Rescattering and Chiral Dynamics in B Decay Susan Gardner SLAC - - PowerPoint PPT Presentation

Rescattering and Chiral Dynamics in B → ρπ Decay

Susan Gardner SLAC & University of Kentucky

✘ Why B → ρπ: Isospin Analysis for α ☞ Isospin violation and ways to mimic it. Why σ? ✘ Evaluating B → π+π−π0 Decay ☞ Scalar and vector form factors consistent with chiral symmetry ✘ Mitigating unwanted contributions in the ρπ phase space

[in collaboration with Ulf-G. Meißner (Univ. Graz) and Jusak Tandean (Univ. of Kentucky)] S.G. and U. Meißner, hep-ph/0112281, PRD, in press; J. Tandean and S.G., hep-ph/0204147.

• S. Gardner, B → ρπ, May, 2002, 1

Testing the Standard Model

• 2
• 1

1 2

• 2
• 1

1 2

|εK| |εK| K0→π0νν |ε,/εK| , K0→π0νν ∆md |Vub/Vcb| sin 2α sin 2α sin 2β sin 2β K+→π+νν K+→π+νν sin γ sin γ ρ η 0.2 0.4 0.6 0.8 1

• 1
• 0.5

0.5 1

sin 2βWA ∆md ∆ms & ∆md |εK| |Vub/Vcb| ρ η

CK M

f i t t e r

[CKMfitter: H¨

• cker, Lacker, Laplace, Diberder, hep-ph/0104062; March, 2002 update]
• S. Gardner, B → ρπ, May, 2002, 2

B → ρπ: Isospin Analysis for α

[Lipkin, Nir, Quinn, Snyder, 1991; Snyder and Quinn, 1993; Quinn and Silva, 2000.] Isospin: ρπ state =

⇒ If = 0, 1, 2; B = ⇒ Ii = 1/2; |∆I| = 1/2, 3/2, 5/2.

Parametrize the amplitudes by A|∆I|,If:

a+− ≡ A(B0 → ρ+π−) = 1 2 √ 3[A3/2,2 + A5/2,2] + 1 2[A3/2,1 + A1/2,1] + 1 √ 6A1/2,0 a−+ ≡ A(B0 → ρ−π+) = 1 2 √ 3[A3/2,2 + A5/2,2] − 1 2[A3/2,1 + A1/2,1] + 1 √ 6A1/2,0 a00 ≡ A(B0 → ρ0π0) = − 1 √ 3[A3/2,2 + A5/2,2] + 1 √ 6A1/2,0 ,

Two key assumptions:

☞ A(B0 → π+π−π0) = f+ a+− + f− a−+ + f0 a00, fi describes ρi → ππ. ☞ The penguin is |∆I| = 1/2 = ⇒ P 00 = 1

2(P +− + P −+) .

• S. Gardner, B → ρπ, May, 2002, 3

B → ρπ: Isospin Analysis for α

CKM unitarity: =

⇒ 2 weak phases: V ∗

ubVud

|V ∗

ubVud| = eiγ

; V ∗

tbVtd

|V ∗

tbVtd| = e−iβ

and α = π − β − γ, so that

eiβa+− = T +−e−iα + P +− eiβa−+ = T −+e−iα + P −+ eiβa00 = T 00e−iα + P 00

With |∆Γ| ≪ Γ, we have q/p = exp(−2iβ) and q ¯

aij/p ∝ exp(−iβ), so that

the isospin analysis determines sin(2α). With the penguin assumption, there are 10 parameters in all, to be determined in an analysis of the Dalitz plot.

• S. Gardner, B → ρπ, May, 2002, 4

B → ρπ: Isospin Violation?

Isospin is merely an approximate symmetry. In the SM,

✘ Isospin violation can generate a |∆I| = 5/2 amplitude. ☞ via O(α) × A1/2,0 or O(ǫ(2)) × A3/2,2; ǫ(2) = √ 3(md − mu)/(4(ms − ˆ m)). ✘ Isospin violation can distinguish the form factors fi. ✘ Penguin contributions of |∆I| = 3/2 character can occur.

These uncertainties can be mitigated in an empirically driven way.

• S. Gardner, B → ρπ, May, 2002, 5

B → ρπ: Isospin Violation?

Isospin is merely an approximate symmetry. In the SM,

✘ Isospin violation can generate a |∆I| = 5/2 amplitude. ☞ via O(α) × A1/2,0 or O(ǫ(2)) × A3/2,2; ǫ(2) = √ 3(md − mu)/(4(ms − ˆ m)). ✘ Isospin violation can distinguish the form factors fi. ✘ Penguin contributions of |∆I| = 3/2 character can occur.

These uncertainties can be mitigated in an empirically driven way.

✘ The |∆I| = 5/2 and |∆I| = 3/2 amplitudes have the same weak phase? ☞ True to the extent that A3/2,2 > A1/2,0. No “|∆I| = 1/2 rule”? ✘ f± vs. f0? Compare hadronic τ and e+e− data. ☞ Note ρ0 − ω mixing is reflective of B → ωπ0 → π+π−π0. ✘ ? Not “enhanced” in any way. ☞ Sources:

electroweak penguins or O(α, ǫ(2)) corrections to the hadronic m.e.’s [Deshpande & He, 1995; S.G., 1999; S.G. & Valencia, 1999]

• S. Gardner, B → ρπ, May, 2002, 5

B → ρπ cf. B → ππ

In B → ρπ decay the combination A3/2,2 + A5/2,2 appears throughout. However, in B → ππ decay

b+− ≡ A(B0 → π+π−) = − 1 √ 3A1/2,0 + 1 √ 6[A3/2,2 − A5/2,2] b00 ≡ A(B0 → π0π0) = − 1 √ 3A1/2,0 −

• 2

3[A3/2,2 − A5/2,2] b+0 ≡ A(B+ → π+π0) = √ 3 2 A3/2,2 + 1 √ 2A5/2,2

Isospin analysis in B → ππ relies on (b+− − b00)/

√ 2 − b+0 = 0

[Gronau & London, 1990]

If A1/2,0/A3/2,2 is small, neglecting the A5/2,2 amplitude can incur significant errors in sin(2α) and can break bounds on the hadronic uncertainty.

[S.G., 1999]

If A5/2,2 and A3/2,2 share the same weak phase, the B → ρπ analysis is insensitive to this effect.

• S. Gardner, B → ρπ, May, 2002, 6

B → ρπ: Mimicking Isospin Violation

What other contributions enter the ρπ phase space? Note recent studies of D− → π−π+π− decay:

D− → π−σ(500) → π−π+π− accounts for ≈half of the total decay rate.

[E791 Collaboration, Phys. Rev. Lett. 86, 765 (2001).]

The B → σπ channel can contribute to the ρπ phase space in

B → ππ+π− and modifies Br( ¯ B0 → ρ∓π±)/Br(B− → ρ0π−) to yield better

agreement with experiment.

[Deandrea & Polosa, Phys. Rev. Lett. 86, 216 (2001).]

Using CLEO [PRL 85, 2881 (2000)] and BaBar data [hep-ex/0107058] yields

R = Br( ¯ B0 → ρ∓π±) Br(B− → ρ0π−) = 2.7 ± 1.2 .

This ratio of ratios is ≈6 if one works at tree level in factorization.

[Bauer, Stech, and Wirbel, 1987.]

How does this contribution impact the B → ρπ → π+π−π0 analysis? The σπ contributes preferentially to the ρ0π0 final state and can break the assumed relation between the penguin contributions.

• S. Gardner, B → ρπ, May, 2002, 7

B → ρπ: Extension to include σπ

The B → σπ channel has definite properties under CP. The B → ρπ analysis can be extended to include this channel, if need be. Defining aσ

00 = A(B0 → σπ0), we have

eiβaσ

00 = T 00 σ e−iα + P 00 σ .

T 00

σ and P 00 σ

= ⇒ four additional hadronic parameters. More observables

are present as well. Including the scalar channel,

A3π ≡ A(B0 → π+π−π0) = f+ a+− + f− a−+ + f0 a00 + fσ aσ

00, where fσ

describes σ → π+π−. The products fif ∗

j contained in |A3π|2 and | ¯

A3π|2 are distinguishable

through the Dalitz plot of this decay — the coefficients of these functions are distinct observables. Sufficient observables exist to determine all the parameters.

• S. Gardner, B → ρπ, May, 2002, 8

Evaluating B → π+π−π0 Decay

The effective, |∆B| = 1 Hamiltonian for b → dq¯

q decay is given by Heff = GF √ 2

• λu(C1Ou

1 + C2Ou 2) + λc(C1Oc 1 + C2Oc 2) − λt 10

• i=3

CiOi

• ,

where λq ≡ VqbV ∗

• qd. C1 ∼ O(1) and C1 > C2 ≫ C3...10.

For a light, narrow resonance, the product Ansatz

Aσ(B → π+π−π) ≡ (σ → π+π−)π|Heff|B = σπ|Heff|BΓσππ ,

should work well. Γσππ describes σ → ππ. Use factorization to compute

σπ|Heff|B, ρπ|Heff|B.

Focus on Γσππ, Γρππ. Compute the effective B → ρπ branching ratio via

M = Aσ(B → π+π−π) + Aρ(B → π+π−π) + ....

• S. Gardner, B → ρπ, May, 2002, 9

Coupled-channel pion and kaon scalar form factors

Earlier analyses (E791) use the σ → π+π− vertex function

Γσππ(x) = gσπ+π−

• 1

x − M 2

σ + iΓσ(x)Mσ

• ; Γσ(x) = Γσ

Mσ √x

• x/4 − M 2

π

• M 2

σ/4 − M 2 π

.

[Blatt & Weisskopf, 1952]

FSI in the J = 0 I = 0 channel are very strong.

☞ cf. r2

Sπ ≃ 0.6 fm2 and r2 V π ≃ 6/M 2 ρ ≃ 0.4 fm2; fixed-order ChPT

fails more rapidly in scalar-isoscalar channel

☞ The σ need not be a “pre-existing” resonance; can be

generated thr. FSI. [Oller & Oset, 1999] Need to invoke a resummation approach, consistent with unitarity (including both ππ and K ¯

K channels) and chiral symmetry....

[Meißner & Oller, 2001]

• S. Gardner, B → ρπ, May, 2002, 10

Coupled-channel pion and kaon scalar form factors

We have Γσππ(x) −

→ χ Γn ∗

1 (x) where

0|¯ nn|ππ = √ 2 B0 Γn

1(s) ,

0|¯ nn|KK = √ 2 B0 Γn

2(s) ,

0|¯ ss|ππ = √ 2B0 Γs

1(s) ,

0|¯ ss|KK = √ 2 B0 Γs

2(s) ,

B0 is the vacuum quark condensate, B0 = −0|¯ qq|0/F 2

π.

Start with the scattering T-matrix [Babelon, Basdevant, Caillerie, Mennessier, 1976; Oller & Oset, 1997]

T(s) = [I + K(s) · g(s)]−1 · R(s) K(s) from O(p2) CHPT Lagrangian, e.g., K(s)11 = (s − M 2

π/2)/F 2 π.

g(s) is the scalar loop integral.

Fix R(s) by matching Γ(s) to the next-to-leading order (one loop) CHPT

ππ and K ¯ K scalar form factors.

[Gasser & Leutwyler, 1985; Meißner & Oller, 2001]

Γ(s) valid from threshold up to energies of about 1.2 GeV.

Resultant form factor consistent with dispersion relation approach.

[Donoghue, Gasser, Leutwyler, 1990]

• S. Gardner, B → ρπ, May, 2002, 11

Coupled-channel pion and kaon scalar form factors

0.2 0.4 0.6 0.8 1 s

1/2 [GeV]

• 20
• 15
• 10
• 5

5 10 15 20 Γσππ [GeV

• 1]

scalar f.f. used by E791; scalar f.f. in a chiral, unitary approach

[ (solid) ReΓσππ ; (dot-dashed) ImΓσππ; (dashed) |Γσππ|]

• S. Gardner, B → ρπ, May, 2002, 12

• S. Gardner, B → ρπ, May, 2002, 13

ππ phase shifts in L = 0, I = 0

0.3 0.4 0.5 0.6 0.7 0.8

E(GeV)

20 40 60 80

δ0

0(degrees)

Hyams et al. Protopopescu et al.

0.3 0.4 0.5 0.6 0.7 0.8 E (GeV) 10 20 30 40 50 60 70 80 90 δ0

0 (degrees)

L=0 I=0 scalar f.f. e791

[LHS: Colangelo, Gasser, Leutwyler, 2001]

• S. Gardner, B → ρπ, May, 2002, 14

• S. Gardner, B → ρπ, May, 2002, 15

Vector form factor

The vector form factor is determined by e+e− → π+π− data in the ρ

• region. Needed analyticity, unitarity constraints are captured by the

Muskhelishvili-Omn` es equation. For s (Mπ + Mω)2,

Fρ(s) = P(s) Ω(s) , P(s) is a real polynomial, and the Omn`

es function is

Ω(s) = exp

• s

π ∞

4M2

π

ds s′ φ1(s′) s′ − s − iǫ

• ,

tan φ1(s) ≡ Im Fρ(s) Re Fρ(s) = tan δ1

1(s) ,

δ1

1(s) is the phase shift of I = 1, L = 1 scattering.

To realize Γρππ(s), define Γρππ(s) ≡ −Fρ(s)/fργ where

fργ = 0.122 ± 0.001 GeV2.

• S. Gardner, B → ρπ, May, 2002, 16

Vector form factor

For F(s) use fit of e+e− → π+π− data based on the Heyn-Lang parametrization.

[S.G. & O’Connell, 1998; Heyn & Lang, 1981]

• cf. with Breit-Wigner form

ΓBW

ρππ(s) =

gρ s − M 2

ρ + iΓρMρ

,

Can repair low-energy behavior by using

ΓRW

ρππ(s) =

gρ s − M 2

ρ + iΠ(s) ,

Π(s) = M 2

ρ

√s p(s) p(M 2

ρ)

3 Γρ ,

where p(s) =

• s/4 − M 2

π. Form used in BaBar book.

• S. Gardner, B → ρπ, May, 2002, 17

Vector form factor

0.4 0.6 0.8 s

1/2 [GeV]

• 20

20 40 60

• Γρππ [GeV
• 2]

0.4 0.6 0.8 s

1/2 [GeV]

• 20

20 40 60

• Γρππ [GeV
• 2]

→ indicates Fρ(s)/fργ form factor.

[ (solid) ReΓρππ ; (dot-dashed) ImΓρππ; (dashed) |Γρππ|]

LHS: Breit-Wigner form; RHS: form used by BaBar For the isospin analysis, Im Γρππ/Re Γρππ is crucial

= ⇒ can confront π − π phase shifts directly.

• S. Gardner, B → ρπ, May, 2002, 18

Vector form factor

500 600 700 800 900 s

1/2 [MeV]

50 100 150 δ1

1

δ1

1(s) is the phase shift of I = 1, L = 1 π − π scattering.

Hyams et al., Nucl. Phys. B 64, 134 (1973) (); Protopopescu et al., Phys. Rev. D 7, 1279 (1973) (△).

Only the Breit-Wigner form fails to confront the data

• S. Gardner, B → ρπ, May, 2002, 19

Results

Effective branching ratios (in units of 10−6) for B → ρπ decay, computed at tree (penguins, after Deandrea et al., 2000) level.

δ [MeV] (f.f.) ¯ B0 → ρ−π+ ¯ B0 → ρ+π− ¯ B0 → ρ0π0 B− → ρ0π− R 200 (BW) 15.1 (15.1) 4.21 (4.21) 0.508 (0.509) 3.50 (3.68) 5.5 (5.2) 300 (BW) 16.4 (16.4) 4.74 (4.74) 0.918 (0.927) 3.89 (4.10) 5.4 (5.2) 200 (RW) 15.1 (15.1) 4.19 (4.19) 0.468 (0.474) 3.49 (3.68) 5.5 (5.2) 300 (RW) 16.4 (16.3) 4.69 (4.68) 0.835 (0.847) 3.87 (4.07) 5.5 (5.2) 200 (∗) 15.3 (15.2) 4.26 (4.26) 0.473 (0.477) 3.49 (3.68) 5.6 (5.3) 300 (∗) 16.4 (16.3) 4.75 (4.75) 0.865 (0.877) 3.85 (4.06) 5.5 (5.2) Including the σ.... δ [MeV] (f.f.) B− → σπ− B− → (ρ0 + σ)π− ¯ B0 → σπ0 ¯ B0 → (ρ0 + σ)π0 R 200 (BW) 2.97 6.77 0.0258 0.516 2.9 300 (BW) 5.17 9.50 0.0457 0.940 2.2 200 (RW) 2.97 6.74 0.0258 0.475 2.9 300 (RW) 5.17 9.45 0.0457 0.855 2.2 200 (∗) 4.60 7.80 0.0396 0.515 2.5 300 (∗) 7.66 11.2 0.0663 0.928 1.9

• S. Gardner, B → ρπ, May, 2002, 20

Results

• 1
• 0.5

0.5 1 cos(θ) 2 4 6 8 10 |M|

2x10 13

• 1
• 0.5

0.5 1 cos(θ) 0.2 0.4 0.6 0.8 |M|

2x10 13

LHS: B− → π−π−π+; RHS: ¯

B0 → π−π+π0

(solid) σπ ; (dashed) ρπ

• S. Gardner, B → ρπ, May, 2002, 21

Results

10 20 10 20 5·10-14 1·10-13 1.5·10-13 2·10-13

s

1+

s

2+ 10 20 5 1 1.5 10 20 10 20 1·10-14 2·10-14 3·10-14 4·10-14 5·10-14

s

s

+0 10 20 1· 2 3 4

B∗ and B0 contributions after Deandrea et al., 2000

Note B → (B∗, B0)π → ππ. May add considerably to the structure of the background?

• S. Gardner, B → ρπ, May, 2002, 22

Results

Effective branching ratios for B → ρπ decays, with δ = 0.3 GeV. All branching ratios are reported in units of 10−6.

Decay mode ρ σ B∗ B0 ρ + B∗ ρ + σ ρ + σ + B∗ + B0 ¯ B0 → ρ−π+ ¯ B0 → ρ+π− ¯ B0 → ρ0π0 B− → ρ0π− 16.0 4.76 0.86 4.06 0.001 0.001 0.065 7.66 0.54 0.13 0.39 2.71 0.009 0.020 0.016 0.107 16.6 4.90 1.35 7.20 15.9 4.80 0.91 11.1 16.3 4.98 1.33 12.7 R 5.1

• 3.0

1.9 1.7 Including the off-shellness of the B∗ meson.... Decay mode ρ B∗ ρ + B∗ ρ + σ ρ + σ + B∗ ¯ B0 → ρ−π+ ¯ B0 → ρ+π− ¯ B0 → ρ0π0 B− → ρ0π− 16.0 4.76 0.86 4.06 0.03 0.01 0.02 0.25 16.0 4.85 0.88 4.43 15.9 4.80 0.91 11.1 16.0 4.88 0.95 10.7 R 5.1

• 4.7

1.9 1.9

• S. Gardner, B → ρπ, May, 2002, 23

Summary and Outlook

✘ We have examined the impact of isospin violation on the B → ρπ analysis. If A3/2,2 and A5/2,2 share the same weak phase, A5/2,2 does not impact the B → ρπ analysis in any way. ✘ The scalar form factor can be determined to good precision.

Our form factor describes the f0(980) as well; its shape in B → f0(980)π → 3π, e.g., should serve to test our approach.

✘ Remarkably, the impact of σπ on the ratio R is huge, whereas

its impact on the ρ0π0 mode is merely significant. Varying the cuts on the helicity angle θ should be helpful.

✘ The σπ channel has definite properties under CP, so that

the isospin analysis in ρπ can be extended to include it, if necessary.

• S. Gardner, B → ρπ, May, 2002, 24