The Radiative Return and Form Factors at Large Q 2 J.H. K UHN, - - PowerPoint PPT Presentation

▶
the radiative return
SMART_READER_LITE
LIVE PREVIEW

The Radiative Return and Form Factors at Large Q 2 J.H. K UHN, - - PowerPoint PPT Presentation

Jun 10, 2023 337 likes •661 views

The Radiative Return and Form Factors at Large Q 2 J.H. K UHN, TTP, KARLSRUHE I Basic Idea II Monte Carlo Generators: Status & Perspectives III Nucleon Form Factors at B-Factories Pion and Kaon Form Factors at large Q 2 IV and

slide-1
SLIDE 1

The Radiative Return and Form Factors at Large Q2

J.H. K¨ UHN, TTP, KARLSRUHE

I Basic Idea II Monte Carlo Generators: Status & Perspectives III Nucleon Form Factors at B-Factories IV Pion and Kaon Form Factors at large Q2 and τ → νK−K0 V Conclusions

1

slide-2
SLIDE 2

I BASIC IDEA

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 2

photon radiated off the initial e+e− (ISR) reduces the effective energy of the collision dσ(e+e− → hadrons + γ) = H(Q2, θγ) dσ(e+e− → hadrons) ◮ measurement of R(s) over the full range

  • f energies, from threshold up to √s

◮ large luminosities of factories compensate α/π from photon radiation ◮ radiative corrections essential (NLO) ◮ advantage over energy scan (BES, CMD2, SND): systematics (e.g. normalization) only

  • nce

High precision measurement of the hadronic cross-section at DAΦNE, CLEO-C, B-factories

slide-3
SLIDE 3

Rough estimates for rates:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 3

π+ π− γ : Eγ > 100MeV √s [GeV ]

  • L [fb−1] #events, θmin = 7◦

1.02 1.35 16 ·106 10.6 100 3.5 ·106 multi-hadron-events (R ≡ 2) √s = 10.6 GeV Q2-interval [GeV ] #events, θmin = 7◦ [ 1.5 , 2.0 ] 9.9 ·105 [ 2.0 , 2.5 ] 7.9 ·105 [ 2.5 , 3.0 ] 6.6 ·105 [ 3.0 , 3.5 ] 5.8 ·105

slide-4
SLIDE 4

Lowest order

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 4

dσ dQ2

  • e+e− → γ + had(Q2)
  • = σ
  • e+e− → had(Q2)
  • × α

πs

  • s2+Q4

s(s−Q2)

  • log(s/m2

e) − 1

  • , no angular cut

s2+Q4 s(s−Q2) log

1+cos θmin

1−cos θmin

  • − s−Q2

s

cos θmin

  • ⇒ differential luminosity:

dL dQ2

  • Q2, s
  • = α

πs

  • · · ·
  • L(at s)

e.g. θmin = 30◦ ; √s = 10.58 GeV ; Q = 1 GeV ; ∆Q = 0.1 GeV

dL dQ2

  • Q2, s
  • ∆Q2 = 7.6 · 10−6 L(at s)

100 fb−1 at 10.58 GeV ⇒ 0.76 pb−1 per scan point at 1 GeV

slide-5
SLIDE 5

Basic Ingredients for Pion Formfactor

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 5

◮ ISR pion form factor ➪ to be tested ◮ FSR radiation from point- like pions (probably

  • verestimated)

◮ additional radiation: collinear (EVA MC)

  • r NLO calculation (PHOKHARA MC)
slide-6
SLIDE 6

II MONTE CARLO GENERATORS

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 6

P H OTONS FROM KARLSRUHE H ADRONICALLY R ADIATED

References etc. → http://cern.ch/german.rodrigo/phokhara

slide-7
SLIDE 7

A

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 7

PHOKHARA 2.0: π+π−, µ+µ−, 4π

  • ISR at NLO: virtual corrections

to one photon events and two photon emission at tree level

2 2

γ γ γ γ

+ +

  • FSR at LO: π+π−, µ+µ−
  • tagged or untagged photons
  • modular structure

❶ LL at a fixed order + subleading terms (1 % ) ❷ Full angular dependence ❸ Momentum conservation ❹ Tagged or untagged photon

slide-8
SLIDE 8

PHOKHARA 3.0

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 8

◮ specifically developed for π+π− (plus photons) ◮ allows for simultaneous emission of photons from initial and final state, including virtual corrections (interference neglected).

  • ⇒ dominated by “two step process”: e+e− → γ ρ (→ γ ππ)

⇒ importance of ππγ as input for aµ

slide-9
SLIDE 9

A

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 9

Large effect for Q2 < m2

ρ eliminated by suitable cuts

  • n π+π− configuration (suppress 2γ events )

a

  • ✁
✂☎✄ ✁

180

  • ✁
✂✝✆ ✁

180

  • ✞

s = 1.02 GeV

e

✟

e

✠ ✡ ☛ ✟ ☛ ✠ ☞

(

☞

) Q2(GeV2)

d

✌

(IFSNLO) dQ2

✍

d

✌

(IFSLO) dQ2

✎

1

1

✏

9

✏

8

✏

7

✏

6

✏

5

✏

4

✏

3

✏

2

✏

1

✏

14

✏

12

✏

1

✏

08

✏

06

✏

04

✏

02

✑ ✏

02

50

✒ ✓ ✔✖✕ ✗ ✓

130

✒

no Mtr cut Mtr cut

c p

✕ ✘ ✙

p

✕ ✚

:

✓

15

✒

e

✛

e

✜ ✢ ✣ ✛ ✣ ✜ ✤

(

✤

) Q2(GeV2)

d

✥

(IFSNLO) dQ2

✦

d

✥

(ISRNLO) dQ2

✧

1 1

★

9

★

8

★

7

★

6

★

5

★

4

★

3

★

05

★

04

★

03

★

02

★

01

✩ ★

01

✩ ★

02

✩ ★

03

✩ ★

04

⇒ Talk by D. Leone

  • r measure photon
slide-10
SLIDE 10

Experimental Perspectives

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 10

BABAR, BELLE

higher Q2 available ⇒ measurement of R(Q2) from threshold up to at least 5 GeV. Examples: ◮ ππ ◮ 4π± ◮ K K ππ ◮ K K K K

slide-11
SLIDE 11

PHOKHARA 4.0

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 11

  • µ+µ−γ with FSR at NLO
  • vacuum polarisation can be switched on
  • nucleon pair production included
slide-12
SLIDE 12

To be done:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 12

  • three mesons: 3π (→ ρπ), KKπ
  • KKππ, 4K
  • narrow resonances

parameters of J/ψ , ψ′ :

  • bservable: Γe

Γf Γtot ; f = µ+µ− , π+π−, 3π , 4π , 4K , ...

compare :

σf σµ+µ−(off resonance) ?

=

σf σµ+µ−(on resonance)

f = µ+µ− , π+π−, 4π , . . . virtual photon only (I=1) f = 3π, K ¯ K, K ¯ Kπ , . . . 3 gluon intermediate state (I=0)

slide-13
SLIDE 13

III NUCLEON FORM FACTORS

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 13

(with Czy˙ z, Nowak, Rodrigo, hep-ph/0403062)

Q2 4m2

N accessible at B-factories

⇒ study e+e− → γ N ¯ N (with N = p or n) hadronic current: Jµ = −ie · ¯ u(q2)

  • F N

1 (Q2) γµ − F N 2 (Q2)

4mN [γµ, / Q]

  • v(q1) ,

Q = q1 + q2 , q = (q1 − q2)/2

  • r

GM = F1 + F2 , GE = F1 + Q2 4m2 F2

slide-14
SLIDE 14

Result:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 14

dσ = 1 2s LµνHµν dΦ2(p1 + p2; Q, k) dΦ2(Q; q1, q2) dQ2 2π , LµνHµν = (4πα)3 Q2

  • |GN

M|2 − 1

τ |GN

E |2

  • ×

32s β2

N(s − Q2)

1 y1 + 1 y2 (p1 · q)2 + (p2 · q)2 s2

  • + 2
  • |GN

M|2 + 1

τ |GN

E |2

1 y1 + 1 y2 (s2 + Q4) s(s − Q2) − 2

  • ,

where y1,2 = s − Q2 2s (1 ∓ cos θγ) , τ = Q2 4m2

N

, β2

N = 1 − 4m2 N

Q2

slide-15
SLIDE 15

A

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 15

Separation of |GM|2 and |GE|2 through angular distribution: LµνHµν = (4πα)3 Q2 (1 + cos2 θγ) (1 − cos2 θγ) × 4

  • |GN

M|2 (1 + cos2 ˆ

θ) + 1 τ |GN

E |2 sin2 ˆ

θ

  • ˆ

θ = angle of nucleon with respect to γ-direction in hadronic rest frame

  • valid for s/Q2 ≪ 1, corrections and “optimal frame” → hep-ph/0403062

⇒ additional rotation by θD = 1 2 arctan

  • 2sγcγ

γ (β2 + c2

γ − s2 γ/γ2)

  • ≈ 1

γ sγcγ 1 + c2

γ

with sγ = sin θγ , β = (s − Q2)/(s + Q2), γ = (s + Q2)/2

  • sQ2
  • Similarity to e+e− → N ¯

N : dσ dΩ = α2βN 4Q2

  • |GN

M|2 (1 + cos2 θ) + 1

τ |GN

E |2 sin2 θ

slide-16
SLIDE 16

A

Implementation on basis of model for form factor:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 16

  • ✁

(GeV

✂

)

✄ ☎ ✆ ☎ ✝ ✞ ✆ ☎ ✟ ✠ ✡ ☛ ☞ ✌ ✍ ✎ ✍ ✎✑✏ ✒ ✎✑✏ ✠ ✎✑✏ ☛ ✎✑✏ ✌ ✎ ✓

(GeV

✔

)

✕ ✖ ✗ ✘ ✖ ✙ ✗ ✚ ✗ ✘ ✖ ✛ ✗ ✜ ✢✑✣ ✤ ✢ ✥ ✣ ✤ ✥ ✤ ✣ ✤ ✤ ✦✑✣ ✤ ✦ ✧ ✣ ✤ ✥ ✤ ✣ ✤ ✤ ✦ ✣ ✤ ✦ ✧ ✣ ✤ ✧ ★ ✣ ✤
slide-17
SLIDE 17

A

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 17

ADONE(73) DM1(79) DM2(83) DM2(90) FENICE(93) FENICE(94) FENICE(98) B :

  • ✁
✂✄ ☎ ✆ ✝ ✞

A :

  • ✁
☎ ✆ ✞

B A

✟ ✠ ✟ ✡ ☛ ☞ ✌ ☞ ✍

(GeV

✎

)

✏

(nb)

✑ ✒ ✝ ✓ ✞ ✔ ✂✖✕ ✑ ✂✖✕ ✝ ✂✖✕ ✞ ✂✖✕ ✗ ✂ ✘ ✕ ✑ ✘ ✕ ✝ ✘ ✕ ✞ ✘ ✕ ✗ ✘ ✙ ✚ ✛ ✜✢✤✣ ✥✦ ✧✩★ ✪ ✢ ✫ ✬ ✭✯✮✱✰ ✲ ✮ ✬ ✜✳ ✢ ✫ ✥ ✫ ✬ ✭✯✴ ✬ ✜ ✵ ✥ ✫ ✶ ✴ ✷ ✸ ✧✩★ ✪ ✦ ✥ ✫ ✬ ✭ ✴ ✬ ✜ ✥ ✥ ✫

At least one photon satisfies:

✹ ✢ ✫ ✬ ✭ ✮ ✰ ✲ ✮ ✬ ✜ ✥ ✢ ✫ ★ ✺ ★ ✻ ✼ ✽ ✾ ✽ ✿ ❀ ❁

(GeV

❁

)

❂ ❃ ❂ ❄ ❅ ❆ ❇ ❈ ❉ ❊ ❋
  • ❍
■ ✜✢ ❏ ✳ ✵ ❑ ✥ ✸ ✜✢ ✻ ▲ ✜✢ ✻ ▼ ✜✢ ✻ ◆ ✜✢ ✻ ❖ ✜✢ ✻ P ✜✢ ✻ ◗

e+e− → p¯ p e+e− → p¯ p γ implementation in PHOKHARA

slide-18
SLIDE 18

Angular distributions of nucleon

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 18

✂✁ ✄ ☎✝✆ ✂✞ ✟ ✠ ✄ ✡☛✌☞ ✍✎ ✑✏ ✒

4.5

✑✏ ✒ ✓ ✔ ✕ ✓ ✔

5

✑✏ ✒ ✓ ✖ ☛ ✗ ✔ ✘ ✆✚✙ ✛ ✆ ✔ ✡ ✍ ☛ ✗ ✎ ✍ ✗ ✔ ✘✢✜ ✔ ✡ ✍ ✍ ✗ ✏ ✣ ✏ ✤ ✥ ✦ ✧ ✦ ★ ✩✪✫ ✘✢✬ ✭ ✮ ✯ ✭ ✰ ✱ ✲ ✳ ✴ ✵ ✶

(nb)

✡ ☛ ☞ ✍ ☛ ✷ ☛ ☞ ✍ ✷ ✡ ✸ ☞ ✍✂✹ ✡☛ ✤ ✺ ✸ ✹ ✡☛ ✤ ✺ ✻ ☞ ✍✂✹ ✡☛ ✤ ✺ ✻ ✹ ✡☛ ✤ ✺ ✍ ☞ ✍✂✹ ✡☛ ✤ ✺ ✍ ✹ ✡☛ ✤ ✺ ✼ ☞ ✍✂✹ ✡☛ ✤ ✺ ✼ ✹ ✡☛ ✤ ✺ ✖ ☞ ✍✂✹ ✡☛ ✤ ✺ ✖ ✹ ✡☛ ✤ ✺ ✽✿✾ ❀ ❁✑❂ ✽✿❃ ❄ ❅ ❀ ❆❇❉❈ ❊❋ ✽✝● ❍

4.5

✽✝● ❍ ■❑❏ ▲ ■ ❏

5

✽✝● ❍ ■ ▼ ❇ ◆ ❏ ❖ ❂◗P ❘ ❂ ❏ ❆ ❊ ❇ ◆ ❋ ❊ ◆ ❏ ❖✢❙ ❏ ❆ ❊ ❊ ◆
  • ❚
  • ❯
❱ ❲ ❳ ❲ ❨ ❩❬❭ ❪✚❖ ❫ ❴ ❵ ❫ ❛ ❜ ❝ ❞ ❡ ❢ ❣

(nb)

❆ ❇ ❈ ❊ ❇ ❤ ❇ ❈ ❊ ❤ ❆ ✐✿❥ ❆❇ ❯ ❦ ❧ ❈ ❊ ❥ ❆❇ ❯ ❦ ❧ ❥ ❆❇ ❯ ❦ ♠ ❈ ❊ ❥ ❆❇ ❯ ❦ ♠✿❥ ❆❇ ❯ ❦ ❊ ❈ ❊ ❥ ❆❇ ❯ ❦ ❊✿❥ ❆❇ ❯ ❦ ♥ ❈ ❊ ❥ ❆❇ ❯ ❦ ♥ ❥ ❆❇ ❯ ❦ ▼ ❈ ❊ ❥ ❆❇ ❯ ❦

lab frame hadronic rest frame (two choices for GM/GE)

slide-19
SLIDE 19

Comments

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 19

  • similar results for neutron pair production
  • NLO corrections from ISR included (corrections ∼ 1–2%)
  • no FSR

thousands of events around 4–5 GeV2 several events up to 7–8 GeV2

slide-20
SLIDE 20

IV MESON FORM FACTORS at LARGE Q2

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 20

(with Bruch, Khodjamirian, hep-ph/0409080)

radiative return will explore large Q2 convenient representation for Fπ : generalized VDM with ρ, ρ′, . . . combined with Veneziano-type tower of resonances (Dominguez) Fπ(s) =

∞

  • n=0

cn m2

n

m2

n − s ,

cn = (−1)nΓ(β − 1/2) √π(1

2 + n)Γ(n + 1)Γ(β − 1 − n)

, m2

n = m2 ρ(1 + 2n) ,

β = free parameter

slide-21
SLIDE 21

Modifications:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 21

  • finite widths
  • parameters of ρ, ρ′, ρ′′ fitted to data
  • Breit-Wigner for ρ, ρ′, ρ′′ with Q2-dependent widths

⇒ reasonable agreement between model and fit

Parameter Input Fit(KS) Fit(GS) dual- PDG value QCDNc=∞ mρ

  • 773.9 ± 0.6

776.3 ± 0.6 input 775.5 ± 0.5 Γρ

  • 144.9 ± 1.0

150.5 ± 1.0 input 150.3 ± 1.6 mω 783.0

  • 782.59 ± 0.11

Γω 8.4

  • 8.49 ± 0.08

mρ′

  • 1357 ± 18

1380 ± 18 1335 1465 ± 25 Γρ′

  • 437 ± 60

340 ± 53 266 400 ± 60 mρ′′ 1700

  • 1724

1720 ± 20 Γρ′′ 240

  • 344

250 ± 100 mρ′′′

  • 2040
  • Γρ′′′
  • 400
  • c0
  • 1.171±0.007

1.098±0.005 1.171

  • β

c0 2.30±0.01 2.16±0.015 2.3(input)

  • cω

0.00184(KS)

  • 0.00195(GS)
  • c1
  • 0.119 ± 0.011
  • 0.069 ± 0.009
  • 0.1171
  • c2
  • 0.0115 ± 0.0064

0.0216 ± 0.0064

  • 0.0246

c3 cn=1

  • 0.0438 ∓ 0.02
  • 0.0309 ∓ 0.02
  • 0.00995
  • ∞
  • n=4

cn

  • 0.01936
  • 0.01936
  • χ2/d.o.f.
  • 155/101

153/101

slide-22
SLIDE 22

A

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 22

1 10 100 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

√s[GeV] |Fπ(s)|2

0.0001 0.001 0.01 0.1 1 10 1 1.5 2 2.5 3 3.5

√s[GeV] |Fπ(s)|2

data point at 3.1 GeV (J/Ψ → ππ) cannot be accomodated

slide-23
SLIDE 23

spacelike region:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 23

good agreement with data and with sum rules

0.2 0.4 0.6 0.8 1

  • 0.3
  • 0.25
  • 0.2
  • 0.15
  • 0.1
  • 0.05

s[GeV2] |Fπ(s)|2

0.0001 0.001 0.01 0.1 1

  • 10
  • 9
  • 8
  • 7
  • 6
  • 5
  • 4
  • 3
  • 2
  • 1

s[GeV2] |Fπ(s)|2

slide-24
SLIDE 24

A e+e− → K+K− , K0 ¯ K0

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 24

isospin symmetry: FK+ = +F (I=1) + F (I=0) FK0 = −F (I=1) + F (I=0) resonances: FK+(s) = +1

2

  • cK

ρ BWρ(s) + cK ρ′BWρ′(s) + cK ρ′′BWρ′′(s)

  • + 1

6

  • cK

ω BWω(s) + cK ω′BWω′(s) + cK ω′′BWω′′(s))

+ 1

3

  • cφBWφ(s) + cφ′BWφ′(s)
  • ,

FK0(s) = −1

2

  • cK

ρ BWρ(s) + cK ρ′BWρ′(s) + cK ρ′′BWρ′′(s)

  • + 1

6

  • cK

ω BWω(s) + cK ω′BWω′(s) + cK ω′′BWω′′(s)

  • + 1

3

  • ηφcφBWφ(s) + cφ′BWφ′(s)
slide-25
SLIDE 25

quark model:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 25

γ∗ ρ K+ K−

1 √ 2fρ

gρKK γ∗ ω K+ K−

1 3 √ 2fω

gωKK

constraint: fρ = fω , gρKK = gωKK ⇒ cρ = cω fit performed with (solid curves)

  • r without (dashed curves) this constraint
slide-26
SLIDE 26

Results:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 26

Parameter Input Fit(1) Fit(2) PDG value mφ

  • 1019.372 ± 0.02

1019.355 ± 0.02 1019.456 ± 0.02 Γφ

  • 4.36 ± 0.05

4.29 ± 0.05 4.26 ± 0.05 mφ′ 1680

  • 1680 ± 20

Γφ′ 150

  • 150 ± 50

mρ 775

  • 775.8 ± 0.5

Γρ 150

  • 150.3 ± 1.6

mρ′ 1465

  • 1465 ± 25

Γρ′ 400

  • 400 ± 60

mρ′′ 1720

  • 1720 ± 20

Γρ′′ 250

  • 250 ± 100

mω 783.0

  • 782.59 ± 0.11

Γω 8.4

  • 8.49 ± 0.08

mω′ 1425

  • 1400-1450

Γω′ 215

  • 180-250

mω′′ 1670

  • 1670 ± 30

Γω′′ 315

  • 315 ± 35

cφ

  • 1.018 ± 0.006

0.999 ± 0.007

  • cφ′

1 − cK

φ

  • 0.018 ∓ 0.006

0.001 ∓ 0.007

  • cK

ρ

  • 1.195 ± 0.009

1.139 ± 0.010

  • cK

ρ′

  • 0.112 ± 0.010
  • 0.124 ± 0.012
  • cK

ρ′′

1 − cK

ρ − cK ρ′

  • 0.083 ∓ 0.019
  • 0.015 ∓ 0.022
  • cK

ω (1)

cK

ρ

1.195 ± 0.009

  • cK

ω (2)

  • 1.467 ± 0.035
  • cK

ω′(1)

cK

ρ′

  • 0.112 ± 0.010
  • cK

ω′(2)

  • 0.018 ± 0.024
  • cK

ω′′

1 − cK

ω − cK ω′

  • 0.083 ∓ 0.019
  • 0.449 ∓ 0.059
  • χ2/d.o.f.
  • 328/242

281/240

slide-27
SLIDE 27

A

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 27

10 100 1000 10000 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05

√s[GeV] |FK+(s)|2

10 100 1000 10000 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05

√s[GeV] |FK0(s)|2

0.001 0.01 0.1 1 10 100 1.2 1.4 1.6 1.8 2 2.2 2.4

√s[GeV] |FK+(s)|2

0.001 0.01 0.1 1 10 100 1.2 1.4 1.6 1.8 2 2.2 2.4

√s[GeV] |FK0(s)|2

slide-28
SLIDE 28

A

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 28

Spectral function separated for I = 0 and I = 1 (useful for electroweak analysis!) ρ(I=0,1)

K ¯ K

(s) = 1 12π

  • FK+(s) ± FK0(s)

2

  • 2 2pK(s)

√s 3

1e-05 0.0001 0.001 0.01 0.1 1 10 1 1.2 1.4 1.6 1.8 2 2.2 2.4

√s[GeV] ρ(I=0)

K ¯ K (s)

1e-05 0.0001 0.001 0.01 1 1.2 1.4 1.6 1.8 2 2.2 2.4

√s[GeV] ρ(I=1)

K ¯ K (s)

significant model dependence above 1.5 GeV (poor data for |FK0|2 !)

slide-29
SLIDE 29

A τ → K−K0ν

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 29

Predictions based on isospin symmetry and I = 1 part of form factor:

  • 1

BR(τ → µ−¯ νµντ)

  • dBR(τ → K−K0ντ)

d

  • Q2

= |Vud|2 2m2

τ

  • 1 + 2Q2

m2

τ

1 − Q2 m2

τ

2 1 − 4m2

K

Q2 3/2 ×

  • Q2 |FK−K0(Q2)|2

and FK−K0 = −FK+ + FK0 ⇒ BR(τ → K−K0ντ) = 0.19 ± 0.01% (0.13 ± 0.01%) to be compared with BR(τ → K−K0ντ) = 0.154 ± 0.016%.

slide-30
SLIDE 30

A Q2 distribution:

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 30

will provide further constraints!

  • 0.5

0.5 1 1.5 2 2.5 3 3.5 4 4.5 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8

  • Q2[GeV]

(data from CLEO)

slide-31
SLIDE 31

V Conclusions

J.H. K¨ uhn, TAU 04 The Radiative Return and Form Factors at Large Q2 31

  • continuous development of PHOKHARA

⇒ radiative corrections ⇒ more channels ⇒ cooperation between theory and experiment crucial

  • nucleon form factors:

GE and GM can be measured for a wide range of Q2

  • pion form factor: structures at large Q2

kaon form factors: K+K− & K0 ¯ K0 ⇒ K−K0 ⇒ prediction for τ → νK−K0