Modeling the QCD contribution to αQED (MZ)

Physics Letters B 294 (1992) 293-297 North-Holland

PHYSICS LETTERS B

Modeling the QCD contribution to aQED (Mz) B. Holdom and Randy Lewis Department of Physics, Universityof Toronto, Toronto, Ontario, Canada MSS 1A7 Received 1 June 1992; revised manuscript received 31 August 1992

In addition to the perturbative QCD contributions to otQEo(Mz) we estimate the nonperturbative, quark mass dependent, contributions using a constituent quark model. The model includes the momentum dependenceof the dynamical quark mass and its application here is similar to its description of basic quantities in low energyQCD.

1.Introduction

e(e + e--,7*--, hadrons) R(s)==- o(e+e___,7.~lx+lx_ )

In terms of the QED vacuum polarization

ilIu,,(P) =ieZH(P z) (p2gu,, -PUP,,),

= - 12re Im//had(S) ,

( 1)

the running QED coupling satisfies I

a(_p2)

I

o~(_q2 ) =4n[H(q2)-H(p2)]

we obtain the well known result [

1

1

a(__p2)

a(--q2)

(2)

~I

E0:0, o, 1 2,L

,3,

a ( 0 ) is essentially the fine structure constant. We shall argue that A provides an excellent test of constituent quark models of low energy QCD. It is well known how to reliably extract A from experimental data. If/-/(p 2) is considered as a function in the complex p2 plane, it will be real-valued for p2< 0 and analytic everywhere except for a branch cut along the positive real axis. We then have

H(p2)=~

ds~

(4)

F for p2 inside a contour, F, which does not cross the branch cut. In terms of the quantity

] had

ds (P2-q2)R(s)

for euclidean momenta (i.e. p2 < 0 and q2 < 0 ). In this paper we shall focus on the following QCD contribution:

(5)

(s_q2)(s_p2) '

(6)

o where, again, p2 < 0 and q2 < 0. To determine A by this method, R ( s ) needs to be determined at all scales, s. A recent detailed computation based on the available experiment results and a modeling of the high energy contribution has been performed by Burkhardt, Jegedehner, Penso and Verzegnassi [ 1 ]. They provide a prediction for A as well as an approximate parameterization of the running coupling at all scales. We argue that a successful constituent quark model should be able to reproduce A via a direct calculation of H ( 0 ) and H ( M 2 ) , bypassing the calculation of R ( s ) . In fact of interest to us are free quark models which are completely incapable of reproducing R (s). This state of affairs is exactly analogous to the determination of parameters in the low energy chiral lagrangian of QCD. We are referring to the coefficients of terms at order p4, LI-Llo in the notation of ref. [2 ]. The five parameters which survive in the chiral symmetry limit, L1, L2, L3, L9, Llo, have been suc-

0370-2693/92/$ 05.00 © 1992 ElsevierSciencePublishers B.V. All fights reserved.

293

Volume 294, number2

PHYSICSLETTERSB

cessfully described by vector and axial-vector meson dominance on one hand and by a free constituent quark model (see below) on the other. These results imply that an effective duality between these two pictures is operational in the low energy expansion, at least to order p4. This paper explores another test of this duality, s i n c e / / ( 0 ) is simply another quantity that appears at order p4 in the expansion. The expectation then is that a constituent quark model should describe d even while totally failing to describe the resonances contributing to R (s). ,J is quite sensitive to the light quark masses. In a perturbative calculation one is tempted to use the current quark masses, but this leads to the absurd result that A diverges in the chiral symmetry limit. The constituent quark masses are more physically relevant, but it is not obvious how to introduce them into a perturbative calculation. We will argue that it is possible to largely disentangle the nonperturbative effect of a constituent quark mass from the perturbative gluonic corrections. The simplest constituent quark model is a nonlinear sigma model coupled to free quarks. This model does a fair job [3] of reproducing L l, L2, L3, Z9, L 1o, but the constituent quark mass remains a free parameter. The problem is that an ultraviolet cutoff must be introduced and thus the relation between the constituent quark mass and a physical quantity such as f~ is obscured. We will instead consider the recently-studied gauged nonlocal constituent (GNC) quark model [ 4 ]. Here a more realistic momentum-dependent dynamical quark mass is introduced and the values for L~, L2, Z3, L9, and most notably Ll0, are significantly improved. The model also yields a cutoff-independent determination of the constituent and current quark masses from the physical decay constants and masses of mesons, and in the process determines L4L8 [ 5 ]. We then have a consistent framework for the estimation of d. We will be encouraged by the similarity of the calculation o f d to the calculation of L~o. We first determine the contribution to A from the GNC model, while ignoring perturbative gluonic corrections. The latter are added as a separate contribution, so that d = d ° + d ~' .

and we conclude by comparing our result to the experimental value.

2. A without perturbative corrections If we considered only the lowest order one quark loop vacuum polarization diagrams the result would be

Nca2 [Jq(M2) - J q ( 0 ) ]

(8)

zl°= ~'--fZn

where q runs over quarks of charge Qq and mass mq, and

Jq(x2) = y+ ]_ ( 4m~'~

\x~/

- [ 1 - ((2m2"~] 4 m 2 ~ \ -N - ~/ -l + - 7 . ,] x z /

(x/l+(4m2/x 2) +1~ (4m~/x2) _ 1]'

× In \ x / 1 +

(9)

Y is a momentum-independent logarithmically-divergent term which cancels out of eq. (8). For the three light quarks, a low energy nonperturbative contribution will show up as a replacement for Jq(0) in eq. (8). We use the GNC model and calculate the appropriate term in the low momentum expansion of the vacuum polarization diagrams in fig. 1. As with the calculation of the L~ parameters, we are calculating a term at order p4 in the derivative expansion. The vertices and propagators in the quark loop

I--~,

/ ,

X ~ ,

/

\

I k

/

I

(7)

We will indicate the uncertainties in our calculation 294

12 November 1992

Fig. I. G N C quark (solidline)and pion and kaon (dashed line) contributionsto Q E D vacuum polarization.

Volume 294, number 2

PHYSICS LETTERS B

diagrams are extracted from the GNC quark model lagrangian [4 ]: ~oNc (x, y) =

¢(x),ffx-y)

X {iTu[Ou -iRu(Y) l --g}V(Y)

- O(x)X(x-y)~(x)X(x,

y)~(y)v(y).

(10)

represents the light constituent-quark triplet and Jt=diag(mu, ma, ms) is the current-quark mass matrix. Notice that the model implies gA = 1 for constituent quarks (as suggested in ref. [6 ] ). × ( x - y ) is the Fourier transform of the dynamical quark mass which we represent in terms of a parameter A, Z(_p2)_

( A + l ) m o3

Am 2 _ p 2

( 11 )

•

The normalization is such that 27(mo2 ) = too, and we show below that mo is a function ofA. X(x, y) is a path-ordered exponential:

X(x,y)=Pexp(-iiFu(z)dzU),

(12)

x

F.(z) =

½i{~(z) [0~ - iR~,(z) l~* (z)

+ ~t(z) [0u - i L u ( z ) ]~(z)}, ~(x) = exp

\ k = 1 Jo

2k~k(X ) . /

(13)

(14)

Here, Tr(2~k) = ~jk, 1 and fo is the pseudoscalar decay constant in the chiral limit. In the limit of vanishing ~¢t, this model has local SU (3) L× SU (3) R symmetry. The left and right handed external gauge fields are Lu= Vu-Au7 s and Ru= Vu+Au? 5 where Vu(x) =vaU(x)2 a, etc. The required one- and two-photon vertices are given in ref. [ 4 ], and their derivation is discussed in detail in ref. [ 7 ]. Although the model considered in the latter reference is different (it does not exhibit the full local chiral symmetry) the resulting photon vertices happen to be the same as in the present model. The quark loop contributions to the dimensionless parameters L~, L2, L3, Lg, Lip are finite and dimensionless and thus are independent of all mass scales. There are also small pion loop renormalization effects which depend on the choice of matching scale 2. 2 is the renormalization scale for matching the pre-

12 November 1992

dicted effective chiral lagrangian onto the GNC model lagrangian. The model must fit these Li's with only the one parameter, A. Table 1, taken directly from ref. [4], compares the GNC results with ;t=2mo to the experimental values. The quark loop contribution to d ° on the other hand depends logarithmically on the mass parameter rno. mo is determined by the model through the following relation, first derived by Pagels and Stokar [ 8 ]: N¢ I r2 ( X 2 - - ½r2X27' ) f 2 = ~ n 2 dr 2 (rZ+Z2) 2

(15)

o

We will fix fo= 84 MeV [5], in which case mo becomes a function of the single parameter, /~. For A = ( 1, 2, 3, 4), mo= (342, 317, 299, 287) MeV. In this way we obtain an expression that replaces Jq (0) in eq. ( 8 ) from the quark loop diagrams in fig. 1. It has the same divergent term, Y, to cancel the divergence of Jq ( M 2). Here we note the similarity to the calculation of Lto which involves the vector minus axial-vector ( V V - A A ) two-point function. The VV quark diagrams which are used in the GNC model to calculate Llo are exactly those in fig. 1. The canceling divergence in the Llo calculation is provided by a single AA diagram involving a trivial axial vertex. The similarity of the two calculations is reassuring. The pseudoscalar loop diagrams of fig. 1 are evaluated from the lowest order chiral lagrangian, with the momentum in this loop cut off at the matching scale 2. To be conservative we allow 2 to vary between 500 MeV and 1 GeV, but this uncertainty proTable 1 The five coefficients of the chiral lagrangian at O ( p 4) that exist in the limit of chiral SU ( 3 ) × SU (3). The experimental values are compared to the GNC predictions for various values of the parameter A. All values ( X 103 ) are renormalized at mn. Coefficient

Ll L2 L3 L9 Llo

Experiment a)

0.9+0.5 1.6_+0.4 --3.6_+ 1.3 7.4_+0.7 --6.0_+0.7

GNC quark model b) A=I

A=2

A=3

A=4

1.29 2.58 -6.15 8.88 --7.10

1.09 2.17 -5.13 7.57 --5.72

0.99 1.98 -4.70 7.02 --5.10

0.93 1.86 -4.45 6.71 --4.73

") Refs. [2,9]. b)Ref. [4].

295

Volume 294, number 2

PHYSICS LETTERS B

duces only a tiny fraction of the error given in eq. (16) below. We attribute most of the uncertainty in our determination ofA ° to the uncertainty in the parameter A. From table 1 we see that a conservative choice for the range in A is 1
(16)

3. Perturbative corrections to d

The previous result has neglected QCD corrections to the logarithmic running of a between Mz and the quark masses. Consider for a moment massless fermions in the regime of perturbative QCD, so that zt~rt(_p2) can be obtained to order otot2 from the following renormalization group equations:

2NoQ

0a

q= l X

[

1+ °q(#2) + k /[

i~as u-fl u

37t

//o =-

z as -

( Yl

,

/~1 or3 8n2

(17) (18)

s,

where q runs over the Nf quark flavours, and k=~45_ 11~(3)_ [ H l, - ]~(3) ]Nf,

(19)

flo= 1 1 - 3ZNf,

(20)

jl~l = 1 0 2 - - ~ U f .

(21)

~(3 ) = 1.202... denotes the Riemann zeta function and the given expression [ 10 ] for k represents the MS renormalization scheme. The renormalization group equations do not incorporate quark mass effects. The contribution from each quark should drop out for _ p 2 less than approximately the mass squared of the quark. To include this effect, recall the explicit result [ 11 ] for constant ors that includes quark masses: 296

Aw,'t

const cts =

12 November 1992

~ NcQ 2 3n

X lnk(xomq)2 ] + - -n

£mq/ +O(a~)

, (22)

Xo = e x p ( ~ ) =2.30 and

H ( M z ~ =ln ( ~

\mq,I

\mq]

- ~+4~'(3)

=lnk(x~mq)2 ] + 2 l n ( X l ) - ~ + 4 ¢ ( 3 ) .

(23)

The term of zeroth order in as in eq. (22) is just the constant mass version (A = oo) of the calculation in the previous section. If for the three light quarks we take mq to be mo = 310 MeV plus the current quark mass, then we obtain a result for A° which is of order 10% lower than the corresponding GNC result of eq. (16). Another way to see the difference between constant and nonconstant mass is to note that the term of zeroth order in ors in eq. (22) reproduces eq. ( 16 ) for the unrealistically low value of mo= 140 MeV. For consistency with (16) we should extract ors corrections in the presence of a nonconstant mass 27(_p2). We will instead use the constant mass calculation to estimate the oq corrections, first since the corrections themselves are small compared to (16) and second since other larger uncertainties show up in our constant mass estimate. The problem is to combine the quark mass effects in eq. (22) with the running oq effects of the renormalization group. We denote the contributions to A of order ot~ by 8ff. From the form of the final expression in eq. (23) we write

NcQ 2 5,d= ~q 3n

X(Iq+Ots((xtmq)2) n

[2 l n ( x , ) - - ~ + 4 ¢ ( 3 ) ] ) ,

(24) with

lq = I(ql ) ( x, mu, Kl md ) + I(q2) ( Kl md, Klms) +/q(3) (/tj1rn,,/tJ 1 mc) + 1 (4) (KI mc, x, mb) + Iq(5)(X~ mb, Mz)

(25)

Volume 294, number 2

PHYSICS LETTERS B

12 November 1992

and

A=3.94+0.12 .

4 , {In(M/A("))'~ I~'°(m, M)=O(M-Xl m,) -~o'n ~ ~ )

To uncover the effect o f the strange quark mass we may consider the case o f vanishing current quark masses for the three light quarks. The analog o f eq. (29) in this case is

ln(m/A c,,)) ] 2 ln(m/A c'))

4fl, {.1 + l n [ 2 l

fl~ ~

-l+ln[21n(M/AC")) 2 ln(M/A c"))

])1

lira/1=3.85+0.3, (26)

0.30<~1/1<0.36,

Ac4)=100MeV,

0.39<~1/1<0.65,

AC4)=200 M e V .

(27)

We find that ~2/1 represents a larger source o f uncertainty. We first note that approximately half the value o f ~1/1 is due to the Iq term in eq. (24). So to estimate ~2/1 we calculate the corresponding Iq term from eqs. (17) and (18) with k ~ 0 and then double the result. This gives ~2A~0.3(0.5) for At4)= 100(200) MeV. Physically, we expect the size o f the perturbative corrections to be limited by the infrared cut-off provided by the constitu6nt mass. We will use this crude estimate of ~2/1 to give a conservative estimate of the full uncertainty in our perturbative contribution to/1, A~'=0.33_+0.3,

A t 4 ) = 100 MeV,

=0.54+0.5,

AC4)=200 M e V .

(28)

The uncertainty here is unfortunately much larger than the uncertainty in our nonperturbative contribution to/1.

4. Conclusions Eq. (28) m a y be combined with eq. (16) to give our final results: /1=3.8_+0.3,

A t 4 ) = 100 MeV,

=4.0_+0.5,

AC4)=200 M e V .

(29)

This may be compared with the result o f the R-ratio method [ 1 ] described in the introduction,

Ac4)=100MeV,

,A/~0

=4.1+0.6,

Iq represents the solution of the renormalization group equations, O(x) is unity for x > 0 and zero otherwise and A t") is the Q C D scale parameter for n quark flayours. We expect this to be correct for some rl o f order 2. For 1.5 < Xl < 2.5 we obtain

(30)

At4)=200 MeV.

(31)

We see that the strange quark mass has little effect. Eq. (29) implies ot -~ ( M 2) = 128.94+0.3, =128.7+0.6,

A c4)= 100 MeV, At4)=200 MeV.

We should emphasize again that our determination o f ot ( M 2) is not intended to be competitive with the standard extraction o f o t ( M 2) from experimental data. Our point has been to show that when the m o m e n t u m dependence of the constituent quark mass is taken into account, a free quark model o f low energy Q C D does well at accounting for the accepted value o f ot ( M 2).

Acknowledgement This research was supported in part by the Natural Sciences and Engineering Research Council o f Canada.

References [ 1] H. Burkhardt, F. Jegedehner, G. Penso and C. Verzegnassi, Z. Phys. C 43 (1989) 497. [2 ] J. Gasser and H. Leutwyler, Nucl. Phys. B 250 (1985) 465. [3] E.g.I.J.R. Aitchison and C.M. Fraser, Phys. Lett. B 146 (1984) 63; Phys. Rev. D 31 (1985) 2605; J. Balog, Phys. Lett. B 149 (1984) 197. [4] B. Holdom, Phys. Rev. D 45 (1992) 2534. [5] B. Holdom, J. Terning and K. Verbeek, Phys. Lctt. B 245 (!990) 612;B 273 (1991) 549 (E). [6] S. Weinberg, Phys. Rev. Lett. 65 (1990) 1181. [ 7 ] J. Terning, Phys. Rev. D 44 ( 1991 ) 887. [ 8 ] H, Pagels and S. Stokar, Phys. Rev. D 20 (1979) 2947. [9] C. Riggenbach,J. Gasser, J.F. Donoghue and B.R. Holstein, Phys. Rev. D 43 ( 1991 ) 127. [ 10] M. Dine and J. Sapirstein, Phys. Rev. Lett. 43 (1979) 668; K.G. Chetyrkin, A.L. Kataev and F.V. Tkachov, Phys. Len. B 85 (1979) 277; W. Celmaster and R. Gonsalves, Phys. Rev. Lett. 44 (1980) 560. [ 11 ] E. de Rafael and J.L. Rosner, Ann. Phys. 82 (1974) 369. 297