Pion formfactor at intermediate momentum transfer in QCD

Volume 114B, number 5

PHYSICS LETTERS

5 August 1982

PION FORMFACTOR AT INTERMEDIATE MOMENTUM TRANSFER IN QCD B.L. IOFFE and A.V. SMILGA

Institute of Theoretical and Experimental Physics, Moscow 117259, USSR Received 14 April 1982

A general method is proposed for the QCD based calculation of formfactors at intermediate Q2 and of the partial widths of the lowqying mesonic resonances. The basic idea is to use the QCD sum rules for the vertex functions and to saturate their Borel transform by the lowest mesonic states. With this method, the pion formfactor at 0.5 < Q2 < 4 GeV2 is calculated. The results are in a good agreement with experiment.

1. Introduction. It has been established that quantum chromodynamics allowed one, in accord with experiment, to calculate the masses of low mesonic [1 ] and baryonic [2] states. It is clear that the next step to consistent description o f the low energy hadronic physics in QCD must be the calculation of the partial widths of different resonances and of their formfactors at intermediate Q2. In this paper we present a general method which enables the solution o f this problem. 2. The method. Let us consider for definiteness the vertex function corresponding to two axial and one electromagnetic current

r~u,x(p', p;q)

= - fd4x

d4y exp [i(p'x -

qY)l

(01T{JA(x),J~I(y),J2+(0)}[0)

(1)

,

where

jA = ~Tu75d ' f~l = eu~,yx u + ed~lTxd ,

(2)

u(x), d(x) are the fields of u- and d-quarks. We adopt that mass squares of external lines p 2 p'2, q2 are negative and [p21~ Ip2'l ~ Iq21 ~ 1 GeV 2. Then the characteristic values of the strong interaction constant are small, Ors(p2 ) ~ 0.3 and in the zeroth approximation in asP~u,x is determined by the simple quark loop (fig. 1) and by the power corrections originating from the mean vacuum values of different quark and gluon operators in the operator expansion. The general expression for F / " 1 2 ' A-(p, t p'; q) is presented as a sum of 14 structure functions fi, i = 1,2, 14, . . each of them being a function o f p 2, p 2 and q2. Write for each of these functions a double dispersion relation in variables p2 and p'2 (s = _ p 2 , s' = _ p ' 2 ) : ""~

•

ft.(s,s';q2)=~ ;dp2;dp 0

0

'2 Pi(p2'p'2;q2)

+subtraction terms.

(3)

(p2 + s) (p'2 + s')

The quantity -4Pi(p2 , p'2;q2) equals to the double discontinuity o f the amplitude fi(p2, p'2, q2) on the cuts 0 ~ (-27rt')36(k2)6 [ ( p - k) 2 ] 6 [ ( p ' - k)2]. (The quark masses are disregarded.) The subtraction terms in (3) are polynomials in one variable s or s' but arbitrary functions in two others (s', q2 or 0 031-9163/82/0000-0000/$02.75 © 1982 North-Holland

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s, q2, respectively). To link the QCD-calculated structure functions fi(s, s'; q2) with the parameters of physical states and to get rid of unknown subtraction terms, let us exploit the Borel transformation procedure proposed in [1] in two variables s and s' simultaneously. Determine the Borel transform of function f(s, s'; q2) as

S'n+l ( __ ~$1 ~ d )nsm+i (d) m ' 2 f(M2'M'2;q2)-qOs'qOsf(s's';q2)=lim-~. -ds f(s,s ;q )

(4) n, m ~oo; s,s'-+oo;

s'/n -+M '2, s/m ~ M 2 .

Borel transformation (4) of eq. (3) yields 1

f

fi(MZ'M'Z;q2) = ~ 0d p

2

;dp2

0

exp

2 Mp'2i] i i 2;q2,

M2

In the following we put M 2 = M '2 and f/(M 2 , M '2 , q2) - f i ( M 2 ' q2). Functions fi(M 2, q 2) satisfy the subsidiary conditions following from the transversality relations Fvu,xqx = p vu,xPvt = Pvu,xPu = 0 since the equal-time commutator terms vanish after double Borel transformation. The rhs of(5) can be represented as a sum over the physical hadronic states. I f M 2 is chosen of order 1 GeV 2, then the rhs of (5) will be dominated by the lowest hadronic state contribution in the given channel while the higher state contribution will be suppressed. Applying to the vertex function Fvu,x(p, p ' ; q) (1) where the lowest state in the axial current channel is the pion, this means that including only one-pion state, the spectral function is Puu ,x(P', P; q) = -1r2(0 [jNA(0)17r-(p'))Or-(p') X 8(p 2) 8(p '2) =

[/'~1(0)In-(p))

rr2f2p'vpu(p + p')xFrr(Q 2) 6(p 2) 6 ( p ' 2 ) ,

(6)

where FTr(Q2) is the pion electromagnetic formfactor, fTr = 133 MeV, Q2 = _q2 and in our approximation of massless quarks we put m 2 = 0. To improve the accuracy of our consideration, and by full analogy with the polarization operator case [1], let us take into account in rhs of (5), in addition to the one-pion state contribution, the continuum contribution as well. It is clear that at large p2, p'2, Q2 the contribution to the double discontinuity of the vertex function is connected with the quark loop, fig. 1. That is why the inclusion of continuum reduces to that starting from certain chosen values p2, p'2 2> So, Oi(p 2, p'2 ; q2) is described by the double discontinuity of the graph fig. 1. In the sum rules for the vertex functions continuum is more important than in the sum rules for the polarization operator because the relative contribution of continuum increases with Q2. This fact is clear a priori from physical reasons and is confirmed by direct calculation. Hence it follows that the calculation of the continuum contribution is necessary also in order to estimate the Q2 region, in which the pion formfactor FTr(Q2) of interest can be found with this approach with a sufficient accuracy. Considering the whole set of spectral functions and saturating the rhs of (5) by the lowest mesonic states, one shored include into the axial channel (in addition to the one-pion state) A 1-meson (JP = 1+). Then, in principle, from analysis of all sum rules one can determine the electric, magnetic and quadrupole formfactors of A 1-meson, two formfactors of transitions ~'virtrr -+ A 1 and improve the calculation of the pion formfactor. In this paper we shall not perform this complete program (such a study will be done separately) and restrict ourselves with an investigation of the sum rule for one structure function and with a rather rough approximation when the rhs of (5) is approximated by contributions of one pion and continuum (i.e., A 1 -meson is referred to continuum).

3. Power corrections. It is clear from dimensional considerations that the most significant power corrections are proportional to vacuum expectations of operators (01G~uGauvl0) and as(01~Fff" ~p~10)where G~v is the gluonic 354

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•

P ,u

lines to vector current.

v

] I

P

Fig. 1. The main diagram for the vertex function I uv,X(p, P; q); d o t - d a s h e d lines correspond to external axial currents, wavy

p,

p

p'

P

p'

p

p'

I

Fig. 2.

field tensor. (The contribution of operators (0l ~ k l 0 ) and (01 ~avp t a ~ Gap~ 10) because of the chiral invafi~mce is proportional to the bare quark masses and vanishes in the massless quark approximation). When considering vacuum expectations (01~P~k" t~P~10) we shall use the factorization hypothesis, i.e. include only vacuum state into the sum over intermediate states En(01~F~ln)(nlt~P~[0) so that matrix elements from any four-quark operators reduce to

~s<01~¢10)2. It is convenient to calculate corrections proportional to (0IG.a.210) in the fixed point gauge, i.e. when the gauge condition is x~Aa~(x) = 0 (see [3] and references therein). In this ~vgauge, first of all, the quark Green's function S(x,y) in the constant external gluonic field Garyis calculated. Afterwards, such a Green's function must be substituted instead of each quark line of fig. 1 diagram. In the chosen gauge and at adopted in (1) definition of Fv~,× (the coordinate origin is at the left ,vertex of fig. 1) only fig. 2 diagrams survive. Just as the ground term - the contribution of the fig. 1 diagram - corrections proportional to (01(G~v)210) are useful to find calculating first the double discontinuity and exploiting relation (5) to get the sum role. After this, the calculations of these corrections, though very cumbersome, can be made directly. The power corrections proportional to (0l~ff]0) 2 can be divided into two types: (1) corrections determined by the hard gluon exchange graphs - fig. 3; (2) corrections with a quark pair production by soft gluon - fig. 4. The calculation of the first type corrections is elementary. When calculating the second type corrections one should account for the following three mechanisms of their emergence: (i) In the fixed point gauge the potential Aa~(x) of the emitted by quarks soft gluon at small x can be expressed via the gluonic field strength tensor G~v and its covariant derivatives. The latter, because of equations of motion are expressed through bilinear combinations of quark fields, so that as a result, four-quark operators appear (fig. 4a).

/77; p

p' ....

p

P'

p

p' (a)

Fig. 3.

p' (b)

Fig. 4.

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(ii) The vacuum expectations (01~/(0) qjk(x) 10) at small x must be expanded in x (see ref. [2], Errata, and ref. [4]) up to the 3-d order which leads to the four-quark operators (fig. 4b). (iii) And finally, the interference of mechanisms (i) and (ii) is possible: (0ITS/(0) ff~(x)10) is expanded in x up to the first power and simultaneously Aa~(x) is expressed through G/~/). a Not dwelling on the calculation details, let us now present the results.

4. Results. We present here the results for one structure function f, proportional to PuPvPx, where P = (p +p')/2; I'uu,x = fPuPvPx + .... For this structure function the contribution of the ground term described by the fig. 1 graph, after the Borel transform (5) is

~ x 3 fdxfdyexp(_x/M f(0)(M2 ' Q2) = 2rr -~ 0 0

2)RQ_47

[3R2(x+Q2)(x+2Q2)_R4_5Q2(x+Q2)3[ '

(7)

where

R(x,y) = [ Q 2 ( 2 x + Q 2 ) + y 2 1 1 / 2 ,

x=p2+p'2,y=p2-p'2.

The power correction proportional to (01(G~v)210) is f ( 1 ) ( M 2 , 0 2) = ( i / 6 M 2) (01(%/70 Ga~uG~v10),

(8)

while the power correction proportional to (01~ ~10) 2 appeared to be f(2)(M2 ' Q2) = (416rr/81) as(01~ffl0)2M-4(1 + ~ Q2/M2).

(9)

The sum of the rhs of (7)-(9) must be due to (5), (6) equalled to the contribution of pion + continuum, so that Frr(a2 ) = (1/2f2) [f(0)(M2 ' 0 2) + f(1)(M2 ' 0 2) + f(2)(M2 ' Q2) _ continuum[.

(10)

As is seen from (7)--(9) at Q2 ~ oo the ground term decreases as Q - 4 , while the power corrections even increase with Q2. This maens that the applicability of our approximation at large Q2 is limited not only by continuum contribution but also by the power corrections. At small Q2 the ground term has a nonphysical singularity ~Q41n Q2 resulting from massless quark propagators and ~ointing out the inapplicability of our formulae at small Q2. The fact that this method is inapplicable at small Q~ could be expected beforehand since in case of slowly varying electromagnetic field F by, the disregarded mean vacuum values of the type (Ol~ouv ~kFuvlO) may appear. The inapplicability of this method in the region of very small Q2 is evident also from consideration of power corrections to other structures where the terms Q-2 and even Q-4 are present. At numerical calculations we have considered two versions of account for the continuum contribution: (1) the continuum contribution is defined by the integration region in (7) p2 + p'2 ~> So ; (2) the continuum contribution is defined by the region p2 or p'2 >So. It appeared that both versions lead to very close results, so that we restrict ourselves here to the discussion of version 1. so was found from condition of the best coincidence of the rhs and lhs of the sum rule in their M 2 dependence at fixed Q2 = 1 - 2 GeV 2. Fig. 5 shows the M 2 dependence of the lhs of the sum rule minus continuum contribution at the following values of the parameters: s o = 1.2 GeV 2, (0 I(C~s/Tr) (G~v)210) = 0.012 GeV 4, as(0[~10)2 = 8 X 10 -5 GeV 6 . If the rhs of the sum rule is dominated by the pion contribution, this quantity must be constant. It is seen from fig. 5 that such a requirement is well fulfilled within the range 0.7 ~ M 2 ~ 1.7 GeV 2 . Let us choose M 2 = 1.2 GeV 2 and using the sum rule at this value o f M 2 find the Q2 dependence of the pion formfactor FTr(Q2). The result is shown in fig. 6 where experimental data are also plotted (see ref. [5] and references therein) and the asymptotic dependence curve [ 6 ] - [ 1 0 ] is presented for comparison, Fir (Q2)asymp = 87tf20:s(Q2)/Q2. The agreement with experiment is good enough. At Q2 > 3 - 4 GeV 2 the accuracy of our results is rather low since at Q2 = 3 GeV 2 the continuum contribution is already rather high (~ 50% from the all rest) and the power corrections are 45% in the lhs of the sum rule. The change o f M 2 weakly affects F~r(Q2 ) in the considered interval Q2, for instance, a t M 2 = 1 GeV 2 the curve Frr(Q 2) does not practically differ from that given in fig. 6. 356

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FTr

0.4

Q2 = 1 GeV 2

0.3 "--

0.3

0 2 = 2 GeV 2

0.2

0.2

0.1

0.1

\

0.5

, 1.0

j 1.5

M2 (GeV2)

Fig. 5. TheM 2 dependence on the lhs of the sum rule (10) at Q2 = 1 GeV 2 (solid line) and Q2 = 2 GeV2 (dashed line). The 1.h. vertical bars mark the M 2 values above which the ratio of the power corrections to the ground term does not exceed 50%. To the left of r.h. vertical bars the ratio of the continuum contribution to all other in the lhs of (10) does not exceed 50%.

\

.. ,, -.

I.

,

I

1

I.

10

2.0

3.0

4.0

50

Q2 (GeV 2

Fig. 6. The pion formfactor Frr(Q2): solid line: the result of our calculations, dashed line: asymptotic dependence [ 6 10], dots: experimental data [5].

Up to now, in order to calculate the pion formfactor weused thevertex function with the axial current. Instead o f them we can consider the pseudoscalar current vertex function where transition matrix elements (0 liK75dl 7r-) are also known <0 liK3,5dlTr-) =f,r [m2/(mu + md)] = --2 (1/fTr) <0l~qJl0) •

(11)

The sum rule determining the pion ~ormfactor in this case is o f the form 3

4 M 4 F l n ( M 2 / A 2 ) ] -8/9 ~b(M2, Q2) (0l , ~ ] 0 ) 2 FTr(Q2), 167r2 k i n (/12/A2) j = f- z:

(12)

where $0 x ~b(M2, Q2) = Q2 f d x exp ( - x / M 2) o o

x 2 _ y2

Jdy -R3

"

In (12) the power corrections are not taken into account because their role in this case is small, and the continuum contribution is transferred to the lhs. (The logarithmic factor results from anomalous dimension o f the pseudoscalar current,/1 is the normalization point,/a ~ 0.5 GeV). The results o f the formfactor calculations do not strongly differ from those presented above for the axial current case, but they are less confident theoretically since the M 2 dependence o f the lhs o f the sum rule is stronger here and the continuum contribution is larger. Besides, as is known [ 11,12], in the pseudoscalar case, noncontrollable non-perturbative corrections due to instantons are much more important. F r o m the known values o f F ( Q 2 ) , by saturating the dispersion relation for FTr(Q 2) by the p-meson contribution 2 • , one can find the constant gprrTr,gpTr, = F ~r(Q2 ) (Q 2 + mp)go/m p2 where go is the P7 transition constant. F r o m our result at Q2 ~ 1 - 2 GeV2 it followsgp,r~r ~ 0.9g o. Being consistent in our aspiration to find all hadron parameters from QCD we are to take theoretical value go obtained in ref. [1], 4trig2 = 0.41. Then gp~rTr = 5.0 in comparison with the experimental value (go ~r,r)exp = 6.1. Of course, this latter result has little new as compared to 357

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the vector dominance approach. However, it demonstrates the way of finding interaction constants and partial widths for the case of other resonances. In the course of our work we were informed that the pion formfactor is being considered also by A.V. Radyushkin and V.A. Nesterenko on the basis of a related approach. We are indebted to them for preliminary information on their results. We are deeply grateful also to M.A. Shifman for useful discussions.

References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]

358

M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448. B.L. Ioffe, Nucl. Phys. B188 (1981) 317;B191 (1981) 591 (E). A.V. Smilga, Yad. Fiz. 35 (1982) 473. E.V. Shuryak and A.I. Vainshtein, Novosibirsk preprint INP-81-106 (1981). C. Bebek et al., Phys. Rev. D17 (1978) 1793. V.L. Chernyak and A.R. Zhitnitsky, Pisma Zh. Eksp. Teor. Fiz. 25 (1977) 544; V.L. Chernyak, Proc. XV LINP Winter School, Vol. I (Leningrad, 1980) p. 65. A.V. Radyushkin, prepfint JINR P2-10717 (Dubna, 1977); A.V. Efremov and A.V. Radyushkin, Phys. Lett. 94B (1980) 245. G.P. Lepage and S.J. Brodsky, Phys. Lett. 87B (1979) 359. G.R. Farrar and D.R. Jackson,Phys. Rev. Lett. 43 (1979) 246. A. Duncan and A.H. MueUer, Phys. Rev. D21 (1980) 1636. B.V. Geshkenbein and B.L. Ioffe, Nucl. Phys. B166 (1980) 340. V.A. Novikov, M.A. Shffman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. BI91 (1981) 301.