Документ взят из кэша поисковой машины. Адрес оригинального документа : http://nuclphys.sinp.msu.ru/conf/epp10/Nesterenko.pdf Дата изменения: Sat Sep 7 16:47:54 2013 Дата индексирования: Fri Feb 28 02:51:03 2014 Кодировка: Windows-1251 Поисковые слова: ultraviolet

PRD71(2005), JPG32(2006), PRD77(2008), NPBPS234(2013), arXiv:1306.4970 [hep-ph]

Hadronic vacuum p olarization function in the framework of disp ersive approach to QCD
A.V. Nesterenko
Bogoliub ov Lab oratory of Theoretical Physics Joint Institute for Nuclear Research, Dubna, Russian Federation 16th Lomonosov Conference on Elementary Particle Physics Moscow, Russia, 22 - 28 August 2013


INTRODUCTION

Hadronic vacuum p olarization function (q 2) plays a central role in various issues of QCD and Standard

Х

Х

Й

Mo del. In particular, the theoretical description of some strong interaction pro cesses and hadronic contributions to electroweak observables is inherently based on (q 2): ћ electron-p ositron annihilation into hadrons ћ hadronic lepton decay ћ muon anomalous magnetic moment ћ running of the electromagnetic coupling
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 1


GENERAL DISPERSION RELATIONS The cross-section of e+e- hadrons: 22 ч = 4 2 3 L ч , s where s = q 2 = (p1 + p2)2 > 0,



Ф?

Х
Ц У Т Ч

ћ

Ф?

?

1 Lч = qчq - gч q 2 - (p1 - p2)ч(p1 - p2) , 2 ч = (2 )4


(p1 + p2 - p) 0 Jч(-q ) J (q ) 0 ,

and Jч =

? f Qf : q ч q : is the electromagnetic quark current. Kinematic restriction: hadronic tensor ч (q 2) assumes non-zero values only for q 2 m2, since otherwise no hadron state could b e excited
A.V.Nesterenko

Feynman (1972); Adler (1974).
2

16th Lomonosov Conference (Moscow, 2013)


The hadronic tensor can b e represented as ч = 2 Im ч , (q 2) ч (q 2) = i eiqx 0 T Jч(x) J (0) 0 d4x = i(qчq - gч q 2) . 2 12 Kinematic restriction: (q 2) has the only cut q 2 m2. Disp ersion relation for (q 2): ( ) 2, q 2) = 1 (q 2 - q 2) (q 0 d 0 2)( - q 2) 2 i C ( - q 0 R ( s) 2 - q 2) = (q ds, 0 2)(s - q 2) m2 (s - q 0
Im

A

C
r
Re

q

2

q

2

0

0

m

2

2 2 where (q 2, q0 ) = (q 2) - (q0 ) and R(s) denotes the measurable ratio of two cross-sections ( R(s) 0 for s < m2 )

1 (e+e- hadrons; s) R ( s) = lim (s + i) - (s - i) = +e- ч+ч-; s) . 2 i 0+ (e
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 3


For practical purp oses it proves to b e convenient to deal with the Adler function (Q2 = -q 2 0) R(s) d (-Q2) D(Q2) = - , D(Q2) = Q2 ds 2 2)2 d ln Q m2 (s + Q
Adler (1974); De Rujula, Georgi (1976); Bjorken (1989).

The inverse relations b etween the functions on hand read s-i 1 d R(s) = lim D(- ) , 2 i 0+ s+i
Radyushkin (1982); Krasnikov, Pivovarov (1982) Q2
B

Im

r
0

(-Q2, -Q2) = - 0

d D( ) Q2 0

m s-i
2

s+i

Re

Nesterenko (2013).

The integration contour in complex -plane lies in the region of analyticity of the integrand.
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 4


The complete set of relations b etween (q 2), R(s), and D(Q2): d R ( ) d = - D ( ) , 2)( - q 2) 2 m2 ( - q -q0 0 s-i 1 1 d R(s) = D(- ) , lim (s + i) - (s - i) = lim 2 i 0+ 2 i 0+ s+i R( ) d (-Q2) D ( Q2 ) = - = Q2 d. 2 2 )2 d ln Q m2 ( + Q Their derivation requires only the lo cation of cut of (q 2)
2 2 (q 2, q0 ) = (q 2 - q0 ) -q 2

and its UV asymptotic. Neither additional approximations nor phenomenological assumptions are involved. Nonp erturbative constraints: ћ (q 2): has the only cut q 2 m2; ћ R(s): vanishes for s < m2, emb o dies 2-terms; ћ D(Q2): has the only cut Q2 -m2, vanishes at Q2 0.
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 5


DISPERSIVE APPROACH TO QCD Functions on hand in terms of common sp ectral density:
2 - q 2 m2 - q0 d 2 2 (q 2, q0 ) = (0)(q 2, q0 ) + ( ) ln , 2 m2 - q 2 - q0 m2 d (0)(s) + (s - m2) ( ) , R(s) = R s Q2 - m2 d D(Q2) = D(0)(Q2) + 2 ( ) , 2 2 Q + m m2 +Q 1d d r ( ) 1 ( ) = Im lim p( - i) = - = Im lim d(- - i), d ln 0+ d ln 0+ 2 with (0)(q 2, q0 ), R(0)(s), and D(0)(Q2) b eing leading-order terms, p(q 2), r(s), and d(Q2) b eing the strong corrections
Nesterenko, Papavassiliou (2005, 2006); Nesterenko (2007-2013).
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 6


ћ The obtained integral representations automatically emb o dy all the aforementioned nonp erturbative constraints ћ Their derivation requires only the general disp ersion relations and the asymptotic ultraviolet b ehavior of (q 2) ћ Neither additional approximations nor mo del-dep endent assumptions were involved The leading-order terms of the functions on hand: 0 2 2 (0)(q 2, q 2) = 1- 1- - , 0 2 2 tan tan 0 tan tan 0 m2 3/2 , R(0)(s) = (s - m2) 1 - s (0)(Q2) = 1 + 3 1 - 1 + -1 sinh-1 1/2 , D
2 where sin2 = q 2/m2, sin2 0 = q0 /m2, = Q2/m2
Feynman (1972); Akhiezer, Berestetsky (1965).
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 7


Perturbative contribution to the sp ectral density: d rpert( ) 1 1d pert( ) = Im lim ppert( - i) = - = Im lim d d ln 0+ d ln 0+ 4 1 2 ( ) = + 2( /2) + 2 0 ln

pert

(- - i).

The following mo del for sp ectral density will b e employed:

Nesterenko (2011-2013).

In the massless limit (m = 0) integral representations read
q2 1 - ( /q 2) d , + ( ) ln 2 2) q0 1 - ( /q0 0 ( ) d d . R(s) = (s) 1 + ( ) , D ( Q2 ) = 1 + 2 s 0 +Q For ( ) = Im dpert(- - i0+)/ two highlighted equations b e2, q 2) = - ln - (q 0 -

come identical to those of the APT

Shirkov, Solovtsov (1997-2007).

But it is essential to keep the threshold m2 nonvanishing.
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 8


ADLER FUNCTION massless limit (m = 0)
1.5

realistic case (m = 0)
1.5

. D( Q )
1 3,4

2

D( Q )

2

2
1.0 1.0

1

2,3,4
0.5 0.5

Q, GeV
0.0

Q, GeV
0.0

.
0.5 1.0 1.5 2.0 2.5

.
0.5 1.0 1.5 2.0 2.5

0.0

0.0

Nesterenko, Papavassiliou (2006); Nesterenko (2007-2009).

Reliability of approaches: ћ Perturbation theory: Q ћ Massless APT: Q
A.V.Nesterenko

1.5 GeV

1.0 GeV
9

ћ Disp ersive approach: entire energy range
16th Lomonosov Conference (Moscow, 2013)


HADRONIC VACUUM POLARIZATION FUNCTION
1.5

(q)

-

2

1.0

0.5

q , GeV
0.0 0.5 1.0 1.5 2.0

2

2

2.5

? Solid curve presents DispQCD result for (q 2) = (0, q 2), whereas its lattice prediction is shown by data p oints.
Della Morte, Jager, Juttner, Wittig (2011); Nesterenko (2013).

DispQCD result is in a go o d agreement with lattice data in the entire energy range.
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 10


INCLUSIVE LEPTON HADRONIC DECAY The interest to this pro cess is due to ћ The only lepton with hadronic decays ћ Precise exp erimental data ћ No need in phenomenological mo dels ћ Prob es infrared hadron dynamics The exp erimentally measurable quantity: ( - hadrons- ) R = - e - ) = R , V + R , A + R , S , ( ?e
J J R ,V = R=0 + R=1 = 1.783 + 0.011 + 0.002, ,V ,V J J R ,A = R=0 + R=1 = 1.695 + 0.011 + 0.002. ,A ,A



П



ЯЮ

ЦУТЧ

ALEPH Collab oration (1998-2008).
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 11


The theoretical prediction for the quantities on hand reads Nc J =1 R ,V/A = |Vud|2 SEW V/A + EW , QCD 2 Nc = 3, |Vud| = 0.9738 + 0.0005, SEW = 1.0194 + 0.0050, EW = 0.0010,
V/A QCD

=2

2 M

m2 V/A

s R f 2 M

V/A

ds ( s) 2 , M

where M = 1.777 GeV, f (x) = (1 - x)2 (1 + 2x), 1 1 V/A V/A V/A R (s) = lim (s + i) - (s - i) = Im lim 2 i 0+ 0+
Braaten, Narison, Pich (1992); Pivovarov (1992).

V/A

(s + i)

Integration by parts leads to
QCD

2) - g ()R m2 + 1 = g (1)R(M

2 i

C1+C2 2 where = m2/M and g (x) = x(2 - 2x2 + x3).
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013)

d g D(- ) , 2 M

12


Im

Im
Only if D(Q2) p ossesses correct prop erties in Q
2
B
C 4
r

A

C
0

1

=

M

2

Re

C3

Re
m

m

2

C

2

M

2

0

2
M

2



QCD

2) - g ()R m2 + 1 = g (1)R(M

2 i

C3+C4

d g D(- ) 2 M

Despite the aforementioned remarks, in the p erturbative analysis the massless limit (m = 0) is assumed, that gives QCD
A.V.Nesterenko

- 1 2 lim = 1 - g -ei D M ei d. 2 0+ - +
16th Lomonosov Conference (Moscow, 2013) 13


Inclusive decay within p erturbative approach: Commonly, p erturbative D(Q2) is directly employed here
2) j , dj pert(Q Q2 j =1 (1) with pert(Q2) = 4 /[0 ln(Q2/2)], 0 = 11 - 2nf/3, and d1 = 1/ .

D ( Q2 )

Dpert(Q2) = 1 +
()

()

In what follows the one-lo op level ( = 1) with nf = 3 active flavors will b e assumed. The one-lo op p erturbative expression for V/A reads QCD V/A pert 4 A1() + A2() d, =1+ 2 + 2) 0 0 (

2 where = ln M /2 , A1() = 1 + 2 cos() - 2 cos(3) - cos(4), A2() = 2 sin() - 2 sin(3) - sin(4).
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 14


Perturbative approach gives Vert Aert , but V = p p exp

A

exp

:

V = 1.224 + 0.050, A = 0.748 + 0.034 [ ALEPH-2008 data ] exp exp
1.6
V QCD

1.6
ALEPH (update 2008)

A QCD

ALEPH (update 2008)

1.2

1.2

0.8

0.8

0.4

A

0.4

C

, GeV
0.0 0.5 1.0 1.5 2.0 2.5 3.0

, GeV
0.0 0.5 1.0 1.5 2.0 2.5 3.0

= 434

+117 -127

MeV

= 1652

+21 -23

MeV

no solution

V-channel: p erturbative approach gives two equally justified solutions, but only highlighted one is usually retained. A-channel: p erturbative approach fails to describ e exp erimental data on inclusive lepton hadronic decay.
A.V.Nesterenko 16th Lomonosov Conference (Moscow, 2013) 15


Inclusive decay within disp ersive approach: Description of the inclusive lepton hadronic decay within DispQCD enables one to prop erly account for ћ effects due to hadronization (m = 0) ћ nonp erturbative constraints on the functions on hand The use of initial expression for V/A with obtained ab ove QCD integral representations eventually leads to
V/A QCD

52 33 d = 1 - V/A 1 + 6V/A - V/A + V/A + H ( ) 2 8 16 M m2 V/A 12 13 2 1 + 1 - V/A - 1 , - 3V/A 1 + V/A - V/A ln 8 32 V/A
V/A

with
A.V.Nesterenko

2 = m2 /M , H (x) = g (x) (1 - x) + g (1) (x - 1) - g V/A
16th Lomonosov Conference (Moscow, 2013)

V/A

Nesterenko (2011-2013).
16


1.6

V QCD

1.6

A QCD

A

ALEPH (update 2008)

C

ALEPH (update 2008)

1.2

1.2

0.8

0.8

0.4

0.4

, GeV
0.00 0.25 0.50 0.75

, GeV
0.00 0.25 0.50 0.75

= 408 + 30 MeV

= 418 + 35 MeV

The comparison of obtained result with exp erimental data yields nearly identical values of the QCD scale parameter in vector and axial-vector channels, that testifies to the self-consistency of the develop ed approach.
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SUMMARY The integral representations for (q 2), R(s), and D(Q2) are derived within disp ersive approach to QCD These representations emb o dy the nonp erturbative constraints and retain the effects due to hadronization The obtained results are in a go o d agreement with lattice data and low-energy exp erimental predictions The develop ed approach is capable of describing exp erimental data on inclusive lepton hadronic decay in vector and axial-vector channels in a self-consistent way
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