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JPG 32, 1025 (2006) NPBPS 186, 207 (2009)

HADRONIC EFFECTS IN LOW-ENERGY QCD: ADLER FUNCTION AND DECAY A.V. Nesterenko
Bogoliub ov Lab oratory of Theoretical Physics Joint Institute for Nuclear Research, Dubna, 141980, Russia

Fourteenth Lomonosov Conference on Elementary Particle Physics Moscow, Russia, 19 - 25 August 2009


INTRODUCTION

Hadronic vacuum p olarization function (q 2) plays a crucial role in various issues of elementary particle

е

е

Щ

physics. Indeed, the theoretical description of some strong interaction pro cesses and of the hadronic contributions to electroweak observables is inherently based on (q 2): З electron-p ositron annihilation into hadrons З hadronic lepton decay З muon anomalous magnetic moment З running of the electromagnetic coupling 1


The cross-section of e+e- annihilation into hadrons reads 2 2 2 LЕ , = 4 3 Е s where s = q 2 = (p1 + p2)2 > 0,



дН

е
ж г в з

З

дО

М

1 LЕ = qЕq - gЕ q 2 - (p1 - p2)Е(p1 - p2) , 2 Е = (2 )4


(p1 + p2 - p) 0|JЕ(-q )| |J (q )|0 , ? f Qf : q Е q : stands

denotes a final hadron state, and JЕ = for the electromagnetic quark current.

It is worth stressing that Е (q 2) exists only for q 2 4m2 , since otherwise no hadron state could b e excited:
R.P.Feynman (1972); S.L.Adler, PRD10 (1974).

2


The hadronic tensor can b e represented as Е = 2 Im Е , Е (q 2) = i eiqx 0 T JЕ(x) J (0) 0 d4x = (qЕq - gЕ q 2)(q 2). The hadronic vacuum p olarization function (q 2) satisfies the once-subtracted disp ersion relation (cut for q 2 4m2 )
2 2 (q 2) = (q0 ) - q 2 - q0

where m denotes the measurable ratio of two cross-sections:

R(s) ds, 2)(s - q 2) 4m2 (s - q 0 135 MeV is the mass of the meson and R(s)



e+e- hadrons; s 1 lim (s - i) - (s + i) = R ( s) = +e- Е+Е-; s) . 2 i 0+ (e It is worth noting here that R(s) 0 for s < 4m2 b ecause of the kinematic restrictions mentioned ab ove:
R.P.Feynman (1972).

3


For practical purp oses it proves to b e convenient to deal with the so-called Adler function D(Q2) (Q2 = -q 2 0):
2 2) = d (-Q ) , D (Q d ln Q2

D(Q2) = Q2

R ( s) ds, 2 )2 4m2 (s + Q



which plays an indisp ensable role for the congruous analysis of the timelike and spacelike exp erimental data:
S.L.Adler PRD10 (1974); A.Rujula, H.Georgi PRD13 (1976); J.D.Bjorken (1989).

The inverse relation b etween D(Q2) and R(s) reads
s-i 1 d R(s) = lim D(- ) 2 i 0+ s+i
A.V.Radyushkin (1982), hep-ph/9907228; N.V.Krasnikov, A.A.Pivovarov PLB116 (1982).

Im

4m

2

s

+
. .

i

.

Re

s

i

4


Although there are no direct measurements of the Adler function, it can b e restored by employing the exp erimental data on R(s) (overall factor N D
exp

Q2 is omitted): ff s0 R ( s) R exp 2 ) = Q2 2 theor(s) (Q ds + Q ds 2 )2 2)2 4m2 (s + Q s0 ( s + Q
c
1.5

D( Q )

2

1.0

0.5

Q, GeV
0.0

.
0.5 1.0 1.5 2.0 2.5

0.0

There is also a numb er of lattice simulations, which generally agree with the shown result:
JLQCD and TWQCD Collabs., PRD79 (2009); QCDSF Collab., NPB688 (2004).

5


On the one hand, p erturbation theory provides an explicit expression for the Adler function valid at Q2 :
2) j. d (Q j =1 j s
()

1.5

D(Q )

2

1.0

0.5

Dpert(Q2) = 1 +
()

Q, GeV
0.0

.
0.5 1.0 1.5 2.0 2.5

0.0

S.G.Gorishny, A.L.Kataev, S.A.Larin PLB259 (1991); L.R.Surguladze, M.A.Samuel PRL66 (1991); P.A.Baikov, K.G.Chetyrkin, J.H.Kuhn, PRL101 (2008).

On the other hand, this p erturbative approximation is inconsistent with the disp ersion relation for D(Q2) due to unphysical singularities of the strong running coupling s(Q2):
(1) Dpert(Q2) = 1 + d1 s (Q2), (1)

1 2 ) = 4 , s (Q 2/2) 0 ln(Q
(1)

where d1 = 1/ and 0 = 11 - 2nf/3.

6


Disp ersion

relation

imp oses

stringent constraints on D(Q2):

D(Q2) = Q2

R ( s) ds 2 )2 4m2 (s + Q



Since R(s) assumes finite values and R(s) const when s , then D(Q2) 0 at Q2 0 (holds for m = 0 only) Adler function p ossesses the only cut Q2 -4m2 along the negative semiaxis of real Q2

PRIMARY OBJECTIVE : to merge these nonp erturbative constraints with p erturbative result for the Adler function.

7


NEW INTEGRAL REPRESENTATION FOR D(Q2) This ob jective can b e achieved by deriving the integral representations for the Adler function and R(s)-ratio, which involve a common sp ectral function. D(Q2) = Q2 R(s) ds 2)2 4m2 (s + Q
s-i
Im

4m

2

s

+
. .

i

.

Re

1 d R ( s) = lim D(- ) 2 i 0+ s+i Parton mo del prediction R0(s) = (s - 4m2 )
R.P. Feynman (1972).

s

i

kinematic restriction on R(s): Q2 D0(Q2) = 2 Q + 4m2 8


D(Q2) =

Q2 + d(Q2) Q2 + 4m2
s-i 1 d R ( s) = lim D(- ) 2 i 0+ s+i

R(s) = (s - 4m2 ) 1 +

d D( ) s D(Q2) = Q2 R ( s) ds 2)2 4m2 (s + Q




Q2 - 4m2 d D(Q2) = 2 1+ D ( ) Q + 4m2 + Q2 4m2

1 D( ) = lim D 2 i 0+

theor

(- - i) - D

theor

d Rexp( ) (- + i) = - d ln 9

A.V. Nesterenko, J. Papavassiliou, JPG32 (2006).


Thus one arrives at the following integral representations:
Q2 - 4m2 d D(Q2) = 2 1+ D ( ) Q + 4m2 + Q2 4m2

R(s) = (s - 4m2 ) 1 + 1 D( ) = lim D 2 i 0+
theor

d D ( ) s
theor



(- - i) - D

d Rexp( ) (- + i) = - d ln

A.V. Nesterenko, J. Papavassiliou, JPG32 (2006).

З all nonp erturbative constraints on D(Q2) are satisfied З congruent analysis of spacelike and timelike pro cesses In the limit m = 0 the obtained expressions b ecome identical to those of the Analytic p erturbation theory
D.V. Shirkov, I.L. Solovtsov, PRL79 (1997); EPJC22 (2001); TMP150 (2007).

10


There is no unique way to compute the corresp onding sp ectral density by making use of p erturbative D
p ert

1.5

D(Q )

2

1.0

(Q2).
0.5

In what follows the one-lo op sp ectral function is adopted: 1 (1)( ) = 1 + 1 ln2 + 2

Q, GeV
0.0

.
0.5 1.0 1.5 2.0 2.5

0.0

A.V. Nesterenko, PRD62 (2000); PRD64 (2001).

ADVANTAGES : З unphysical p erturbative singularities are eliminated З additional parameters are not intro duced З reasonable agreement with D
exp

(Q2) for all energies 11


INCLUSIVE LEPTON DECAY

The inclusive semileptonic ( - hadrons R = ? ( - e- e

branching ratio: - ) = R ,V + R ,A + R ,S. )

Its nonstrange part asso ciated with vector quark currents: Nc R ,V = |Vud|2SEW QCD + EW = 1.764 Б 0.016 2
OPAL Collab., EPJC7 (1999); ALEPH Collab., EPJC4 (1998), RMP78 (2006).

In this equation Nc = 3, |Vud| = 0.9738 Б 0.0005, EW = 0.0010, SEW = 1.0194 Б 0.0050, M = 1.777 GeV, and
QCD

=2

2 M

0

s2 s ds 1- 2 1 + 2 2 R(s) 2 . M M M 12


Perturbative approach :
(1) 2 QCD = 1 + d1 s (M )

= (678 Б 55) MeV,

nf = 2

E. Braaten, S. Narison, A. Pich, NPB373 (1992).

Current analysis : 41 (1) 2 2 QCD = 1 + d1TL (M ) - + f ( )(1) M d - d1 0 m2 f ( ) = 3 - 2 2 + 2, = , = f () 0.048, 2 M 4 (1) 2) (1)( ) d , TL (s) = (s - m m = m0 0 s З massive case: = (941 Б 86) MeV З massless limit: = (493 Б 56) MeV
A.V. Nesterenko, NPBPS186 (2009).
(1) TL (m2 ),

1 d1 = , + m
-

13


SUMMARY

З New integral representations for the Adler function and R(s)-ratio are derived З These representations p ossess app ealing features:
З З З З

unphysical p erturbative singularities are eliminated additional parameters are not intro duced the 2-terms are automatically taken into account reasonable description of D(Q2) in entire energy range

З The effects due to the pion mass play a substantial role in the analysis of the inclusive lepton hadronic decay 14