Insisting on the role of experimental data€¦ · Pere Masjuan 9 A word on systematics •Consider...

Insisting on the role of experimental data:the pseudoscalar-pole piece to (gµ-2) and

the |Vub| from B→(π,η,η’)lν

Pere MasjuanAutonomous University of Barcelona

([email protected])

Work done in collaboration withRafel Escribano, Pablo Sanchez-Puertas and Sergi González-Solís

Universitat Autònomade Barcelona

• Introduction to Stieltjes functions

• A tale of convergence

• Applications:

• Low-energy parameters and HLBL

• Noise filtering and Vub

• Conclusions

Outline

3Pere MasjuanPere Masjuan

Convergence theorem

Hadron 2017, Salamanca, Sept. 28th

Stieltjes function

(dispersion relation)Example:

f(x) =

Z 1/R

0dt

⇢(t)

xt� 1(⇢(t) > 0) ⇠ f(x) =

1

⇡

Z 1

s0

dt

Imf(t)

t+ x

⇢(t) = 1R = 1 , t = 1/s

f(x) =

Z 1/R

0dt

⇢(t)

xt� 1

=

Z 1

1ds

1

s

1

x� s

=

1

x

log(1� x)

f(x) =X

n=0

(�x)n

n+ 1

(some boring maths)


Padé Approximants

are polynomials

Q(z)f(z) +R(z) = O �zq+r+1

�Padé approx:

R(z), Q(z)

f(z) =X

k=0

akzk PN

M (z) =

PNn=0 rnz

n

PMm=0 qnz

n

PNM (z) = r0 + (r1 � r0q1)z + (r2 � r1q1 + r0q

21 � r0q2)z

2 +O(z3)

P 01 (z) =

a01� a1

a0z

f(z) = a0 + a1z + a2z2 +O(z3)

then its PA

Let f(z) be a Stieltjes function and

and has a contact with f(z) of order N+M+1

{Examples: P 1

1 (z) =a0 +

a21�a0a2

a1z

1� a2a1z


(some boring maths)


are polynomials

Q(z)f(z) +R(z) = O �zq+r+1

�Padé approx:R(z), Q(z)

limN!1

PNN+1(z) f(z) lim

N!1PNN (z)

Stieltjes theorem:

(others: Montessus, Pommerenke, Nutall, Baker, Chisholm...)

Example: f(z) =1

zlog(1� z)

f(z) = �X

k=0

zk

k + 1= �1� z

2� z2

3� z3

4� z4

5+O(z6)

Padé Approximants


(some boring maths)


Example: f(z) =1

zlog(1� z)

-1.0 -0.5 0.0 0.5 1.0-0.10

-0.05

0.00

0.05

0.10

Rel

ativ

e er

ror

z

ÊÊÊ ÊÊ ÊÊ Ê ÊÊ Ê Ê Ê·· ·· ·· · ·

0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

Im(z

)

Re(z)

for N=1,2,3,4,5PNN (z)

poleszeros

are polynomials

Q(z)f(z) +R(z) = O �zq+r+1

�Padé approx:R(z), Q(z)

Padé Approximants(some boring maths)



Im(z

)

Re(z)

for N=0,1,2,3,4

poleszeros

PNN+1(z)

Rel

ativ

e er

ror

z-1.0 -0.5 0.0 0.5 1.0

-0.10

-0.05

0.00

0.05

0.10

ÊÊÊ ÊÊ Ê ÊÊ ÊÊÊ ··· ··· · ··

0 1 2 3 4-1.0

-0.5

0.0

0.5

1.0

Example: f(z) =1

zlog(1� z)

Q(z)f(z) +R(z) = O �zq+r+1

�Padé approx:R(z), Q(z) are polynomials

Padé Approximants(some boring maths)



limN!1

PNN+1(z) f(z) lim

N!1PNN (z)

N

Ê

ÊÊ Ê Ê

Ê

ÊÊ Ê Ê

0 1 2 3 4 5 6

-40

-20

0

20

40

Rel

ativ

e er

ror x

100 z=-10

Example: f(z) =1

zlog(1� z)

Q(z)f(z) +R(z) = O �zq+r+1

�Padé approx:R(z), Q(z) are polynomials

Message: convergence pattern gives systematic error

Padé Approximants



A word on systematics

•Consider a model for η TFF•Generate a pseudodata set emulating the physical situation (SL+TL)•Build up your PA sequence•Fit and compare

VMD

15

Ê

Ê

Ê

Ê

ÊÊ Ê Ê Ê

Ú

Ú Ú Ú ‡‡

P10 P11 P12 P13 P14 P15 P16 P17 P18 FhggH0L0.250

0.255

0.260

0.265

0.270

0.275

0.280P11 P22 P33 P44 FhggH0L

FhggH0L

Ê

Ê

Ê

Ê

ÊÊ

Ê Ê Ê

Ú

Ú Ú Ú ‡‡

P10 P11 P12 P13 P14 P15 P16 P17 P18 bh0.250

0.255

0.260

0.265

0.270

0.275

0.280

0.285

0.290P11 P22 P33 P44 bh

bh

Ê Ê

Ê

Ê

Ê

Ê

ÊÊ

Ú

Ú Ú Ú ‡‡

P10 P11 P12 P13 P14 P15 P16 P17 P18 ch0.250

0.255

0.260

0.265

P11 P22 P33 P44 ch

ch

Ê

Ê

Ê

Ê

Ê

ÊÊ

Ú Ú Ú ‡‡

P12 P13 P14 P15 P16 P17 P18 dh0.20

0.21

0.22

0.23

0.24

0.25

0.26P22 P33 P44 dh

dhFig. 7 Convergence pattern for the P

L

1 (red circles) and P

N

N

(green triangles) sequences to Fhgg (Q2) for Q

2 = 0 and its firsts three derivatives bh ,chand dh . The black squares show the results for each parameter from the model in Eq. (B.3), prolonged by the dashed line.

9. H. J. Behrend et al. [CELLO Collaboration], Z. Phys. C49 (1991) 401.

10. J. Gronberg et al. [CLEO Collaboration], Phys. Rev. D57 (1998) 33 [hep-ex/9707031].

11. R. Arnaldi et al. [NA60 Collaboration], Phys. Lett. B677 (2009) 260 [arXiv:0902.2547 [hep-ph]].

12. H. Berghauser et al. [A2 Collaboration], Phys. Lett. B701 (2011) 562.

13. M. Hodana et al. [WASA-at-COSY Collaboration], EPJWeb Conf. 37 (2012) 09017 [arXiv:1210.3156 [nucl-ex]].

14. P. Aguar-Bartolome et al. [A2 Collaboration], Phys.Rev. C 89 (2014) 4, 044608 [arXiv:1309.5648 [hep-ex]].

15. P. Kroll, Eur. Phys. J. C 71 (2011) 1623[arXiv:1012.3542 [hep-ph]].

16. A. E. Dorokhov, A. E. Radzhabov and A. S. Zhevlakov,Eur. Phys. J. C 71 (2011) 1702 [arXiv:1103.2042 [hep-ph]].

17. S. J. Brodsky, F. G. Cao and G. F. de Teramond, Phys.Rev. D 84 (2011) 033001 [arXiv:1104.3364 [hep-ph]].

18. S. J. Brodsky, F. G. Cao and G. F. de Teramond, Phys.Rev. D 84 (2011) 075012 [arXiv:1105.3999 [hep-ph]].

19. Y. N. Klopot, A. G. Oganesian and O. V. Teryaev, Phys.Rev. D 84 (2011) 051901 [arXiv:1106.3855 [hep-ph]].

20. X. G. Wu and T. Huang, Phys. Rev. D 84 (2011) 074011[arXiv:1106.4365 [hep-ph]].

21. S. Noguera and S. Scopetta, Phys. Rev. D 85 (2012)054004 [arXiv:1110.6402 [hep-ph]].

22. I. Balakireva, W. Lucha and D. Melikhov, Phys. Rev. D85 (2012) 036006 [arXiv:1110.6904 [hep-ph]].

23. D. Melikhov and B. Stech, Phys. Rev. D 85 (2012)051901 [arXiv:1202.4471 [hep-ph]].

24. P. Kroll and K. Passek-Kumericki, J. Phys. G 40 (2013)075005 [arXiv:1206.4870 [hep-ph]].

25. D. Melikhov and B. Stech, Phys. Lett. B 718 (2012) 488[arXiv:1206.5764 [hep-ph]].

26. C. Q. Geng and C. C. Lih, Phys. Rev. C 86

(2012) 038201 [Erratum-ibid. C 87 (2013) 3, 039901][arXiv:1209.0174 [hep-ph]].

27. Y. Klopot, A. Oganesian and O. Teryaev, Phys. Rev. D87 (2013) 036013 [arXiv:1211.0874 [hep-ph]].

Dealing with Data



Padé ApproximantsApplications

extract low-energy parametersfrom experimental data

“noise filtering “impose unitary constraints

⇡0, ⌘, ⌘0

e± e±

e⌥ e⌥

�⇤

�⇤TFF

q1

q2

F (q21 , q22)

d�(B ! ⇡`⌫`)

dq2=

G2F |Vub|2

192⇡3m3B

�3/2|F+(q2)|2

B ! ⇡`⌫`Semileptonic B → ⇡`⌫` decay

d�(B → ⇡`⌫`)dq2 = G2

F �Vub�2192⇡3m3

B

((m2B +m2

⇡ − q2)2 − 4m2Bm2

⇡)3�2�F+(q2)�2Determination of Vub

Main source of uncertainty within the form factor F+(q2)

` = e, µP = ⇡0, ⌘, ⌘0

P ! `+`�P ! `+`��

e+e� ! e+e�P



Padé ApproximantsApplications

extract low-energy parameters from experimental data (no data available)

ImF (s) = �3(s)P (s)|FV (s)|2

�(s) =

r1� 4m2

s

⇡0, ⌘, ⌘0

e± e±

e⌥ e⌥

�⇤

�⇤TFF

q1

q2

F (q21 , q22)

TFF is a Stieljes function(neglecting multipion rescattering)

Assuming:

polynomial with positive slope

π-vector FF

F. Stollenwerk, C. Hanhart, A. Kupsc, U. G. Meissner andA. Wirzba, Phys. Lett. B 707 (2012) 184


` = e, µP = ⇡0, ⌘, ⌘0

P ! `+`��

e+e� ! e+e�P

From theory: poorly knownFrom experiment: many Coll. + Data

Fit to Space-like data: CELLO’91, CLEO’98, BABAR’09 and Belle’12

PN1 (Q2)

PNN (Q2)

up to N=5

up to N=3

Pere Masjuan

P01 P11 P21 P31 P41 P51 PDG0.020

0.025

0.030

0.035

0.040

a !

[P.M, ’12]

12

Accurate description of the low-energy region making full use of available experimental data

EPJ Web of Conferences

ÊÊÊÊ

Ê

ÚÚÚÚÚÚÚÚÚÚÚÚÚÚ

Ú‡‡‡‡‡‡‡

‡‡‡ ‡‡ ‡

‡‡

‡

‡

ÏÏÏÏÏÏÏÏ Ï

Ï ÏÏ Ï

Ï

ÏP33HQ2LP16HQ2L

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Q2@GeV2D

Q2 Fpgg*HQ2 L

@GeVD

!!!

!

"

""""""""

"

"

"

## ####

#

#

# #

#

P22!Q

2"

P15!Q

2"

P12!Q

2"

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

Q2 #GeV2$

Q2F!"#"!Q

2"#G

eV$

!

!

!

!

!

"

"

""

"

#####

#

#

##

$$

$$ $

$$

$ $

$

$

P11!Q

2"

P15!Q

2"

P16!Q

2"

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

Q2 #GeV2$

Q2F!'"#"!Q

2"#G

eV$

Figure 1. ⇡0 (left upper panel), ⌘ (right upper panel), and ⌘0 (lower panel) TFFs. Green-dot-dashed lines showour best PL

1 (Q2) fit, and black-solid lines show our best PNN(Q2) fit. Black-dashed lines display the extrapolation of

the PNN(Q2) at Q2 = 0 and Q2 ! 1. Experimental data are from CELLO (red circles), CLEO (purple triangles),

and BABAR (orange squares) Colls. [8]. The ⇡0 figure contains also data from BELLE (blue diamonds) [9]; andthe ⌘0 figure data from L3 (blue diamonds) [10].

Table 1. ⇡0, ⌘, and ⌘0 slope bP, curvature cP, asymptotic limit, and contribution to HLBL.

bP cP limQ2!1 Q2FP�⇤�(Q2) aHLBL;Pµ

⇡0 0.0324(22) 1.06(27) · 10�3 2 f⇡ 6.49(56) · 10�10

⌘ 0.60(7) 0.37(12) 0.160(24)GeV 1.25(15) · 10�10

⌘0 1.30(17) 1.72(58) 0.255(4)GeV 1.27(19) · 10�10

and obtain, in such a way, the derivatives of the FP�⇤�(Q2) at the origin of energies in a simple,systematic and model-independent way [5, 6].

Since the analytic properties of TFFs are not known, the kind of PA sequence to be used is notdetermine in advance. We consider two di↵erent sequences and the comparison among them shouldreassess our results. The first one is a PL

1 (Q2) sequence inspired by the success of the simple vectormeson dominance ansatz [5], and the second one is a PN

N(Q2) sequence which satisfy the pQCDconstrains Q2FP��⇤ (Q2) ⇠ constant. After combining both sequence’s results, slope and curvatureresults are shown in Table 1, where limQ2!1 Q2FP�⇤�(Q2) from the PN

N(Q2) is also shown.The low-energy parameters obtain with this method can be used to constrain hadronic models with

resonances used to account for the hadronic light-by-light scattering contribution part (HLBL) of the

Our proposal: use Padé Approximants[P.M.’12; P.M., M. Vanderhaeghen’12; R. Escribano, P.M., P. Sanchez-Puertas, ’13, ’15]

CELLO

CLEO

BABAR

BELLE

π0-TFF


η-TFF�⌘!��Fit to Space-like data: CELLO’91, CLEO’98, BABAR’11+

PN1 (Q2)

PNN (Q2)

up to N=4

up to N=2

Pere Masjuan 11

[R.Escribano, P.M., P. Sanchez-Puertas, ’13]

fitted poles range !!!!!spp ! "0.71–0.77# GeV and !!!!!sp

p !"0.83–0.86# GeV, as can be seen in Fig. 4. For comparison,we also show as orange and blue bands what wouldcorrespond to the effective VMD meson resonancemeff [39], using m! ! 0.775 GeV, !! ! 0.148 GeV,m" ! 0.783 GeV, !" ! 0.008 GeV, m# ! 1.019 GeV,and !# ! 0.004 GeV. The bands represent the range ofsuch mass values due to the half-width rule [40–42], i.e.,meff $ !eff=2. We obtain meff ! 0.732"71# GeV for the $case and meff ! 0.822"58# GeV for the $0, with errors dueto the half-width rule. Notice that raising the poles lowersthe LEPs (slope and curvature) and vice versa. As shown,fitting spacelike data does not produce an accurate deter-mination of the resonance poles as already indicated in

Refs. [25,26,43,44]. Thus, we do not recommend to applythis method for such determinations. That includes the useof VMD fits to determine the resonance parameters. Analternative model-independent procedure of extractingthese parameters using PAs can be found in Ref. [45].To reproduce the asymptotic behavior of the TFFs, we

have also considered the PNN"Q2# sequence (second row in

Tables I and II). The results obtained are in nice agreementwith our previous determinations. The best fits are shownas black solid lines in Fig. 1. We reach N ! 2 for the $ caseand N ! 1 for the $0. Since these approximants containthe correct high-energy behavior built in, they can beextrapolated up to infinity (black dashed lines in Fig. 1) andthen predict the leading 1=Q2 coefficient:

P22 Q 2

P15 Q 2

P12 Q 2

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

Q2 GeV2

Q2 F

Q2

GeV

P11 Q 2

P15 Q 2

P16 Q 2

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

Q2 GeV2

Q2 F

'Q

2G

eV

FIG. 1 (color online). $ (left panel) and $0 (right panel) TFF best fits. Blue dashed lines show our best PL1 "Q2# when the measured two-

photon partial decay widths are not included in the fits, green dot-dashed lines show our best PL1 "Q2# when the two-photon widths are

included, and black solid lines show our best PNN"Q2# in the latter case. Black dashed lines display the extrapolation of the PN

N"Q2# atQ2 ! 0 and Q2 ! !. Experimental data points are from CELLO (red circles) [32], CLEO (purple triangles) [33], L3 (blue diamonds)[34], and BABAR (orange squares) [35] Collaborations.

TABLE I. Low-energy parameters for the $ and $0 TFFs obtained from the PA fits to experimental data without including the measuredtwo-photon partial decay widths. The first column indicates the type of sequence used for the fit and N is the highest order reached withthat sequence. The last row shows the weighted average result for each LEP. We also present the quality of the fits in terms of %2=DOF(degrees of freedom). Errors are only statistical and symmetrized.

$ TFF $0 TFFN b$ c$ F$&&"0# GeV!1 %2=DOF N b$0 c$0 F$0&&"0# GeV!1 %2=DOF

PN1 "Q2# 2 0.45(13) 0.20(12) 0.235(53) 0.79 5 1.25(16) 1.57(42) 0.339(17) 0.70

PNN"Q2# 1 0.36(6) 0.13(4) 0.201(28) 0.78 1 1.19(6) 1.42(15) 0.332(15) 0.68

Final 0.45(13) 0.20(12) 0.235(53) 1.25(16) 1.57(42) 0.339(17)

TABLE II. Low-energy parameters for the $ and $0 TFFs obtained from the PA fits to experimental data including the measured two-photon partial decay widths. The first column indicates the type of sequence used for the fit and N is the highest order reached with thatsequence. The last row shows the weighted average result for each LEP. We also present the quality of the fits in terms of %2=DOF. Errorsare only statistical and symmetrized.

$ TFF $0 TFFN b$ c$ %2=DOF N b$0 c$0 %2=DOF

PN1 "Q2# 5 0.58(6) 0.34(8) 0.80 6 1.30(15) 1.72(47) 0.70

PNN"Q2# 2 0.66(10) 0.47(15) 0.77 1 1.23(3) 1.52(7) 0.67

Final 0.60(6) 0.37(10) 1.30(15) 1.72(47)

ESCRIBANO, MASJUAN, AND SANCHEZ-PUERTAS PHYSICAL REVIEW D 89, 034014 (2014)

034014-4

limQ2!!

Q2F!"!""Q2# $ 0.160"24# GeV;

limQ2!!

Q2F!0"!""Q2# $ 0.255"4# GeV: (4)

We emphasize once more the importance of including themeasured two-photon partial widths in the fits, that for thecase of the ! TFF allows us to reach N $ 2 and then reducethe uncertainty drastically. Otherwise, we would haveremained at N $ 1 with errors 5 times larger.Finally, our combined weighted average results from

Table II, taking into account both types of sequences,give

b! $ 0.60"6#stat"3#sys; c! $ 0.37"10#stat"7#sys;

b!0 $ 1.30"15#stat"7#sys; c!0 $ 1.72"47#stat"34#sys; (5)

where the second error is systematic (of the order of 5% and20% for bP and cP, respectively). When the spread ofcentral values considered for the weighted averaged resultis larger than the error after averaging, we enlarge this error

to cover that spread5 [36]. Equation (5) represents the mainresults of this work. For the case of the !0, with the PN

N"Q2#sequence we could only reach N $ 1, which turns out to bethe first element on the PL

1 "Q2# sequence. The first elementof each sequence is the worst and should not be taken forfinal averaged results.For the !, the slope of the TFF obtained in Eq. (5) can be

compared with b! $ 0.428"89# from CELLO [32] andb! $ 0.501"38# from CLEO [33]. The TFF was alsomeasured in the timelike region with the results b! $0.57"12# from Lepton-G [46], b! $ 0.585"51# from NA60[47], b! $ 0.58"11# from A2 [48], and b! $ 0.68"26#from WASA [49]. Recently, the A2 Collaboration reportedb! $ 0.59"5# [50], the most precise experimental extractionup to date. For the !0, the slope in Eq. (5) can be comparedwith b!0 $ 1.46"23# from CELLO [32], b!0 $ 1.24"8# fromCLEO [33], and b!0 $ 1.6"4# from the timelike analysis bythe Lepton-G Collaboration (cited in Ref. [39]). One should

P11 P21 P31 P41 P51 CELLO0.2

0.3

0.4

0.5

0.6

0.7

b

P11 P21 P31 P41 P51 P61 CELLO

1.0

1.5

2.0

b'

FIG. 2 (color online). Slope predictions for the ! (left panel) and !0 (right panel) TFFs using the PL1 "Q2# up to L $ 5 for the ! and

L $ 6 for the !0, respectively (blue circles). The internal bands correspond to the statistical error of the different fits and the external onesare the combination of statistical and systematic errors determined as explained in the main text. The CELLO determination is alsoshown for comparison (empty red squares).

P11 P21 P31 P41 P51 CELLO0.0

0.1

0.2

0.3

0.4

0.5

c

P11 P21 P31 P41 P51 P61 CELLO

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

c'

FIG. 3 (color online). Curvature predictions for the ! (left panel) and !0 (right panel) TFFs using the PL1 "Q2# up to L $ 5 for the ! and


5We thank C. F. Redmer for discussions on the averageprocedure.

! AND !0 TRANSITION FORM FACTORS … PHYSICAL REVIEW D 89, 034014 (2014)

034014-5

limQ2!!

Q2F!"!""Q2# $ 0.160"24# GeV;

limQ2!!

Q2F!0"!""Q2# $ 0.255"4# GeV: (4)










P11 P21 P31 P41 P51 CELLO0.2

0.3

0.4

0.5

0.6

0.7

b

P11 P21 P31 P41 P51 P61 CELLO

1.0

1.5

2.0

b'



P11 P21 P31 P41 P51 CELLO0.0

0.1

0.2

0.3

0.4

0.5

cP11 P21 P31 P41 P51 P61 CELLO

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

c'





034014-5

CELLO

CLEO

BABAR

limQ2!1

Q2F⌘�⇤�(Q2, 0) = 0.160(24)GeV


η’-TFFFit to Space-like data: CELLO’91, CLEO’98, L3’98, BABAR’11+�⌘0!��

PN1 (Q2)

PNN (Q2)

up to N=5

up to N=1 limQ2!1

Q2F⌘0�⇤�(Q2, 0) = 0.254(4)GeV

Pere Masjuan 14


fitted poles range !!!!!spp ! "0.71–0.77# GeV and !!!!!sp

p !"0.83–0.86# GeV, as can be seen in Fig. 4. For comparison,we also show as orange and blue bands what wouldcorrespond to the effective VMD meson resonancemeff [39], using m! ! 0.775 GeV, !! ! 0.148 GeV,m" ! 0.783 GeV, !" ! 0.008 GeV, m# ! 1.019 GeV,and !# ! 0.004 GeV. The bands represent the range ofsuch mass values due to the half-width rule [40–42], i.e.,meff $ !eff=2. We obtain meff ! 0.732"71# GeV for the $case and meff ! 0.822"58# GeV for the $0, with errors dueto the half-width rule. Notice that raising the poles lowersthe LEPs (slope and curvature) and vice versa. As shown,fitting spacelike data does not produce an accurate deter-mination of the resonance poles as already indicated in

Refs. [25,26,43,44]. Thus, we do not recommend to applythis method for such determinations. That includes the useof VMD fits to determine the resonance parameters. Analternative model-independent procedure of extractingthese parameters using PAs can be found in Ref. [45].To reproduce the asymptotic behavior of the TFFs, we

have also considered the PNN"Q2# sequence (second row in

Tables I and II). The results obtained are in nice agreementwith our previous determinations. The best fits are shownas black solid lines in Fig. 1. We reach N ! 2 for the $ caseand N ! 1 for the $0. Since these approximants containthe correct high-energy behavior built in, they can beextrapolated up to infinity (black dashed lines in Fig. 1) andthen predict the leading 1=Q2 coefficient:

P22 Q 2

P15 Q 2

P12 Q 2

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

Q2 GeV2

Q2 F

Q2

GeV

P11 Q 2

P15 Q 2

P16 Q 2

0 10 20 30 400.00

0.05

0.10

0.15

0.20

0.25

0.30

Q2 GeV2

Q2 F

'Q

2G

eV

FIG. 1 (color online). $ (left panel) and $0 (right panel) TFF best fits. Blue dashed lines show our best PL1 "Q2# when the measured two-

photon partial decay widths are not included in the fits, green dot-dashed lines show our best PL1 "Q2# when the two-photon widths are

included, and black solid lines show our best PNN"Q2# in the latter case. Black dashed lines display the extrapolation of the PN

N"Q2# atQ2 ! 0 and Q2 ! !. Experimental data points are from CELLO (red circles) [32], CLEO (purple triangles) [33], L3 (blue diamonds)[34], and BABAR (orange squares) [35] Collaborations.

TABLE I. Low-energy parameters for the $ and $0 TFFs obtained from the PA fits to experimental data without including the measuredtwo-photon partial decay widths. The first column indicates the type of sequence used for the fit and N is the highest order reached withthat sequence. The last row shows the weighted average result for each LEP. We also present the quality of the fits in terms of %2=DOF(degrees of freedom). Errors are only statistical and symmetrized.

$ TFF $0 TFFN b$ c$ F$&&"0# GeV!1 %2=DOF N b$0 c$0 F$0&&"0# GeV!1 %2=DOF

PN1 "Q2# 2 0.45(13) 0.20(12) 0.235(53) 0.79 5 1.25(16) 1.57(42) 0.339(17) 0.70

PNN"Q2# 1 0.36(6) 0.13(4) 0.201(28) 0.78 1 1.19(6) 1.42(15) 0.332(15) 0.68

Final 0.45(13) 0.20(12) 0.235(53) 1.25(16) 1.57(42) 0.339(17)

TABLE II. Low-energy parameters for the $ and $0 TFFs obtained from the PA fits to experimental data including the measured two-photon partial decay widths. The first column indicates the type of sequence used for the fit and N is the highest order reached with thatsequence. The last row shows the weighted average result for each LEP. We also present the quality of the fits in terms of %2=DOF. Errorsare only statistical and symmetrized.

$ TFF $0 TFFN b$ c$ %2=DOF N b$0 c$0 %2=DOF

PN1 "Q2# 5 0.58(6) 0.34(8) 0.80 6 1.30(15) 1.72(47) 0.70

PNN"Q2# 2 0.66(10) 0.47(15) 0.77 1 1.23(3) 1.52(7) 0.67

Final 0.60(6) 0.37(10) 1.30(15) 1.72(47)

ESCRIBANO, MASJUAN, AND SANCHEZ-PUERTAS PHYSICAL REVIEW D 89, 034014 (2014)

034014-4

limQ2!!

Q2F!"!""Q2# $ 0.160"24# GeV;

limQ2!!

Q2F!0"!""Q2# $ 0.255"4# GeV: (4)










P11 P21 P31 P41 P51 CELLO0.2

0.3

0.4

0.5

0.6

0.7

b

P11 P21 P31 P41 P51 P61 CELLO

1.0

1.5

2.0

b'



P11 P21 P31 P41 P51 CELLO0.0

0.1

0.2

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0.4

0.5

c

P11 P21 P31 P41 P51 P61 CELLO

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

c'





034014-5

limQ2!!

Q2F!"!""Q2# $ 0.160"24# GeV;

limQ2!!

Q2F!0"!""Q2# $ 0.255"4# GeV: (4)










P11 P21 P31 P41 P51 CELLO0.2

0.3

0.4

0.5

0.6

0.7

b

P11 P21 P31 P41 P51 P61 CELLO

1.0

1.5

2.0

b'



P11 P21 P31 P41 P51 CELLO0.0

0.1

0.2

0.3

0.4

0.5

c

P11 P21 P31 P41 P51 P61 CELLO

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

c'





034014-5

CELLO

CLEO

BABAR


NEW DETERMINATION OF THE ! TRANSITION FORM . . . PHYSICAL REVIEW C 89, 044608 (2014)

TABLE I. Results of this experiment for the ! TFF, |F!|2, as a function of the invariant mass m(e+e!).

m(e+e!) (MeV/c2) 45 ± 5 55 ± 5 65 ± 5 75 ± 5 85 ± 5 95 ± 5|F!|2 0.999 ± 0.031 0.988 ± 0.029 1.005 ± 0.030 0.999 ± 0.031 1.051 ± 0.034 1.014 ± 0.036m(e+e!) (MeV/c2) 110 ± 10 130 ± 10 150 ± 10 170 ± 10 190 ± 10 210 ± 10|F!|2 1.014 ± 0.028 1.019 ± 0.037 1.071 ± 0.041 1.153 ± 0.044 1.083 ± 0.046 1.161 ± 0.056m(e+e!) (MeV/c2) 230 ± 10 250 ± 10 270 ± 10 290 ± 10 310 ± 10 330 ± 10|F!|2 1.312 ± 0.068 1.214 ± 0.076 1.342 ± 0.094 1.393 ± 0.113 1.487 ± 0.144 1.406 ± 0.170m(e+e!) (MeV/c2) 350 ± 10 370 ± 10 390 ± 10 410 ± 10 430 ± 10 450 ± 10|F!|2 1.851 ± 0.235 2.086 ± 0.306 1.918 ± 0.433 2.05 ± 0.61 2.56 ± 0.87 2.83 ± 1.58

measurement is

"!2 = (1.95 ± 0.15stat ± 0.10syst) GeV!2, (3)

which is in very good agreement within the errors with allrecent results reported in Refs. [7–9]. As seen in Fig. 10, the|F!(mll)|2 results of this work are in similar good agreementwithin the error bars with the data points from Refs. [7,8].

The uncertainty reached for the "!2 value in the presentwork is smaller than those of all previous measurements basedon the ! " e+e!# decay, is of a similar magnitude as theNA60 value from peripheral In–In data [8], and still yields tothe latest, preliminary result of the NA60 from p-A collisions[9].

In Fig. 10, the results of this work for |F!(mll)|2 are alsocompared to three different theoretical predictions. Becauseall models assume that |F!(mll = 0)|2 = 1, for a bettercomparison, the fit to the data points from Fig. 9(b) is rescaledby setting its normalization parameter to p0 = 1 and leavingits second parameter p1, reflecting the slope parameter "!2,unchanged. The calculation by Terschlusen and Leupold (TL)combines the vector-meson Lagrangian proposed in Ref. [26]and recently extended in Ref. [27], with the Wess-Zumino-Witten contact interaction [23] (see also Ref. [28] for thecorresponding case of the $0 TFF). Their calculation agreesvery well with the standard VMD form factor. As seen, the TLcalculation [shown in Fig. 10(a) by a dash-dotted line] goesslightly lower than the pole-approximation [Eq. (2)] fit to the

present data, whereas it fully describes the data points withinthe error bars.

The second calculation is based on a model-independentmethod using Pade approximants that was developed forthe $0 TFF in Ref. [29]. Using spacelike data (CELLO[30], CLEO [31], BABAR [32]), this method provides aparametrization that is also suited to describe data in themll range from zero to

#0.4 GeV/c2, and thus provides a

model-independent prediction for the timelike TFF [24]. Overthe full mll range, this calculation [shown in Fig. 10(a) by ared dashed line with an error band] practically overlaps withthe pole-approximation fit to the present data points.

In another recent calculation [25] by the Julich group,the connection between the radiative decay ! " $+$!# andthe isovector contributions of the ! " # # $ TFF is exploitedin a model-independent way, using dispersion theory (DT).This calculation [shown in Fig. 10(b) by a dotted line withan error band] goes slightly above the fit to the presentdata.

Currently, the VMD models that are used to calculatethe contribution of the hadronic light-by-light scatteringto (g ! 2)µ include only %, &, and ' resonances. Thesecontributions are calculated with " = (774 ± 29) MeV closeto the %-meson mass. This value of " was determined from afit to spacelike data measured by the CLEO collaboration [31]down to the momentum transfer q2 = !1.5 GeV2, which is faraway from q2 = 0 GeV2. The " value from CLEO disagreeswith the VMD value, " = 745 MeV. It also disagrees with

]2) [GeV/c-l+m(l0 0.1 0.2 0.3 0.4 0.5

2 | !|F

1

(a)This Work: DataThis Work: Fit (p0=1)

A2, 2011

TL calculation

approxim.ePad

]2) [GeV/c-l+m(l0 0.1 0.2 0.3 0.4 0.5

2 | !|F

1

(b)This Work: Data

This Work: Fit (p0=1)

NA60, In-In

DT calculation

FIG. 10. (Color online) Results of this work (solid squares) for the ! TFF, |F!(mll)|2, compared to other recent measurements and theoreticalpredictions: former data of the A2 Collaboration [7] [open circles in (a)] and the NA60 in peripheral In-In data [8] [open squares in (b)],calculations of Ref. [23] [dash-dotted line in (a)], Ref. [24] [red dashed line with an error band in (a)], and Ref. [25] [dotted line with an errorband in (b)]. The solid line is the fit from Fig. 9(b) rescaled so that p0 = 1.

044608-9

• Study Dalitz decays η(‘)→γ*γ→e+e-γ

• Prediction of the time-like from space-like data

15

[A2 Coll. PRC89 2014]

A2@MAMI

time-like TFFPredictive method!

Pere Masjuan

Ê Ê ÊÊ

ÊÊ

Ê

Ê

0.0 0.2 0.4 0.6 0.8

1

2

5

10

20

50

MHe+e-L @GeVêc2D

H»FHq2 L»L2

[BESIII Coll. PRD92 (2015)]

⌘0 ! e+e��

⌘ ! e+e��

BESIIIPA predict


η-TFF�⌘!��Fit to Space-like data [CELLO’91, CLEO’98, BABAR’11]+

+ Time-like data [NA60’09, A2’11, A2’13]

PN1 (Q2)

PNN (Q2)

up to N=7

up to N=2

Pere Masjuan


ÚÚÚÚÚÚÚÚÚÚÚ

Ú‡‡ ‡ ‡ ‡ ‡

‡‡

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ü

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ü

ü

¯¯¯¯¯¯¯¯¯¯¯

¯¯

-0.20 -0.15 -0.10 -0.05 0.00

-0.15

-0.10

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0.00

P22 HQ 2 L

P17 HQ 2 L

0. 10. 20. 30. 40.

-0.1

0.

0.1

0.2

Q2 @GeV2D

Q2 »F h

gg* HQ2L»@

GeVD

·

· ·

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P11 P21 P31 P41 P51 P61 P71

0.50

0.55

0.60

0.65

b h

16

limQ2!1

Q2F⌘�⇤�(Q2, 0) = 0.177(15)GeV


η’-TFF

PN1 (Q2) up to N=7

Pere Masjuan

[R.Escribano, S. Gonzalez-Solis, P.M., P. Sanchez-Puertas, ’15]

17

ÚÚÚÚÚ

Ú

Ú

ÚÚ

ü üü

ü üü

üü ü

üü

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Ï

Ï

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0 5 10 15 20 25 30 350.00

0.05

0.10

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0.20

0.25

0.30

Q2 @GeV2D

Q2 Fh'gg* HQ2L@G

eVD

BaBar

CLEO

CELLO

L3

Joint Fit SL+TL @P11HQ2LDJoint Fit SL+TL @P16HQ2LDJoint Fit SL+TL @P17HQ2LD

ÚÚ

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P11 P12 P13 P14 P1

5 P16 P1

71.0

1.1

1.2

1.3

1.4

1.5

1.6

b h'

SL fitJoint fit SL+TL

Crucial to extract the most precise η-η’ mixing

�⌘!��Fit to Space-like data [CELLO’91, CLEO’98, BABAR’11]++ Time-like data [BESIII’15]

’

[R.Escribano, S. Gonzalez-Solis, P.M., P. Sanchez-Puertas, ’15]


PS-TFF

18Pere Masjuan


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0.00

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P17 HQ 2 L

0. 10. 20. 30. 40.

-0.1

0.

0.1

0.2

Q2 @GeV2D

Q2 »F h

gg* HQ2L»@

GeVD

space-like and time-like data

Low-energy parametersup to the third derivative!

mixing parameters of the η-η’ system

Rare decays

CELLOCLEO

BABARNA60

A2

fq, fs,�

PS continuum on quarkonium

region

e�

e�

�⇤

� beautiful synergy experiment - theory

P

(g-2)μ

�P!l+l� �P!4l


⇥ F⇡0�⇤�⇤(q21 , (q1 + q2)2)F⇡0�⇤�⇤(q22 , 0)

q22 �M2⇡

T1(q1, q2; p)

+F⇡0�⇤�⇤(q21 , q

22)F⇡0�⇤�⇤((q1 + q2)2, 0)

(q1 + q2)2 �M2⇡

T2(q1, q2; p)

!

q1q1+q2

q2

q2 q1

q1

q1+q2q2 q1

q1+q2

q1+q2

q2

Use data fromthe pion Transition Form Factor

19

The anomalous magnetic moment of the muonThe role of experimental data

From Knecht and Nyffeler, ’01:

aHLBL;⇡0

µ = �e6Z

d4q1(2⇡)4

Zd4q2(2⇡)4

1

q21q22(q1 + q2)2[(p+ q1)2 �m2][(p� q2)2 �m2]


20

The anomalous magnetic moment of the muonThe role of experimental data

` = e, µP = ⇡0, ⌘, ⌘0

Using largest set ever:- Space-like region

[L3,CLEO,CELLO,BABAR,BELLE]- Time-like region

[NA48,A2,NA62+PDG]

P ! `+`�

P ! `+`��

e+e� ! e+e�P

[P.M., Sanchez-Puertas ’17]

[13 different coll.]

+

15

a

HLbL;l.q.µ C

01 C

12 [a

minP ;1,1] C

12 [a

maxP ;1,1]

l.q. 60.4(1.5)(0.5)[1.6] 57.2(1.8)[1.8] 57.3(1.4)(1.0)[1.8]

l.q. norm 67.4(1.7)(0.5)[1.8] 63.8(2.0)[2.0] 63.9(1.6)(1.1)[1.9]

TABLE IV. The analog results to those for a

HLbL;⇡0

µ (10�11

units) in Table II, but employing the light-quark TFF. Seedetails in the text.

Appendix E: Beyond pole contribution

There is at present a debate on how to deal with theHLbL tensor high-energy behavior dictated by the OPE(cf. discussion in [3] with respect to [19] and the sum-mary talk by Vainshtein in [6]). Whereas we do not wantto enter this debate, in this appendix we discuss how ourapproach could be used by both approaches [3, 19]. Thefirst approach [19] proposes to modify the ⇡

0-pole con-tribution to a

µ

as such that certain OPE constraint tothe hV V V V i Green’s function is satisfied. Its modifica-tion results on setting the external TFF (the gray blobconnected to the external photon in Fig. 2) to a constantone. Following such prescription and using our descrip-tion for the pseudoscalar TFFs, we obtain the results fora

HLbL;Pµ

shown in Table V. The larger errors obtained

a

HLbL;Pµ C

01 C

12 [a

minP ;1,1] C

12 [a

maxP ;1,1]

⇡

0 84.9(1.8)(2.6)[3.2] 82.8(1.7)[1.7] 80.9(1.3)(0.5)[1.4]

⌘ 29.1(1.0)(0.3)[1.0] 27.3(1.4)[1.4] 26.9(1.5)(1.0)[1.8]

⌘

0 30.4(1.0)(0.5)[1.1] 26.8(1.1)[1.1] 25.8(0.7)(0.9)[1.1]

Total 144.4[3.5] 136.9[2.5] 133.6[2.5]

TABLE V. The results for aHLbL;Pµ in units of 10�11 according

to the procedure in Ref. [19]. The errors and labeling areidentical to those in Table II.

now are proportional to the larger central values with,essentially, the same proportionality than our main re-sult. Accounting for the systematic error as we did inSection V, we would obtain

a

HLbL;Pµ

= 135(11)⇥ 10�11, (E1)

to be compared with the result from Ref. [19] aHLbLµ

=(76.5+18+18)⇥10�11 ! 114⇥10�11. This comparisonillustrates again the potential large systematic errors —beyond 10% — typical of resonance models. Actually,the result from [19] was used in the Glasgow consensus toobtain the reference value 10.5(2.6). If we would replacetheir pseudoscalar-pole contribution by our Eq. E1, thefinal result would be

a

HLbLµ

= 126(25)⇥ 10�11, (E2)

one sigma larger, and in better agreement with ballparkestimates.

The second approach [3] provides instead a model forthe pseudoscalar contribution (not only the lightest pseu-doscalar poles), related to the hV V P i Green’s function.

It would be possible within our approach to reconstructan analogous Green’s function. In this scenario to sat-isfy all the constraints imposed by the OPE, one shouldstart directly with the C

12 (Q

21, Q

22, (Q1 +Q2)2) since the

N = 0 is too limited. Then, we cannot provide a sys-tematic error and check on convergence. Besides, furtherconstraints on this hV V P i Green function would be de-sired. For these reasons, we decide not to give a value forsuch scenario. In any case, we remind that this approach,as well as the previous one, were inspired in the ⇡

0 TFFmodel in Ref. [10] which, as said, entails non-accountedsystematic errors.

Appendix F: Stieltjes functions

A function is said to be of the Stieltjes kind if it admitsan integral representation [74]

f(q2) =

Z 1/R

0

d�(u)

1� uq

2, (F1)

where �(u) is any bounded and nondecreasing func-tion [74]. To see that such is the case for the isovec-tor contribution to the TFF in Refs. [65, 84, 115], let

R = 4m2⇡

, and define d�(u) = const.⇥ q

2

⇡

ImF (1/u)u

; mak-ing the change of variables u = 1/s, Eq. (F1) returns theonce-substracted dispersive representation of the isovec-tor contribution discussed in Ref. [84], and also exploitedin Refs. [65, 115], once ImF (s) = �

3(s)P (s)|FV

(s)|2 isidentified. Since �(s) =

p1� 4m2

⇡

/s, P (s) is a linearpolynomial with positive slope and F

V

(s) the ⇡

± vectorFF, then ImF (s) is a positive function, the requirementof �(u) to be nondecreasing is fulfilled and the conver-gence of PAs to the TFF is guaranteed.22

Appendix G: Dispersion Relations

In this appendix we develop our statements concerningpotential drawbacks of dispersive approaches for extend-ing the TFF representation into the SL region beyondenergies of the order of 1 GeV. For this purpose, we em-ploy a simplified approach inspired from Ref. [65]. Specif-ically, we take the definition in Eq. (17) of that referencefor the once-subtracted dispersion relation for the ⌘ TFF,while we adopt a simpler but reasonable description forthe ⇡

± vector form factor based on Refs. [116, 117].The result obtained for the TL region (from q

2 = 0 upto q

2 = m

2⌘

), accessible in the ⌘ ! �

¯ Dalitz decays,is illustrated in the upper panel of Fig. 7, and shows anice agreement with existing data even though the ap-proximations performed; this nice overlap contrasts with

22 If the function f(z) is a Stieltjes function, its n

th-subtractedversion is a Stieltjes function as well [74].

adding the rest from Glasgow Consensus

vs

aHLBL,⇡0

µ = 81.8(1.7)[4.0] · 10�11

aHLBL,⌘µ = 27.1(1.8)[2.2] · 10�11

aHLBL,⌘0

µ = 26.3(1.1)[4.6] · 10�11

aHLBL,GCµ = 105(26) · 10�11

aHLBLµ = 121(15) · 10�11


Dominant piece of HLBL under control

B→π FF

Pere Masjuan 21

d�(B ! ⇡`⌫`)

dq2=

G2F |Vub|2

192⇡3m3B

�3/2|F+(q2)|2

Overview of FF parameterizations

The spectrum in is given by:q2

The relevant form factor for the decay is defined ( ):B ! ⇡`⌫` m` ! 0

h⇡(p⇡)|V µ|B(pB)i = F+(q2)

✓pµB + pµ⇡ � m2

B �m2⇡

q2qµ

◆

( as a Ref. Minireview from PDG)

Semileptonic B → ⇡`⌫` decay

d�(B → ⇡`⌫`)dq2 = G2

F �Vub�2192⇡3m3

B

((m2B +m2

⇡ − q2)2 − 4m2Bm2




we want Vub

B→π FF

Pere Masjuan 22

d�(B ! ⇡`⌫`)

dq2=

G2F |Vub|2

192⇡3m3B

�3/2|F+(q2)|2


F (q2) =ResF (q2 = sp)

q2 � sp+

1

⇡

Z 1

sth

ds0ImF (s0)

s0 � q2 � i"

Obey a dispersion relation → Stieltjes function → convergence theoremsth = (mB +m⇡)

2

0 < q2 < (mB �m⇡)2

ResF+(q2 = m2

B⇤) / mB⇤fB⇤gB⇤B⇡

B ! ⇡`⌫` with

The spectrum in is given by:q2

The relevant form factor for the decay is defined ( ):B ! ⇡`⌫` m` ! 0

h⇡(p⇡)|V µ|B(pB)i = F+(q2)

✓pµB + pµ⇡ � m2

B �m2⇡

q2qµ

◆

( as a Ref. Minireview from PDG)

Semileptonic B → ⇡`⌫` decay

d�(B → ⇡`⌫`)dq2 = G2

F �Vub�2192⇡3m3

B

((m2B +m2

⇡ − q2)2 − 4m2Bm2




(exp. data below threshold)

F+(q2) =

F+(0)

1� q2/m2B⇤

...

we want Vub

B→π FF

Pere Masjuan 23

d�(B ! ⇡`⌫`)

dq2=

G2F |Vub|2

192⇡3m3B

�3/2|F+(q2)|2


F+(q2) =

r11� q2/m2

B⇤+

r21� q2/m2

B⇤0

F+(q2) =

F+(0)

1� q2/m2B⇤

VMD (1 parameter):

Becirevic, Kaidalov ’99 (2 param):

or

Ball, Zwicky ’04 (2 param):

F+(q2) = F+(0)

✓1

1� q2/m2B⇤

+rq2/m2

B⇤

(1� q2/m2B⇤)(1� ↵q2/m2

B⇤)

◆

F+(q2) =

F+(0)

(1� q2/m2B⇤)(1� ↵q2/m2

B⇤)


B→π FF

Pere Masjuan 24Hadron 2017, Salamanca, Sept. 28th

d�(B ! ⇡`⌫`)

dq2=

G2F |Vub|2

192⇡3m3B

�3/2|F+(q2)|2


F+(q2) =

1

P (q2)�(q2, q20)

1X

n=0

ak(q20)[z(q

2, q20)]k z(q2, q20) =

pt+ � q2 �

pt+ � q20p

t+ � q2 +p

t+ � q2

t+ = (mB +m⇡)2

P (q2) = z(q2,m2B⇤)

F+(q2) =

1

1� q2/m2B⇤

KX

n=0

bk(t0)[z(q2, q20)]

k

Boyd, Grinstein, Lebed ’95,’97 (z-parameterization):

Bourrely, Caprini, Lellouch ’09 (alternative z-parameterization):

“unitary constraint”KX

n=0

bk(t0) 1KX

n=0

bk(t0) ⇠ 0

(in practice)

B→π FF

Pere Masjuan 25

d�(B ! ⇡`⌫`)

dq2=

G2F |Vub|2

192⇡3m3B

�3/2|F+(q2)|2


F+(q2) =

1

P (q2)�(q2, q20)

1X

n=0

ak(q20)[z(q

2, q20)]k

F+(q2) =

1

1� q2/m2B⇤

KX

n=0

bk(t0)[z(q2, q20)]

k

F+(q2) =

F+(0)

(1� q2/m2B⇤)(1� ↵q2/m2

B⇤)

F+(q2) = F+(0)

✓1

1� q2/m2B⇤

+rq2/m2

B⇤

(1� q2/m2B⇤)(1� ↵q2/m2

B⇤)

◆

1)

II)

III)

IV)

All of them have something in common


B→π FF

Pere Masjuan 26

d�(B ! ⇡`⌫`)

dq2=

G2F |Vub|2

192⇡3m3B

�3/2|F+(q2)|2


F+(q2) =

1

P (q2)�(q2, q20)

1X

n=0

ak(q20)[z(q

2, q20)]k

F+(q2) =

1

1� q2/m2B⇤

KX

n=0

bk(t0)[z(q2, q20)]

k

F+(q2) =

F+(0)

(1� q2/m2B⇤)(1� ↵q2/m2

B⇤)

F+(q2) = F+(0)

✓1

1� q2/m2B⇤

+rq2/m2

B⇤

(1� q2/m2B⇤)(1� ↵q2/m2

B⇤)

◆

1)

II)

III)

IV)

All of them are Padé approximants

Partial Padé

Partial Padé

Padé Type

Padé Type

[Baker,’95]


B→π FF

Pere Masjuan 27

d�(B ! ⇡`⌫`)

dq2=

G2F |Vub|2

192⇡3m3B

�3/2|F+(q2)|2


F+(q2) =

1

P (q2)�(q2, q20)

1X

n=0

ak(q20)[z(q

2, q20)]k

F+(q2) =

1

1� q2/m2B⇤

KX

n=0

bk(t0)[z(q2, q20)]

k

F+(q2) =

F+(0)

(1� q2/m2B⇤)(1� ↵q2/m2

B⇤)

F+(q2) = F+(0)

✓1

1� q2/m2B⇤

+rq2/m2

B⇤

(1� q2/m2B⇤)(1� ↵q2/m2

B⇤)

◆

1)

II)

III)

IV)

All of them are Padé approximants

Partial Padé

Partial Padé

Padé Type

Padé Type

[Baker,’95]

Non of them usePadé Theory


B→π FF

Pere Masjuan 28Hadron 2017, Salamanca, Sept. 28th

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ÏÏ

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Ï

ÏÏ

——

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Ú Ú

ÚÚ Ú

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Ú

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ÚÚ

ÙÙ

——

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dBêdq2â106@Ge

V-2 D

Belle 2013 tagged B0 H13 binsLBelle 2013 tagged B- H7 binsL

This work @P11D

Figure 2: Individual fits to BaBar11 (left-top panel), BaBar12 (right-top panel), Belle11(left-down panel) and Belle13 (right-down panel) experimental data sets on the B æ fi¸‹¸

decay spectra.

to be minimized reads

‰2 = ‰2data +

q5p,q=1 �HPQCD07

p (CovHPQCD07pq )≠1�HPQCD07

q

+q12

p,q=1 �MILC08p (CovMILC08

pq )≠1�MILC08q

+q3

p,q=1 �UKQCD15p (CovUKQCD15

pq )≠1�UKQCD15q

+q12

p,q=1 �MILC15p (CovMILC15

pq )≠1�MILC15q ,

(19)

where ‰2data corresponds to Eq. (17) and with

�latticek = F lattice

+ (q2) ≠ P MN (q2) , (20)

accounting for the lattice QCD form factor predictions. The results derived from theminimization of Eq. (19) are collected in Table 2. Di�erent to Table 1, in this tablewe show the results as obtained with each element of the corresponding sequences

14

PA sequence stopped: complex-conjugated poles! → Apply Stieltjes convergence!

(noise filtering)


B ! ⇡`⌫` FF from lattice+experiment

Ê

Ê

Ê

Ê

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Á

Á

Á

Á

Á

Á

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ÁÁ

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-2

-1

0

1

2

3

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s

Deviation of BaBar 2011 dataFit to all dataFit to BaBar 2011 data only

‡‡ ‡

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‡

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‡

‡

‡

‡

‡· ··

·

·

·

·

·

·

·

··

‡‡

··

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-1

0

1

2

3

q2 @GeV2D

s

Deviation of BaBar 2012 dataFit to all dataFit to BaBar 2012 data only

Ï

Ï

Ï

Ï ÏÏ

Ï

ÏÏ

Ï

Ï

Ï

Ï

·

·

·

· ··

·

· ·

·

·

·

·

ÏÏ

··

——

0 5 10 15 20 25 30-3

-2

-1

0

1

2

3

q2 @GeV2D

s

Deviation of Belle 2011 dataFit to all dataFit to Belle 2011 data only Ú

Ú

Ú

Ú

Ú

Ú

ÚÚ

Ú

Ú

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Ú

Ú

Ù

Ù

Ù

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Ù

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Ù

Û

Û

Û

Û

Û

Û

ÛÛ

Û

ÛÛ

Û

Û

ı

ı

ı

ı

ı

ı

ı

ÙÚÙÚ

ıÛıÛ

——

0 5 10 15 20 25 30-3

-2

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2

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q2 @GeV2D

s

Deviation of Belle 2013 dataFit to all dataFit to Belle 2013 data only

Figure 3: Deviation, in ‡, of each experimental datum with respect to our combined andindividual fits. Solid and empty geometric markers account, respectively, for the fits asgiven in Figs. 1 and 2.

going up to P 21 , P 2

2 , T 21 and P 3

1,1, respectively, together with the final results, given inthe last column, including both statistical and systematic uncertainties. While thestatistical error is directly obtained from the fit, we derive the systematic ones byascribing the di�erence of central values of the element we have stopped the sequenceand the preceding ones. We would like to notice that fixing the Bú pole the statisticalerror remains quite stable while, on the contrary, the systematic ones increases. No-ticeably, including lattice data into the fit allow us to determine |Vub| with improvedprecision reducing the associated statistical uncertainty by ≥ 80% with respect tothe previous case when only the decay spectra were fitted (cf. Table 1). In addition,the sequences with two poles, P L

2 and P M1,1, have been enlarged, respectively, by one

and two elements with respect to the previous case. Again, we find some extraneouspoles placed for the diagonal P 2

2 and P 21,1 approximants placed, respectively, far away

from the origin and pair up with a close-by zero in the denominator. One interestingresult that can be read from Table 2 is that all of the sequences we have consideredseems to converge in a hierarchical way. The coe�cients of the P 2

1 , P 22 , T 2

1 and P 31,1

approximants are gathered in Table 3 and a graphical account of the corresponding

15



B ! ⇡`⌫` FF from lattice+experiment

F+(0) = 0.263(9)

|Vub| = 3.52(8)⇥ 10�3

�2/DOF = 1.31

Ê Ê

ÊÊ

Ê

Ê

‡‡ ‡

‡

‡

‡

‡ ‡‡

‡ ‡

‡

Ï

Ï

Ï

Ï ÏÏ Ï

ÏÏ

Ï

Ï

Ï

Ï

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Ú

Ú

Ú

Ú

Ú

Ú Ú

ÚÚ Ú

Ú

Ú

Ù

Ù

Ù

Ù

Ù

Ù

Ù

È

È

È

È

ÊÊ

‡‡

ÏÏ

ÚÚ

ÙÙ

ÈÈ

——

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2

4

6

8

10

12

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dBêdq2â106@Ge

V-2 D

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This work @P12DP 22

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‡

‡

‡

‡‡

‡‡

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Ï

Ï

Ï

Ï

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Ú

0 5 10 15 20 25 300

2

4

6

8

10

12

q2 @GeV2D

F +BpHq2 L Our Fit P12

MILC 2015

RBCêUKQCD 2015MILC 2008

HPQCD 2007

Figure 5: B æ fi form factor as obtained from a combined fit to experimental data andlattice predictions on the form factor shape (cf. Fig. 4) with a P 2

1 (q2) approximant (blacksolid curve).

confidence region ellipse relating |Vub| and F Bfi+ (0) with a sizable correlation...Update

and further discuss the figure.We have also performed additional fits including the CLEO 2007 data in the ‰2.

This option is further discussed in Appendix 6 and the results presented in Table 9.As a final exercise, we would also like to explore the e�ect of fitting all experi-

mental data together with di�erent settings regarding the lattice form factor calcu-lations. These are HPQCD+RBC/UKQCD, HPQCD+RBC/UKQCD+MILC08 andHPQCD+RBC/UKQCD+MILC2015 and the results collected, respectively, in thesecond, third and fourth columns of Table 5 for the P 2

1 (q2) approximant (the othersequences lead similar results). While the second and third columns lead similar re-sults, the fourth column, including the updated MILC 2015 form factor calculations,clearly shifts upwards(downwards) the |Vub|(F Bfi

+ (0)) value and gives smaller statis-tical uncertainties. Upon comparison with last column, we conclude that MILC 2015basically drives the form factor from a theoretical perspective.

Discuss prospects of improvement in Belle-II...

Comment on P M3 Padés

5 B æ ÷(Õ)¸‹¸ decays and ÷-÷Õ mixingIn the previous section, the F Bfi

+ (q2) form factor have been parameterized by means ofPadé approximants applied to the B æ fi¸‹¸ di�erential branching ratio distribution

19

P 22

“unitary constraint is significant”

(c.c. poles)

F+(0) = 0.260(9)

|Vub| = 3.57(8)⇥ 10�3

�2/DOF = 1.11

(real poles)

WRO

NG


for the first time: all data + all lattice (preliminary)

31Pere Masjuan

• We presented a method, a TOOLKIT

• based on analyticity and unitarity

• Simple although further studies ongoing

• Approaches yes (improvable), assumptions no

• Systematic:

• easy to update with new data (or derivatives)

• error from approach

• Predictive (checkable)

Conclusions


Thanks!

Insisting on the role of experimental data€¦ · Pere Masjuan 9 A word on systematics •Consider...

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