Hyperon Transition FormFactors
Carlos Granadosin collaboration with
Stefan Leupold and Elisabetta Perotti
The George Washington UniversityUppsala University
HYP2018Portsmouth, VAJune 28, 2018
Motivation
Studying Hyperons
I Complement and extend current understanding of the structure ofnucleons, and of N-∆ at various energy scales
I low energy:check convergence of 3-flavor χPT,I intermediate: phenomenology on nucleon structure where light
quarks are replaced by strange quarks,I high: scaling laws dependence on quark mass.
I Motivate experimental work addressing intrinsic properties ofhyperons
I Only magnetic moments are known
I Provide theoretical input for hyperon detection in pp (PANDA) andpp (HADES).
Σ− Λ form factorsDalitz Decay
e
e’
p
d2Γ
dsdz=
1
(2π)364m3Σ
λ12 (m2
Σ, s,m2Λ)
√1− 4m2
e
s|M3|2
I Σ/Σ∗ produced from e.g., ppcollisions alternative tounfeasible fix targetexperiments (Hyperon electronscattering).
I Decay rate prediction fromΣ/Σ∗ → Λe+e− amplitude,M3.
Σ− Λ form factorsDalitz Decay
e
e’
p
|M3|2 =e4
s2((mΣ −mΛ)2 − s)|GE(s)|2(mΣ + mΛ)2
(1−
(1− 4m2
e
s
)z2
)+|GM(s)|2(s(1 + z2) + 4m2
e (1− z2))
I Σ/Σ∗ production from ppcollisions at PANDA
I Decay rate prediction fromΣ/Σ∗ → Λe+e− amplitude,M3.
I Access transition form factorsat very low virtuality,√s ∼ (mΣ −mΛ) ≈ 77MeV
I Helicity structure from angulardistribution of Λ decay
Theory approaches on hyperon FF
I Full χPT calculation. Noexplicit decuplet, no vectormeson
I Heavy Baryon χPT
Kubis,Meissner (2001)
Kubis,Hermmert,Meissner (1999)
I Dispersion Theory + ChPT(including Decuplet states)
Granados, Leupold, Perotti (2017)
Alarcn, Hiller Blin, Vicente Vacas, Weiss
(2017)
(1) (2) (3)
(6*)
+
(5*)
+
(7*)
(9)
(10)
(5) (6)
+
(7) (8)
(4)
(11) (12)
Σ
Λ
π
π
Σ− Λ form factors
jµ =
((γµ +
mΛ −mΣ
q2qµ)
F1(q2)− iσµνqνmΛ + mΣ
F2(q2)
)
GE ≡ F1 +q2
(mΛ + mΣ)2F2
GM ≡ F1 + F2
I Compute form factors from〈0 |jµ|ΣΛ〉 throughdispersion relations. Useanalyticity to expand to thetransition region.
Unitarity and dispersion relations
I FromS†S = 1
and
S = 1 + iT ,
2ImTfi =∑
X
T †fXTXi
B AT B XT† X ATIm = ∑X
I Dispersion relations,
T (s) = Pn−1(s) + sn
∫ ∞−∞
ds ′
π
ImT (s ′)
s ′n(s ′ − s + iε)
Dispersion Relations
I From 2-pion inelasticity,
Im
Σ
Λ
Σ
Λ
π
π π
π
GE/M (q2) = GE/M (0) +q2
12π
∫ ∞4m2
π
ds
π
TE/M(s)p3c.m.(s)FV∗
π (s)s3/2(s − q2)
I T and FV , 2-pion amplitudes projected in J = 1
π Form Factor and ππ scattering
π−
π+
π−
π+
π−
π+
Sebastian P. Schneider, Bastian Kubis, Franz
Niecknig, Phys.Rev.D86:054013,2012
0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8
10-2
10-1
100
101
102
√s [GeV]
|FV π(s)|2
Belle data [25]Ref. [23]Ref. [24]Fit
FVπ (s) ≈ Ω(s)
= exp
s
∫ ∞4m2
π
ds ′
π
δ(s ′)
s ′(s ′ − s)
Pion-Baryon ScatteringDispersion Relations
I From 2-pion inelasticity of a scattering amplitude with right hand cut,
Im
ImT = (K + T ) e−iδ sin δ
Pion-Baryon ScatteringDispersion Relations
Im
T (s) = K (s) + Ω(s)
Pn−1(s) + sn
∫ ∞4m2
π
ds ′
π
sin δ(s ′)K (s ′)
|Ω(s ′)| (s ′ − s)s ′n
I Left hand cut K (s) can be computed from 3-flavormeson-baryonχPT .
I Ω(s) and δ(s ′) are extracted from ππ-scattering data.
Pion-Baryon scattering and Chiral PT
π
π
Λ
Σ(∗)
≈π
π
Λ
Σ(∗)
Σ/Σ∗ +π
π
Λ
Σ(∗)
L(1)8 = i〈BγµDµB〉+
D
2〈B γµ γ5 uµ,B〉
+F
2〈B γµ γ5 [uµ,B]〉
L(1)10 =
hA
2√
2εade gµν (Tµ
abc uνbd Bce + Bec u
νdb T
µabc )
I Use L(1)8+10 to compute
left hand cut amplitudesfrom polar componentsof scattering amplitude
I Calculate octet(Born)and decuplet(Σ∗-resonance)exchange diagrams.Pole componentunaffected by spuriousspin 1/2 components.
Coupling constants, D = 0.80, F = 0.46,
hA = 2.3± 0.1 and b10 = (1.1± 0.25)GeV−1
Contact terms and NLO ChPT
π
π
Λ
Σ(∗)
PE0 = PE
Born + PEres ,
PM0 = PM
Born + PMNLOχPT − KM
res,low ,
I Prescription dependent contactterms.To be absorbed by subtractionterms in dispersion relationsPn−1(s)
I Match to contact terms fromNLO Lagrangian for the octetbaryon sector
Contact terms and NLO ChPT
L(2)8 = bD〈Bχ+,B〉 + bF 〈B[χ+,B]〉 + b0〈BB〉〈χ+〉
+ b1〈B[uµ, [uµ,B]]〉 + b2〈Buµ
, uµ,B〉
+ b3〈Buµ, [uµ,B]〉 + b4〈BB〉〈uµuµ〉
+ ib5
(〈B[uµ
, [uν, γµDνB]]〉
− 〈B←−D ν [uν
, [uµ, γµB]]〉
)+ ib6
(〈B[uµ
, uν, γµDνB]〉
− 〈B←−D νuν
, [uµ, γµB]〉
)+ ib7
(〈Buµ
, uν, γµDνB〉
− 〈B←−D νuν
, uµ, γµB〉
)+ ib8
(〈BγµDνB〉 − 〈B
←−D νγµB〉
)〈uµuν〉
+i
2b9 〈Buµ〉〈uν
σµνB〉
+i
2b10 〈B[uµ
, uν ], σµνB〉
+i
2b11 〈B[[uµ
, uν ], σµνB]〉
+ d4〈Bf µν+ , σµνB〉 + d5〈B[f µν
+ , σµνB]〉 .
I Prescription dependent contactterms.To be absorbed by subtractionterms in dispersion relationsPn−1(s)
I Match to contact terms fromNLO Lagrangian for the octetbaryon sector
I Study dependence on lowenergy constant b10
PM0 = PM
Born + PMNLOχPT − KM
res,low ,
Born and Intermediate Σ∗ Exchange AmplitudesI LH cut amplitudes for octet exchange,
KEBorn =
3
2
DF√3F 2
π
mΣ
xBB2
(((mΣ + mΛ)2 − s
)(mΣ −mΛ) + 2A(mΣ + mΛ)
)×(arctan xB − xB )
KMBorn =
3
2
DF√3F 2
π
mΣ
xBB2A(mΣ + mΛ)((x2
B + 1) arctan xB − xB ),
with A = (−m2Σ + m2
Λ + 2m2π − s)/2, B = −2ipc.mpz , and
xB = B/A.I Similar structures for Σ∗ exchange, KE
Res ∼ (arctan xR − xR ), andKM
Res ∼ ((x2R + 1) arctan xR − xR )
I Contact terms,
PMBorn = PE
Born = −2DF√3F 2
π
PMNLOχPT =
4b10√3F 2
π
(mΣ + mΛ)
PEres ≈ h2
A
24√
3F 2π m
2Σ∗
(m2Σ∗ + mΣ∗ (mΣ + mΛ) + mΣ mΛ)
KMres,low =
h2A
24√
3F 2π
(−m2Σ∗ + 4mΣ∗mΣ −m2
Σ) (mΣ∗ + mΣ)
m2Σ∗ (mΣ∗ −mΣ)
.
Born and Intermediate Σ∗ Exchange Amplitudes
-1500
-1000
-500
0
500
1000
1500
2000
2500
3000
3500
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Re
TM
[GeV
-2]
√s [GeV]
full Bornfull NLO
full NLO+resbare Bornbare NLO
bare NLO+res
-100
-50
0
50
100
150
200
250
300
350
400
450
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Re
TE [G
eV-2
]
√s [GeV]
full Bornfull NLO+res
bare Bornbare NLO+res
I Helicity amplitudes for ΣΛ in the sub-threshold region.
I ρ- meson visible in full amplitudes
I Decuplet exchange appreciable. Near cancellation of electric(spinnon-flip) amplitude.
Born and Intermediate Σ∗ Exchange Amplitudes
-500
0
500
1000
1500
2000
2500
3000
3500
4000
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Im T
M [G
eV-2
]
√s [GeV]
full Bornfull NLO
full NLO+res
-300
-250
-200
-150
-100
-50
0
50
100
0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
Im T
E [G
eV-2
]
√s [GeV]
full Bornfull NLO+res
I Helicity amplitudes for ΣΛ→ ππ in the sub-threshold region.
I ρ- meson visible in full amplitudes
I Decuplet exchange appreciable. Near cancellation of electric(spinnon-flip) amplitude.
Form factors at photon point
Use unsubtracted dispersion rel. to find Electric charge, magneticmoment and electric(magnetic) radius of Σ− Λ transition,
κ?=
1
12π
∞∫4m2
π
ds
π
TM (s) p3c.m.(s)FV∗
π (s)
s3/2,
0?=
1
12π
∞∫4m2
π
ds
π
TE (s) p3c.m.(s)FV∗
π (s)
s3/2.
Form factors at photon point
Λ [GeV] quantity Born NLO NLO+res χPT
1 GM (0) −0.438 5.55 2.58 1.98 (exp.)2 −0.65 5.98 2.66
1 〈r2M〉 [GeV−2] 0.453 33.7 17.9 18.6
2 0.613 35.2 18.8
1 GE (0) −0.432 - 0.0026 02 −0.562 - −0.031
1 〈r2E 〉 [GeV−2] −3.13 - 0.866 0.773
2 −2.91 - 1.044
Table: Comparison to χPT (Kubis,Meissner 2001) using hA = 2.3,b10 = 1.1 GeV−1.
Paremeters
b10 quantity NLO NLO+res χPT
0.85 GM (0) 4.47 1.15 1.98 (exp.)1.35 7.49 4.17
0.85 〈r2M〉 [GeV−2] 27.4 10.9 18.6
1.35 43.1 26.7
Table: Comparison to χPT using Λ = 2 GeV, hA = 2.3 and varying the valuefor b10 (in units of GeV−1).
quantity hA = 2.2 hA = 2.4 χPT
GM (0) 2.94 2.36 1.98 (exp.)〈r2
M〉 [GeV−2] 20.2 17.3 18.6GE (0) −0.076 0.016 0
〈r2E 〉 [GeV−2] 0.708 1.40 0.773
Table: Comparison using Λ = 2 GeV, b10 = 1.1 GeV−1 and varying the value forhA.
Electric and Magnetic Form Factors
-0.04
-0.035
-0.03
-0.025
-0.02
-0.015
-0.01
-0.005
0
0.005
0.01
-1 -0.8 -0.6 -0.4 -0.2 0
GE
q2 [GeV2]
small hA, cutoffradius adjust.
large hA, cutoff
-1
-0.5
0
0.5
1
1.5
2
2.5
-1 -0.8 -0.6 -0.4 -0.2 0
GM
q2 [GeV2]
large hA, small b10, cutoffsmall hA, b10, large cutoff
large hA, av. b10, small cutoffsmall hA, av. b10, large cutoff
large hA, b10, small cutoffsmall hA, large b10, cutoff
I GE is very small over a large range.
I GM can be measured at low energies
I Dalitz decay region, hardly visible
I Large uncertainty in GM driven by uncertainty in b10
Summary
I Compute EM Transition form factors of hyperons through a modelindependent approach combining dispersion relations and NLOχPT :
I Results on scattering amplitudes point to very significantcontributions from decuplet exchanges
I Results on hadronic corrections to Electric and Magnetic formfactors:
I Small Electric form factorI Uncertainties dominated by weakly constrained parameters of theχEFT Lagrangian.
Outlook
I Compare results with current similar approaches,e.g.,Alarcon,Hiller,Vacas,Weiss NPA(2017)
I Include Kaon inelasticities.
I Compute amplitudes for decuplet baryons in final or initial state ofdecay, e.g., Σ∗ → Λe+e− decay (ongoing)Junker,Leupold,Perotti
I Parallel approach to left hand cut amplitudes from lattice QCD.Dispersion+χPT to Dispersion+Lattice
I Tackle next to leading order QED corrections