Is the strange quark heavy or light?

Volume 189, number 3 PHYSICS LETTERS B 7 May 1987

IS T H E STRANGE QUARK HEAVY OR LIGHT?

C. AYALA, E. BAGAN and A. BRAMON Departament de Fistca Te6rtca, Umversttat Aut6noma de Barcelona, 08193 Bellaterra (Barcelona), Spare

Received 12 November 1986

The (mvanant) strange quark mass, rhs, and vacuum condensate, (gs)o, are discussed in the context of SVZ-sum rules saturated by the scalar x-meson The value obtained for ths (rhs- ~ 360 MeV) ts found to approach its constituent mass value, while ( gs ) o is predicted to be negligibly small. These two features, which are charactenstxc of the c- and b- (~.e., heavy) quarks, are new and unexpected for the s-quark. Our results strongly depend on the x-meson parameters, for whxch further analyses seem of Interest.

Current quark masses appearing in the quantum chromodynamics (QCD) lagrangian are responsible for the different types o f symmetry breaking in this theory. These masses, as well as the (gluon and) quark vacuum condensates, turn out to be the essen- tial parameters in the successful and well-known approach to QCD proposed by Shifman, Vainshtein and Zakharov (SVZ) [1]. The values of those parameters have been discussed by several authors [ 1-20] making use of different SVZ- or QCD-sum rules. Concerning current quark mass values there is general agreement in considering q = u , d as light quarks and Q = c, b as heavy quarks. Indeed, the current masses mu, md are found to be much smaller than typical QCD or hadronic mass scales, such as AOcD or the corresponding constituent u- and d-quark masses (given essentially by Mp/2~-M,o/2~-390 MeV). By contrast, the current c- and b-quark masses are found to approach their constituent mass values, My~2 and Mr~2, and are much higher than any QCD mass scale. Estimates o f quark vacuum condensates tend to reinforce this picture. Indeed, for heavy quarks one expects [1,5] the following (approxx- mate ~) relation between Q-quark and gluonic vacuum condensates

( Q Q ) o = - (1/12mQ) ( ( ( o q / n ) G G ) o

- 10 -3 GeV4/mQ, (1)

~ See ref. [21 ] for more details

which is not contradicted by Q = c, b quarks having negligible vacuum condensates and large mQ. By contrast light-quark condensates are typically one order of magnitude less negative than predicted by the naive application of eq. (1).

The situation is less clear-cut for the strange quark. Recent and detailed determinations of its invariant current mass, rhs, lead to [6,7]

rhs=288+ 47 M e V ,

=251 +26 M e V , (2)

i.e., values between AQCD= 150 MeV and the corresponding s-quark constituent mass M¢/2 ~ 510 MeV. Concerning the s-quark vacuum condensate, most authors [ 1,2,8,9 ] assume the SU (3)-symmetric value (gs ) o = (Clq) o while others allow for moderated [3,4,10,11 ] or more drastic [ 12-15] violations up to (gS)o=0.5(~ lq)o . In all the cases, however, the value attributed to (gs )o is two to four times more negative (notice the unpleasant reversal of the situation with respect to the u, d case) than predicted by the RHS of eq. (1) with the mass values (2). For all these reasons - and, possibly, the traditional suc- cesses of SU(3)-f lavour symmetry - one tends to regard the strange quark as a light rather than a heavy quark.

The main purpose of this paper is to point out that this could not be the case. We have reconsidered the determination of u-, d- and s-quark masses in the

0370-2693/87/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )

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SVZ-approach obtaining values somewhat above the standard ones. In particular, we find rhs larger than the values quoted m eq. (2) and approaching its constituent mass value, M,/2. Similarly, our analysis leads to a negligible s-quark vacuum condensate, I (QQ)o l = I (gS)ol <<l (¢lq)o]. If confirmed, our findings would open the lnmguing conjecture that these two properties of the strange quark - shared also by c- and b-quarks - could have a deep and general origin and that the s-uark itself should be classi- fied as a heavy rather than a light object.

Before the advent of SVZ- or QCD-sum rules, informanon concerning current quark mass values could be obtained at two distinct levels of reliability. On one hand, current algebra (CA) techniques allowed independent and reliable determinations of the different mass ratios. For instance, from pseudoscalar meson [22] or wider sets [2] of data one finds, respectively,

r_ m d - m u md-- mu

B

2ms - m d - m u 2ms

=0.013_+0.002, 0.012_+0.001. (3a)

Similar analyses of the pseudoscalar meson spec- trum lead also to [23,24]

md + mu r+ - - - -0.031 _+0.007, 0.038. (3b)

2ms

On the other hand, the determination of the absolute values of these masses proved to be a much more dif- ficult task. The well-known CA estimates of quark masses

mu-~4MeV, rnd~7MeV,

ms= 120-150 MeV, (4)

were obtained by Leutwyler [2,25] from the ratios (3) and the crucial assumption of exact SU (6)-symmetry for n and p states. The rehability and accuracy of the latter assumpnon is known to be very scarce [2 ] but the first etimates (4) became classic results and strongly lnfiuenced subsequent work.

One of the greatest merits of SVZ- or QCD-sum rules is the possiblhty of determining current quark masses m a more solid and QCD-inspired context. As is well known, these sum rules exploit the idea of Q 2-duahty [ 26 ] between two versmns of the vacuum

polarization induced by the different currents Jr (x) appearing in the two-point functions

Hrr (Q2 = _ q 2 )

--ifd4xe'qx(OIT{Jr(X)jF,(O)}lO) . (5)

The QCD version of a given sum rule contains the mass and vacuum condensate parameters we are interested in. Its corresponding phenomenological version contains the mass MR and decay constant fa of the (one or two) lowest resonances saturating the sum rule in the narrow w~dth approximation, FR-~0. Duality establishes the generic equivalence between these two versions of a given sum rule or, more spe- cifically, between their improved (usually by a Borel- or Laplace-transformation) expressmns. Quark masses and condensates can thus be related to the resonance parameters MR and fR. Since the latter parameter ~s experimentally known only for a reduced set of low-mass states (fp, for pseudoscalar mesons P = n, K; fv, for vector mesons V = p, co, ~; and, with much larger uncertainties, fs, for scalar mesons S = 8,

), the relevant sum rules are restricted to two-point functions (5) involving (pseudo-) scalar and (axial-) vector currents. Neglecting the Q2_ (or Borel mass M2-) dependence, all the interesting sum rules can be grouped in two sets of opposite parity

Hpp=~ffs, HIA=Hs, H 2 A - D s , HpA,

Hss - ~u, H~v = H, H2vv --- D, /-/sv, (6)

where the two indices 1, 2 stand for the transverse and longitudinal tensor combinations [ 1,4]. The last sum rule m each set, Hpa and Hsv, can be related to the three preceding ones by a Ward identity thus containing no further reformation. The sum rule H~v (H~,A) has to be saturated by both vector V and salar S resonances (axlal-vetor A and pseudoscalar P states) and contains quark masses and condensates only as non-dominant correctmns. For these reasons they are appropriate to discuss the (axial-) vector propernes - such as (AI and) p, ~ masses and cou- plings - but not for determining light quark masses and condensates (see below). By contrast, these parameters appear m the dominant terms of the two pairs of remaining sum rules to which we now turn our attennon.

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The pair of sum rules/-/pp---- V5 and H s s - V lead to well-known expressions proportional to the sum and difference of quark masses. Particularizing to the u- and s-quarks, those sum rules are most conveniently written as

2

ms+m"fR - ~ 4 n __~ exp ( _ m ~/2M2 )

X (Pro (M 2) + N P ( M 2)

1 i ) - l/2 M2 dt tF(t) e x p ( - t / M 2)

lc

, (7)

where the + signs refer to R= K and 1¢, respectively, with fK.~ defined through the expressmns

(010~,Au(O) I K) = (ms + mu) (0 ligysu I K)

,

<OlO~,v~'(O) l~c >=(ms-m.)<OlxgulK >

=x/2f .M 2 . (8)

In eq. (7) Pm(M 2) is the perturbative part given by

P~(M 2) - 1 - G ~ +4.82adTr + 18.5 (adz02 ,

with

G,~ =- [rnZ~ + m2~ + (m~T- m~)Z]/M 2 ,

whxch up to terms in (order) 2 is in agreement with refs. [16,17]; NP (M 2) is the non-perturbative contribuUon

N P ( M 2) =- { 7r2 [q/+ (0) - 3~:~ (0)

+ ½ ( ( otI~)GG ) ]/M 4 ,

with

q/_+ (0) = - (ms + m.) <~s_+ QU>o,

playing only a negligible role; containing

and the integral

F(t) = 1 + (oq/~)[~ z-21n(t /M2)]

+ ( a J~ )2 1 2 6 .7 - 31.7 ln(t/M 2)

+4.25 ln 2( t/M2) ] ,

takes into account the contribuUons of resonances other than the single, lowest mass one included in eq. (7). Consequently the threshold tc should be above ME and large enough in order to achieve the dominance of the purely perturbatlve contribution of QCD. A similar treatment can be given to the final pair of sum rules /TZAA and//2vv. One now obtains two expressions which contain the sum and the difference of quark condensates and depend on the ratio defined through eq. (7). For convemence, these sum rules can be written as

(0U)o + (gS)o 3 (ms+-mu~ fR - 87r2 k ~ ,/M2

x (P c (M 2 ) - N P ( M 2 )

- 1 i d t F ( t ) exp(- t /M2) ) M z tc

D 2 C I fRmu ) \ MaR exp( \ m s +

with a perturbatlve contrabution now given by

Pc(M 2) - 1 + G + + 6.82,xs/u + 53.4(,xs/u) z ,

where

G~ - [2/(r~q +mu)l

(9)

X [(rn q3/M 2) ln(rn ~/M 2) - (q--*u) ]

+ (1 + yE)(rnq T fflu) 2]M2

- ( 1 - yE)( mZq + m~)/M 2 ,

and y E = 0.577 is the Euler constant. At first sight, the most reliable information on the

strange quark mass seems to be deducible from the sum rule for/-/pp given by eq. (7) saturated by the K meson, whose decay constant fK is accurately known. In this way (or with minor variations) some authors [ 16,18] have obtained (for A - 100 MeV)

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rh~ +rh~ =305 MeV, 240 MeV. (10a)

The criticism to this type of procedure seems obvious: one cannot trust perturbative QCD expressions for the small thresholds, tc ~ I GeV 2, involved when saturating by one low-mass state like the K-meson. For this reason, many authors introduce a second, K ' - state (or more) in the sum rule. This leads to [16,7,18]

rh~ + rh~

=315MeV, 313+49MeV,

300 + 40 MeV. (10b)

The problem is now that the K radial excitations play a substantial role but their decay constants are unknown. They have to be introduced in a model- dependent way or eliminated making use of less reliable sum rules. Therefore, in spite of the reasonable agreement between the results quoted in eqs. (10), independent estimates of rh~ seem desirable.

To this aim we propose to consider the sum rule for Hss given in eq. (7). It has to be saturated by the scalar ~¢-meson, thus solving the previously encountered problem of the low value of t¢ associated to a single resonance. Indeed, the ~:-meson mass is [27] M~ - 1.35 GeV and one should expect threshold values in the range tc= 3-4 GeV 2. For tc= 3.5 GeV 2 and [27,28 ] A = 150 MeV the M2-dependence of the RHS of eq. (7) has been plotted in fig. 1. Also shown are the curves corresponding to other reasonable thresholds tc = 3.3 GeV 2 and 3.7 GeV 2. The stabihty in M 2, as well as in the value of tc, is quite satisfactory. It allows us to deduce

rh~- rh~ = (7.0 + 0.7)f~, (11)

which constitutes one of our main results. The error quoted in eq. (11) accounts for the uncertainties associated with M 2 and t¢ dependence, as well as on the use of inappropriate values of rh,, eqs. (2) and (10), m the neghgibly small (at least for large M 2) mass corrections appearing on the RHS of eq. (7). For future reference we also quote the analogue result [ 29 ] corresponding to the Hss sum rule for rhd--thu. The saturanon ~s now performed with the 8-meson of M~ = 0.98 GeV and threshold values around t¢ = 2 GeV 2. The results are also shown m fig. 1 from which

one can deduce

rha - fftu = (5.4 + 0.3)f~. (12)

As previously stated, the main problem in eq. (11 ) is the poor knowledge one has of the decay constant f~. We now attempt an estimate o f f , following the lines and notation of refs. [9,30]. Weak, semilep- tonic K~ 3 decays proceed involving two independent form factors. The vector form factor f+ (t) has a t- dependence dominated by the K* pole and at t = 0 takes the value jr+ ( 0 ) = 0.97, not far from the exact SU(3) result, f + ( 0 ) = l , as predicted by the Ade- mollo-Gatto theorem. The scalar form factor

fo( t) =d( t)/( M 2 - M ~ )

=f+ (0)(1 +2o/M~ z ) ,

has a t-dependene which is compatible (the experi- mental values for 2o are rather uncertain [ 27 ]) with the attractive idea of K-meson dominance leading to 2o =M~/M~ =0.011. in this case, one can write

d(t) - (M~ - M ~ )fo (t)

= (x/2/x/-3)g~K~f~M~/CM~ - t ) , (13)

where, apart from an SU (2) Clebsch-Gordan and the t-dependent factor, one hasf~ defined in eq. (8) and g~K~ = 5 GeV, as follows from the ~:-~K~ decay width

)=g~K~ [P~[/8~rM2 ---250 MeV [27]. At F(~:---,K~ 2 t=0 , wherefo(0)=f÷ (0)=0.97, eq. (13) implies

f~ = 50 MeV. (14)

With this value, our final prediction from eq. (11 ) is

rhs-rhu = 350 MeV, (15)

a value affected by rather large errors (typically some 20-30%) but certainly larger than most previous estimates.

Three among these prevaous estimates deserve special attention. Our result (15) is compatible with the lower bound coming from the similar analys~s of ref. [ 9 ] but it is larger than the value quoted in ref. [ 10 ], where more sophlstiated models for resonance saturation are used. A third comparison can be made with ref. [7], where one predicts f~ =38 MeV making use of the standard value rhs - rhu= 300 MeV. These numbers are in perfect agreement with eq. (11 ) thus supporting our sum rule analysis. Additional

350


<E I

,E f

I I I I 2 3 4 5

A = 1 5 0 M e V

I l I I 6 7 8 9

M 2 (OeV 21

Fig 1. M2-dependence of the ratio (ffzs-~u)/f~ (higher curves) and (~d--Vh,)/f8 (lower curves), eq. (7), for A= 150 MeV and the continuum threshold tc as md~eated

confidence on our results can be obtained from a parallel analysis of J~ appearing in eq. (12). We have performed different estimates leading to [29] f~ = 2.1 MeV and, through eq. (12), to

ffZd--~u----- 11 MeV, (16)

which are values somewhat larger than the standard ones [7,9]. The ratio between this result and eq. (15) gives r = 0.014 in good agreement with the CA result eq. (3a). There is a final attractive feature in eq. (15), namely, the predicted similarity between the current and constituent s-quark masses, which resembles the situation encountered for the heavier c- and b-quarks. At this point, an analysis of the s-quark vacuum condensate seems to be in order.

As previously discussed, eq. (9), corresponding to the H2vv sum rule saturated by the ~:-resonance, is appropriate for that purpose. Using eq. (11 ) and threshold values around to=4 GeV 2 one finds the stable results shown in fig. 2 or, equivalently, the estimate (A = 150 MeV)

<~U>o - <~S>o -~ -0 .15 GeV2f~ (GeV)

- - 0 . 0 0 7 5 GeV 3 , (17)

where eq. (14) has been used in the last steps. This estimate has to be compared with the value for < Ou) o = <dd) o coming from the well-known and safe CA relation

(rhd +~.) <~U>o = - ~ M ~ , (18)

and the results rhd+rh,___22 MeV or 340 MeV, following from the CA ratios (3) and our estimates eqs. (15) or (16). One obtains

( Q u > o = - 0 . 0 0 7 7 or -0 .0057 GeV 3 ,

thus implying I < ~s > o I << 1 <fm > o I when comparing with eq. (17). This smallness of the s-quark vacuum condensate constitutes our second main result.

It is interesting to analyze the consequences that our results on the s-quark mass and condensate can have in other significant sum rules of the set (6). One of them is the so-called kaon PCAC sum rule, largely discussed by several authors [ 14,19 ], which follows from saturating the/72AA sum rule, eq. (7), with the K-meson. It establishes the equivalence between one positive term containing the known factors f ~ and the sum of two positive and almost equal contri- butions: one given by - (m~+m,) (<gs>o+ <flU>o) and the other proportional to m 2. The first contn-

351


0 24

>

(._9 02

o16

A

IZ3 V 0.12

I

A ~f) 0.08

109

V

0 0 4

=================================

A

I I I 1 I 2 5 4 5 6

= 1 5 0 MeV

I I I I 7 8 9

M 2 (GeV 2)

Fig. 2 M2-dependence of the ratio ( ( gs ) o- ( Ou ) 0)/f,:, eq. (9) for A = 150 MeV and different continuum thresholds to. All umts are m GeV

butlon is reduced by a factor o f two in our analysis, while the second one is enhanced by a similar factor. The global equivalence is roughly preserved and no contradiction appears with other analyses. The same happens in the so-called 0-meson sum rule from which very different and controversial values for m, have been obtained [ 11,12,17]. In this case one has to deal with the combined effect o f a negative term proportional to m ~ and a second negative term containing ms (gs ) o. Small values o f ms combined with largely negative values of (gs ) o are equivalent to our solution with large ms and negligible condensate.

A comparison of our results with those coming from lattice measurements could also be o f interest. QCD-lattlce results for the (bare) s-quark mass range typically [ 31-33 ] from 50 to 80 MeV and are rather sensitive (a factor of two [34]) to the use o f the alternative Kogut-Sussklnd or Wilson formulations. When invariant masses are presented [ 32] the above values increase up to 1200-200 MeV. In any case, these lattice measurements are in better agreement with the small (conventional) SVZ-results than with the larger rhs value obtained in our present analysis. Concerning the s-quark condensate no QCD-lattice measurements seem available even if those for the u-

and d-quark condensates (or the related pion-decay constant f~ ) are quite promising [ 31-33 ].

In conclusion, the SVZ-sum rules giving relevant information on the strange quark mass and vacuum condensate have been briefly discussed. Two of those sum rules seem particularly interesting due to their saturation in terms of the rather high mass K-resonance, which requires a correspondingly large con- t inuum threshold allowing for accurate and reliable QCD calculations. One obtains reasonably stable results implying an invariant current mass for the s- quark, rh,_~ 360 MeV, larger than most previous estimates. Similarly, a negligible value for the s-quark vacuum condensate, I <gs> I<<1 <tau>01 --- I ( d d ) o [, follows from our analysis. The accuracy of our two results is rather scarce due to their crucial dependene on the K-meson decay constant, which, unfortunately, is not precisely known. For this reason, further work on this scalar r -meson and their dominance of the scalar form factor in Kl 3 decays is desirable. I f it confirms the present day results used in our analysis, an Interesting picture seems to emerge: u and d quarks, having current masses much smaller than the constituent ones and non-negligible vacuum condensates, are genuinely light quarks; by

352


c o n t r a s t the s -quark , w h o s e c u r r e n t m a s s is f o u n d to

a p p r o a c h i ts c o n s t i t u e n t m a s s v a l u e a n d w h o s e vac-

u u m c o n d e n s a t e ha s a neg l ig ib le va lue , b e h a v e s ( a n d

s h o u l d b e t r e a t e d ) l ike t he so-ca l led h e a v y q u a r k s c

a n d b.

O n e o f t he a u t h o r s ( A . B . ) t h a n k s C A I C Y T for

f i n a n c i a l s u p p o r t a n d A. T a r a n c 6 n for usefu l

c o m m e n t s .

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Is the strange quark heavy or light?