization :
( exch.)
3/2 3/2 =  -3/2 -3/2 =1/2
others ij=0
 
S q
Jacques Van de Wiele’s result
pp ppe+e- interference effects
Interference between all graphs including either a Delta or a nucleon
p
p
+
p
,p
p
p
p
, N
+ …..
ee+
e-
, N
p
p
cf Kaptari and Kämpfer,….
In PLUTO: factorization of NN → N cross section and (→Ne+e-):
p
p1
No Bremstrahlung
two exit protons are distinguishable
p2
p
+
e+
q2=M2inv(e+e-)=M*
e-
Origin of high dilepton mass tails
p+p 1.25 GeV
HSD
PLUTO
C+C, 1.0 A GeV
no medium effects
10
-2
10
-3
10
-4
10
-5


10
-6
Brems. NN
Brems. pN
All
10
-7
10
-8

10
-2
HSD

0.0
0.2
0.4
0.6
0.8
C+C, 1.0 A GeV
in-medium effects: CB+DM
tail at high dilepton mass:
absent in PLUTO HSD:
?
10
Dalitz
HADES
absent in pp and pn ?
Dalitz
Dalitz
10
Dalitz
Different  mass
distributions ?
-3
2
1 AGeV
1/GeV /c ]
12C+12C

PLUTO
HSD:

p Dalitz
 Dalitz
 Dalitz
 Dalitz
HADES
2
1/Np dN/dM [1/GeV /c ]
HSD
-4

p




Delta mass [email protected]
distribution
in PLUTO:
GeV
 Dmitriev’s mass distribution parametrisation
+ fromform-factors
pp°
but with Moniz vertex
 M  
2
p
M2
M
M 
2
 M r2
3

 M 2 M 
2
2
 k   k  
  

2 
k
k



 r
Teis = 300 MeV/c
M
Teis M   r r
M
2
r
2
2
2
Mass distribution
W. Przygoda’s talk

 from e+e-p
M k
DmitrievM   r r  
M  kr 
3
 kr2   2 
 2 2 
 k  
M(e+e-) >140 MeV/c2
Dmitriev = 200 MeV/c
M(MeV/c2)
d/dM
high mass dilepton yield is sensitive to
high  mass
E. Morinière, PHD thesis
q2=0.02 (GeV/c)2
q2=0.2 (GeV/c)2
N- Dalitz decay dilepton
yield:
Quite well known,
can be improved with
ingredients of the calculation
our data
Strong interaction model
1) N N N  :
• Mass dependent width Breit-Wigner, with possible cut-offs
• model for t dependence or angular distribution
QED
N
2 )   N e- e +
• exact field theory calculation
• 3 independent amplitudes:
e.g. Electric, Magnetic and Coulomb
N
Exact
calculation,
p
But offshell
+ effects?
*
N
e+
q2 =
0
M*
2>
e-No sensitivity
at E=1.25GeV,
 Dalitz
decay
Important at E=2.2 GeV
or in p-p E=0.8 GeV/c
QCD
3) electromagnetic form factors
GM(q2),GE(q2),GC(q2)
Normalized yi
Tingting’s talk
+
simulation
total
pp →pnp
E=1.25 GeV
• Experiment
• Simulation total
• Simulation 
• Simulation 
Ania Kozuch’s talk
• Simulation N*
pp → pp p° E=1.25 GeV
6 events
Sexperiment
=1.39*10
Marcin
Wisniovski
6 events
Ssimulation
pp=1.37*10
→ pp p°
pp → pn p+
E=2.2 GeV
Very good
agreement
pp→ppe+e-, pn →ppe+eChallenging data ?
Tetyana
Witold
„pure ”
+ (p,e+,e-) invariant mass
dilepton angle, helicity angle,…
Conclusion
 A lot of different models to describe
HADES data
 Different results
, but we need to
understand the reasons
 Some investigations for  Dalitz decay
 A lot of other questions about other
processes
 Let’s start the discussions…
Results of simulations for  Dalitz
decay
Possibility to reduce p° background to 20%
1500 e+e-p events
In HADES acceptance
7 days of beam time
Better sensitivity to
discriminate pp bremstrahlung
M
corrected pp data,
comparison to transport model
calculation
IQMD
Δ→e+e-N seems to explain e+e- yield in p+p at 1.25 GeV
 Dalitz decay in transport codes:
p+p and pn at 1.25 GeV
Isospin effects
Transport code
or calculation
Form factors
(different conventions)
Effective form factors
at q2=0 using
convention of
Jones and Scadron
Reference papers
HSD
before 2007
GM=2.7
GE=GC=0
GM=3.3
GE=GC=0
WOLF,Nucl.Phys.A517(1990)615
HSD
after 2007
GM=2.7
GE=GC=0
GM=3.3
GE=GC=0
Ernst,Phys.Rev C58,447(1998)
RQMD
e-VMD
GM=3.
GE=GC=0
Krivoruchenko Phys.Rev.D 65, 017502
IQMD
G=2.72
GE=GC=0
GM=3.33
GE=GC=0
WOLF,Nucl.Phys.A517(1990)615
Zetenyi and
Wolf
g1=1.98,
g2=0,g3=0
GM=3.33
GE=GC=0
Zetenyi, nucl-th 0202047
Kaptari and
Kämpfer
g1= 5.4
g2=6.6
g3=7
GM=3.2
GE=0.04
GC=0.19
Kaptari,Nucl.Phys. A764 (2006)338
analytic continuation to Time-Like region:
3) Intrinsic form factor:
Space Like:
 
g Q2 
1
(1  a 2 Q 2 ) 2
Analytic continuation :
Q2  - q2 ei
Time Like:
 
g q2 
1
(1  a 2 ei q 2 ) 2
phase :
removes singularity at q2=1/a2 (~ 3.45 (GeV/c)2)
  =53° fitted to elastic nucleon form factors Time Like data
 same value taken for N -  transition

Space Like:
Time Like: