Photon radiation in e+e− → hadrons at lowenergies with carlomat 3.1

Fred Jegerlehnerfjeger@physik.hu-berlin.de

Work done by Karol KołodziejEur.Phys.J. C77 (2017) no.4, 254

RMC WG meeting Mainz, 30 June 2017

F. Jegerlehner RMC WG Mainz, September 2012

Outline of Talk:

v Effective field theory: the Resonance Lagrangian Approach, broken HLSv A minimal version: VMD + sQED solving the τ vs e+e− puzzlev Global fit of BHLS parameters and prediction of Fπ(s)v SM prediction of aµ using BHLS predictions for aLO,had

µ

v Lessons and Outlook

F. Jegerlehner RMC WG Mainz, September 2012 1

Introduction

Study of photon radiation at tree level incl. some form factors for low energy hadronproduction in e+e−–annihilation

F. Jegerlehner and K. Kołodziej, Eur. Phys. J. C 77 (2017) no.4, 254doi:10.1140/epjc/s10052-017-4816-7 [arXiv:1701.01837 [hep-ph]].

l K. Kołodziej, carlomat 3.1 version of the program dedicated to the descriptionof processes of electron-positron annihilation to hadrons at low centre of massenergies, http://kk.us.edu.pl/carlomat.html.

l K. Kołodziej, Comput. Phys. Commun. 180 (2009), 1671. arXiv:0903.3334[hep-ph] http://dx.doi.org/10.1016/j.cpc.2009.03.011

o Implements HLS Feymanrules with couplings from global HLS fit, combined withsQED for photon radiation of pions.

o Some features complementary to generator like PHOKHARA, I think.

F. Jegerlehner RMC WG Mainz, September 2012 2

Hiddel Local Symmetry Feynman Rules

HLS Lagrangian and global fit couplings see:

M. Benayoun, P. David, L. DelBuono and F. Jegerlehner,

‘‘An Update of the HLS Estimate of the Muon g-2,’’

Eur.\ Phys.\ J.\ C {\bf 73} (2013) 2453

doi:10.1140/epjc/s10052-013-2453-3

[arXiv:1210.7184 hep-ph]].}

F. Jegerlehner RMC WG Mainz, September 2012 3

:= −i gρππ (p− − p+)µρ 0µ π+

π−

:= 2 i g2ρππ gµν

ρ+µ

ρ− ν

π+

π−

:= 2 i e gρππ gµν

ρ 0µ

A ν

π+

π−

:= −i e (p− − p+)µAµ π+

π−

:= 2 i e2 gµνAµ

A ν

π+

π−

:= e2 gπγγ εµναβ k1αk2βπ0

Aµ[k1]

A ν[k2]

:= −i e gωγπ εµναβ pγαpωµ

Aβ ω ν

π0

:= −i e gρ0γπ0 εµναβ pγαpρµ

Aβ ρ0 ν

π0

:= −i e gρ±γπ∓ εµναβ pγαpρµ

Aβ ρ± ν

π∓

:= −e gγπππ εµναβ p0νp+αp−β

Aµ π+

π0

π−

:= gωρπ εµναβ pωαpρβ

ω µ ρ0 ν

π0

:= gωρπ εµναβ pωαpρβ

ω µ ρ± ν

π∓

:= − gωπππ εµναβ p0νp+αp−β

ω µ π+

π0

π−

gρππ = 6.505(3)gωππ = 0.408(61)gπγγ = − Nc

24π2Fπ(1− c4)

gγπππ = − Nc12π2F 3

π

[1− 3

4(c1 − c2 + c4)

]

gωπππ = − 3Ncg16π2F 3

π(c1 − c2 − c3)

gρ±π∓γ =G2

gρ0π0γ =G2

gωπ0γ =3G2

G = − g4 π2 fπ

c3+c42

gωρπ = C2 ,

C = −NC g2

4 π2 fπc3,

Nc = 3,Fπ = 92.42 MeV,g ≡ gHLS = 5.578(1),c4 ≈ c3, (c3 + c4)/2 ≈ c3 = 0.920(4),c1 − c2 = 1.226(26),fργ = 2 gργγ f

2π ,

fωγ =13 fργ

Comparing the τ+PDG prediction (red curve) of the pion form factor in e+e− annihilation in the ρ−ωinterference region

F. Jegerlehner RMC WG Mainz, September 2012 8

The HLS model at work. The left panel shows the global HLS model fit of the ππ channel togetherwith the data from Novosibirsk, Frascati and Beijing. The right panel shows the P–wave π+π−

phase–shift data and predictions from Leutwyler, Colangelo and F.J.-Szafron together with thebroken HLS phase–shift

F. Jegerlehner RMC WG Mainz, September 2012 9

sQED:

Aµ

π+

π−

≡ ie(p+ − p−)µ

Aµ

Aν

π+

π−

≡ 2ie2gµν

F. Jegerlehner RMC WG Mainz, September 2012 10

Is sQED model viable?

How photons couple to pions? This is obviously probed in reactions like γγ → π+π−, π0π0.Data infer that below about 1 GeV photons couple to pions as point-like objects (i.e. to thecharged ones overwhelmingly), at higher energies the photons see the quarks exclusivelyand form the prominent tensor resonance f2(1270). The π0π0 cross section in this figure is

enhanced by the isospin symmetry factor 2, by which it is reduced in reality.

F. Jegerlehner RMC WG Mainz, September 2012 11

Normalization as in reality: σ(π0π0)/σ(π+π−) = 12 at the f2(1270) peak

(art work by Mike Pennington)

F. Jegerlehner RMC WG Mainz, September 2012 12

γ

γ

π+

π−

π+, π0

π−, π0

u, d

u, d

Di-pion production in γγ fusion. At low energy we have direct π+π− production and by strongrescattering π+π− → π0π0, however with very much suppressed rate. Above about 1 GeV, resolved

qq couplings seen.

Strong tensor meson resonance in ππ channel f2(1270) with photons directly probe the quarks!

F. Jegerlehner RMC WG Mainz, September 2012 13

Pion Form Factor Implemented

E(MeV)

|Fπ(E

)|2

900800700600500400

45

40

35

30

25

20

15

10

5

0

The EM charged pion form factor calculated by direct calls to the subroutine for Fπ(s) (points) andwith the MC program generated automatically (solid line).

F. Jegerlehner RMC WG Mainz, September 2012 14

e+

e−

π+(p′)

π−(p)

γ(k)

γ∗(q)

(1)

e+

e−

π+(p′)

π−(p)

γ(k)γ∗(q)

(2)

e+

e−

π+(p′)

π−(p)

γ(k)

γ∗(q)

(3)

FSR Feynman diagrams of process e+e− → π+π−γ in the LO. The relevant four momenta are indi-cated in parentheses and blobs represent the charged pion form factor.

e+

e−

π+(p′)

π−(p)

γ∗(q) γ∗(q′)

µ+

µ−

γ

(1)

e+

e−

π+(p′)

π−(p)

γ∗(q) γ∗(q′)

µ+

µ−γ

(2)

ISR Feynman diagrams of process e+e− → π+π−γ in which the choice of four momentum transfer inthe charged pion form factor, represented by the red blob, may be ambiguous.

F. Jegerlehner RMC WG Mainz, September 2012 15

Non-SM Vertices Implemented

Aµ(q) V ν(q)≡ −efAV (q

2) gµν , V = ρ0, ω, φ, ρ1, ρ2

The γ − V-mixing terms considered in the present work; ρ1 and ρ2 stand for ρ(1450) and ρ(1700),respectively.

V µ(q)

π+(p1)

π−(p2)

≡ ifV π+π−(q2)(p1 − p2)µ,

V = ρ0, ω, φ, ρ1, ρ2

(a)

π0(q)

ωµ(p1)

ρ0 ν(p2)

≡ fπ0ωρ0(q2)εµναβp1αp2 β

(b)

π0(q)

Aµ(p1)

Aν(p2)

≡ e2fπ0AA(q2)εµναβp1αp2 β

(c)

π0(q)

Aµ(p1)

V ν(p2)

≡ iefπ0AV (q2)εµναβp1αp2 β

V = ρ0, ω

(d)

Triple vertices of the HLS relevant for the present work.

F. Jegerlehner RMC WG Mainz, September 2012 16

π0(p1)

Aµ(q)

π+(p2)

π−(p3)

≡ −efAπ0π+π−(q2)εµναβp1 νp2αp3 β

(a)

π0(p1)

ωµ(q)

π+(p2)

π−(p3)

≡ −efωπππ(q2)εµναβp1 νp2αp3 β

(b)

ρ0µ

Aν

π+

π−

≡ 2iefρ0Aπ+π−(q2)gµν

(c)

Quartic vertices of the HLS model taken into account in the present work.

F. Jegerlehner RMC WG Mainz, September 2012 17

Results

Process ISR full LO

e+e− → # diags σ σ|ε(k)=k # diags σ σ|ε(k)=k

π+π−γ 2 2.041(4)e+4 1.04(1)e–28 5 2.249(4)e+4 2.73(2)e–28π+π−π0γ 32 409(1) 2.21(3)e–30 156 481.5(6) 3.011(1)e–2π+π−µ+µ−γ 26 4.344(9)e–2 4.62(5)e–34 107 6.449(8)e–2 6.42(5)e–34π+π−π+π−γ 36 2.029(5)e–3 2.14(3)e–35 200 3.320(5)e–3 3.03(2)e–35π+π−γγ 6 1.445(14)e+3 1.22(4)e–29 44 2.131(8)e+3 2.08(3)e–29π+π−µ+µ−γγ 90 1.127(7)e–3 1.16(4)e–35 1272 2.535(8)e–3 9.56(1)e–19π+π−π+π−γγ 120 4.68(3)e–5 4.6(1)e–37 2772 1.303(4)e–4 4.969(4)e–15

The cross sections in pb at√

s = 1 GeV and the corresponding U(1) gauge invariance tests. Thenumbers in parentheses show the MC uncertainty of the last decimals.

The differential cross sections of process at√

s = 0.8 GeV,√

s = 1 GeV and√

s = 1.5 GeV:

F. Jegerlehner RMC WG Mainz, September 2012 18

√s = 0.8GeV

e+e− → π+π−γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

0.70.60.50.40.30.20.10

1600

1400

1200

1000

800

600

400

200

0

√s = 1GeV

e+e− → π+π−γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

10.90.80.70.60.50.40.30.20.10

100

90

80

70

60

50

40

30

20

10

0

√s = 1.5 GeV

e+e− → π+π−γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

2.521.510.50

12

10

8

6

4

2

0

The differential cross sections of process e+e− → π+π−γ as functions of invariant mass of the π+π−-pair. The solid line and the shaded histogram represent the ISR cross section, obtained with theMC program and with the known analytic formula, respectively, and the dashed line represents the

full LO result. The difference between solid and dashed lines represents the FSR correction.

LO e+e− → π+π−γ with external Fπ(s) vs. HLS model with fixed couplings exhibiting 26 Feynmandiagrams.

F. Jegerlehner RMC WG Mainz, September 2012 19

√s = 0.8GeV

e+e− → π+π−γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

0.70.60.50.40.30.20.10

2500

2000

1500

1000

500

0

√s = 1GeV

e+e− → π+π−γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

10.90.80.70.60.50.40.30.20.10

100

90

80

70

60

50

40

30

20

10

0

√s = 1.5GeV

e+e− → π+π−γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

2.521.510.50

12

10

8

6

4

2

0

The differential cross sections of e+e− → π+π−γ as functions of invariant mass of the π+π−-paircomputed in a model with the charged pion form factor (solid lines) and in the HLS model with fixedcouplings (dotted lines).

The differential cross sections of processes e+e− → π+π−π0γ, e+e− → π+π−µ+µ−γ and e+e− →π+π−π+π−γ corresponding to those plotted next as functions of the invariant mass of the π+π−π0-,π+π−µ+µ−-, or π+π−π+π−-system.

F. Jegerlehner RMC WG Mainz, September 2012 20

√s = 0.8GeV

e+e− → π+π−π0γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

0.650.60.550.50.450.40.350.30.250.20.15

250

200

150

100

50

0

√s = 1GeV

e+e− → π+π−π0γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

10.90.80.70.60.50.40.30.20.1

9

8

7

6

5

4

3

2

1

0

√s = 1.5GeV

e+e− → π+π−π0γ

Q2(GeV2)

dσ

dQ

2

(nb

GeV

2

)

2.521.510.50

8

7

6

5

4

3

2

1

0

The differential cross sections of e+e− → π+π−π0γ as functions of invariant mass of the π+π−π0-system.

√s = 0.8 GeV

e+e− → π+π−µ+µ−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

0.650.60.550.50.450.40.350.30.250.2

0.4

0.35

0.3

0.25

0.2

0.15

0.1

0.05

0

√s = 1 GeV

e+e− → π+π−µ+µ−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

10.90.80.70.60.50.40.30.2

1.2

1

0.8

0.6

0.4

0.2

0

√s = 1.5 GeV

e+e− → π+π−µ+µ−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

2.42.221.81.61.41.210.80.60.40.2

3

2.5

2

1.5

1

0.5

0

The differential cross sections of e+e− → π+π−µ+µ−γ as functions of invariant mass of the π+π−µ+µ−-system.

F. Jegerlehner RMC WG Mainz, September 2012 21

√s = 0.8GeV

e+e− → π+π−π+π−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

0.650.60.550.50.450.40.350.3

0.01

0.009

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0

√s = 1GeV

e+e− → π+π−π+π−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

10.90.80.70.60.50.40.3

0.07

0.06

0.05

0.04

0.03

0.02

0.01

0

√s = 1.5GeV

e+e− → π+π−π+π−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

2.42.221.81.61.41.210.80.60.40.2

1.2

1

0.8

0.6

0.4

0.2

0

The differential cross sections of e+e− → π+π−π+π−γ as functions of invariant mass of the π+π−π+π−-system.

F. Jegerlehner RMC WG Mainz, September 2012 22

In following Fig, we make the same comparison for the cross sections of process e+e− → π+π−µ+µ−γ

to be compared with previous e+e− → π+π−γ plot. This time carlomat 3.1 generates 627 Feynmandiagrams in the HLS model with the fixed couplings.

√s = 0.8 GeV

e+e− → π+π−µ+µ−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

0.650.60.550.50.450.40.350.30.250.2

1.4

1.2

1

0.8

0.6

0.4

0.2

0

√s = 1 GeV

e+e− → π+π−µ+µ−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)10.90.80.70.60.50.40.30.2

1.4

1.2

1

0.8

0.6

0.4

0.2

0

√s = 1.5 GeV

e+e− → π+π−µ+µ−γ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

2.42.221.81.61.41.210.80.60.40.2

3

2.5

2

1.5

1

0.5

0

The differential cross sections of e+e− → π+π−µ+µ−γ as functions of invariant mass of the π+π−µ+µ−-system computed in a model with the charged pion form factor (solid lines) and in the HLS modelwith fixed couplings (dotted lines).

F. Jegerlehner RMC WG Mainz, September 2012 23

To illustrate the effect of the FSR in processes with two photons, the ISR (shaded histograms) andfull LO (dashed lines) differential cross sections We see that this time the FSR effect is much bigger.

√s = 1GeV

e+e− → π+π−γγ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

10.90.80.70.60.50.40.30.20.10

8000

7000

6000

5000

4000

3000

2000

1000

0

√s = 1GeV

e+e− → π+π−µ+µ−γγ

Q2(GeV2)dσ

dQ

2

(pb

GeV

2

)

10.90.80.70.60.50.40.30.2

0.016

0.014

0.012

0.01

0.008

0.006

0.004

0.002

0

√s = 1GeV

e+e− → π+π−π+π−γγ

Q2(GeV2)

dσ

dQ

2

(pb

GeV

2

)

10.90.80.70.60.50.40.3

0.0009

0.0008

0.0007

0.0006

0.0005

0.0004

0.0003

0.0002

0.0001

0

The differential cross sections of e+e− → π+π−γγ, e+e− → π+π−µ+µ−γγ and e+e− → π+π−π+π−γγ

at√

s = 1 GeV as functions of invariant mass of the π+π−-, π+π−µ+µ− and π+π−π+π−-systems,respectively.

F. Jegerlehner RMC WG Mainz, September 2012 24

Conclusionr we implemented e+e− → hadrons at low centre-of-mass energies in carlomat 3.1 with pionformfactor

r we have illustrated possible applications of the program by considering a photon radiation off theinitial and final state particles for a few potentially interesting processes involving charged pion pairs.

r we have also discussed some problems related to the U(1) electromagnetic gauge invariance thatmay arise if the momentum transfer dependence is introduced in the couplings of the HLS model ora set of the couplings implemented in the program is incomplete.

Thank you for your attention!

F. Jegerlehner RMC WG Mainz, September 2012 25