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

qq

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