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Systematics of strong interaction radii for hadrons
Volume 245, number 3, 4
PHYSICS LETTERS B
16 August 1990
Systematics of strong interaction radii for hadrons B. P o v h a n d J. H i i f n e r Max-Planck-lnstimt fiir Kernphysik Heidelberg, D-6900 Heidelberg, FRG and lnstitut fiir Theoretische Physik der Universitiit, D-6900 Heidelberg, FRG Received 21 May 1990
Hadron-proton total cross sections and slope parameters can be used to determine strong interaction radii of hadrons. Those radii agree with charge radii, where available. We present values for the RMS radii of A, E, E, p, ~band J/0. A quark model with constituent quarks accounts for the absolute values as well as the flavour dependence of the radii if one attributes a mean squared radius (r~) = m( 2to the constituent quarks q with effectivemass mq. We discuss possible origins of this relation.
The radius of a hadron is not an unambiguously defined observable but depends on the probe used in the experiment. The interaction of electrons with a hadron h yields the mean squared charge radius (r2¢h)h. We shall show how a strong interaction mean squared radius (r~)h 2 can be extracted from h a d r o n - p r o t o n (hp) collisions. The charge radius of a spin zero hadron is obtained from the form factor Fh(q) for elastic electron-hadron collisions for small momentum transfers q2 IG(q)l 2 = 1 - ~ 1q 2 2( r c h ) h + O ( q ) 4.
(1)
Because of the nature of the strong interaction there exists no generally agreed procedure for the definition of a strong interaction radius. Wu and Yang [ 1] have proposed that high-energy elastic hadron proton cross sections dtr~tp/dq -~ are related to the charge form factors of projectile and target hadron via d (_el Ihp
= const. If.(q)12lFp(q)[ 2.
(2)
This relation, if true, implies that the slope parameter bhp, which determines the q2 dependence of do-~l/dq 2 ~ exp(-bhpq 2) at small q: is related to the RMS charge radii via _1 2 2 bhp--~((rch)hq-(rch)p).
(3)
We define a mean squared strong interaction radius
by the relation 2
__
1
(rst)h - 3[bhp- ~b~,],
(4)
where b,;o, is a mean value for the slope parameter from pp and ~p scattering. The hypothesis by Wu and Yang [1] requires that strong interaction radii are equal to the charge radii. This equality can only hold approximately since (Gt), 2 is different for a hadron and antihadron and also depends on the CM scattering energy x/~. Both effects are absent in eq. (2), but are included in more refined versions of theories for high-energy elastic collisions, like the Pommeron exchange model [2]. We have compared charge radii with strong interaction radii deduced at v~ = 16 GeV. We have formed averages of the strong interaction radii for h and h. The result is displayed in fig. l a for p, ~r and k, for which charge radii as well as slope parameters are available. The comparison shows close agreement between (r~,)h 2 and (rch)h 2 and supports the hypothesis by Wu and Yang. In the foreseeable future there is little hope to have data on charge radii for other hadrons like ~ , w or p±. For any information on their sizes as well as on sizes of neutral particles like A, ~b, J/0, we have to rely on strong interaction results, though their significance is less clear. Unfortunately, eq. (4) is not a very convenient relation to determine radii for two reasons: (i) Since (r~,)h is proportional to a difference of two quantities, the
0370-2693/905 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
653
Volume 245, number 3, 4
1.0
PHYSICS LETTERS B I
(a)
/ / /
E
/
,,...-,
2 2 h = h
0.5 V / /
0
i
i
i
0
I
i
i
i
I
1.0
I
50
i
0.5 /
I
(b)
4O
..¢,- p / _--
E
3o
b
20
~
-Vs = 16 GeV
/ ,,
K+:~ K-
'~-
/
10 +?-// / 0
i
0
,
I
0.5
,
I
i
i
]
1.o.
2]
Fig. l(a). Relation between the strong interaction hadronic radii defined in eq. (4) and the charge radii for proton, pion and kaon. (b) Relation between the hadron-proton total cross section and the strong interaction mean squared radius of a hadron, defined by eq. (4).
errors can be sizable and (ii) experimental values of slope parameters are not available for all hadrons. Total cross section data are more abundant. Recently we have discovered [3] an empirical linear relation between the total cross section ~r~p and the quantity 2 (r~,)h both quantities taken at the same value of s. For small hadrons, a linear relation between ~r~hpand (r~)h is derived by G u n i o n and Soper [4] within the Low model [5] of two gluon exchange. We shall use the empirical systematics, fig. lb, to determine strong interaction radii from total cross section measurements via the relation (rst)h
=
~ ~ t(o ts ) (r~,)p ~ ( ~ )
2 3 (r~,)p - ~b~,I (s).
(5)
The values of the strong interaction radii together 654
16 August 1990
with electromagnetic radii (where available) are listed in table 1. It is for the first time that such an extended list for hadronic radii is available. Although the notion of a strong interaction radius is less well founded than that of the charge radius the data presented in table 1 may provide important insight into the systematics of hadronic sizes. We discuss them in detail. (i) The values for electromagnetic and strong interaction radii are very similar (as displayed already in fig. la), except of course for neutral particles like the K ° for which the hadronic radius is expected to be as large as the one for K ±. (ii) The strangeness content of a hadron has a clear effect on its radius. Hadrons become smaller when they contain more strange valence quarks. More quantitatively: The values of (r~t)h decrease by nearly constant steps 6(r 2) = (0.08 + 0.03) fm 2 when the number of strange valence quarks increases by one unit. This is so for the J=½+ baryons in the chain p ~ (A, Z ) ~ 7=, for the 0- mesons ~r ~ K and for the 1mesons p--, ~b, where in the latter case the difference corresponds to two steps since the + is predominantly an sg system. (iii) The comparison of the radii for 7r and p mesons displays a strong spin dependence. As expected on general grounds, the pion which more b o u n d has a smaller radius. How can the absolute values and the systematics of the data be understood? We briefly report on the vector d o m i n a n c e model and quote results from a relativistic bag model but have a more detailed discussion within the non-relativistic quark model. In the vector dominance model the charge radii of the baryons and mesons are related to the masses of the vector mesons p, to and + into which the photon converts before interacting with the hadron. In its most naive version (point like hadron and single pole dominance) one has 2
---- (r~h)p__
6
~
= 1 { 6 + 6 '~ 2
2
(6) "
It fits well the ~r± and K ± and K ° data for charge radii but fails for the baryons. In strong interactions other mesons should play the role of p, to, +. Therefore in the vector dominance model it is not obvious that strong interaction and charge radii should be similar. The relativistic bag model [16] starting with massless
Volume 245, number 3, 4
PHYSICS LETTERS B
16 August 1990
Table 1 Experimental values for the mean squared charge and strong interaction radii (r 2) in fm 2 of hadrons and predictions from various models. The experimental values for the strong interaction radii have been deduced from the cross section measurements of refs. [6-11] using eq. (5). The errors are those from the cross section measurements only and not from the possibility that eq. (5) may hold only approximately. The charge radii are from refs. [12-15]. The values of (r 2) for the vector dominance model are calculated from eq. (6) The predictions for the relativistic bag model have been evaluated from the formulae and Table 111 of ref. [16]. The values from the non-relativistic mode contain a contribution from the wave function calculated as described in the text and the contribution A, eq. (9), from the apparent size of the constituent quarks. Hadron
Experiment strong interaction
p A, 52 f~ ~r p, to K÷ K° cb
3/0
Theory charge
vector dominance model
0.67 ± 0.02 0.58 ± 0.02 0.50 ± 0.02
0.67 ± 0.02
0.40 0.34 0.29
0.41 ± 0.02 0.52 ~0.05 0.35 ± 0.02
0.44 ± 0.01
0.40
0.21 ±0.02 0.04± 0.02
0.34 ± 0.05 -0.054+0.026
0.53 0.49 0.46 0.51 0.24 0.47 0.21 -0.011 0.38
0.32 -0.09 0.23
light and massive strange quarks r e p r o d u c e s m e a n s q u a r e d radii and their flavour d e p e n d e n c e semiq u a n t i t a t i v e l y i.e. on the level o f 20% to 40%. In most cases the p r e d i c t e d v a l u e is smaller than the experim e n t a l one. The d e p e n d e n c e on the strangeness content o f the h a d r o n s has the right trend (except for the ~b) but is too weak. The non-relativistic q u a r k m o d e l [17] is r e m a r k a b l y successful in the p r e d i c t i o n o f h a d r o n i c spectra. O n e starts f r o m a non-relativistic h a m i l t o n i a n ( t h o u g h velocities are fairly high) in which constituent quarks with effective masses m, interact via effective forces. The c o n s t i t u e n t quarks are a s s u m e d point-like. F e w c a l c u l a t e d values for h a d r o n i c radii are published. In o r d e r to be able to c o m p a r e with the i n c r e a s e d b o d y o f d a t a in table 1, we h a v e c a l c u l a t e d w a v e functions a n d their radii by the v a r i a t i o n a l m e t h o d . We h a v e n e g l e c t e d all spin d e p e n d e n c e s . T h e n the h a m i l t o n i a n for a m e s o n is 2 1 /4 = P- - ~ s - + K r , 2tz r
relativistic bag model
(7)
w h e r e / z is the r e d u c e d mass. A similar h a m i l t o n i a n is a s s u m e d for the baryons. We have used masses o f
non-relativistic quark model wave function
A
(r 2)
0.27 0.24 0.21 0.18 0.14 0.14 0.13 -0.019 0.11 0.04
0.40 0.31 0.23 0.14 0.40 0.40 0.27 -0.085 0.14 0.02
0.67 0.55 0.44 0.32 0.54 0.54 0.40 -0.10 0.25 0.06
0.3 G e V and 0.5 G e V for the light and strange constituent quarks, respectively. F u r t h e r m o r e a ~ = 0 . 5 and K = 1 G e V / f m where b o t h values are a s s u m e d i n d e p e n d e n t o f strangeness. The calculated values o f (r 2) are given in table 1 in the c o l u m n " w a v e funct i o n " . The values are too small by m o r e than a factor two. This result is not c h a n g e d by m o r e s o p h i s t i c a t e d calculations [17]. F u r t h e r m o r e , a l t h o u g h the predicted radii b e c o m e smaller with increasing strangeness content, this t r e n d is too w e a k c o m p a r e d to experiment. This deficiency o f the non-relativistic q u a r k m o d e l has b e e n k n o w n b e f o r e [18], but b e c o m e s even m o r e visible with the i n c r e a s e d b o d y o f data presented in this paper. It has b e e n s u s p e c t e d for s o m e time that the a s s u m p t i o n that c o n s t i t u e n t quarks are point-like might be r e s p o n s i b l e for the d i s p r e p a n c y b e t w e e n theory and data. For instance in the c a l c u l a t i o n o f widths for w e a k decays o f m e s o n s it has b e e n o b s e r v e d that one s h o u l d not use 14,(0)12, the nonrelativistic w a v e f u n c t i o n o f the relative m o t i o n o f the two q u a r k s at r = 0 but rather at r = 1/rnq. This p h e n o m e n o l o g i c a l p r o c e d u r e is s u p p o r t e d in theoretical w o r k by Poggio et al. [19] and K r a s e m a n n [20]. We f o l l o w the suggestion o f the earlier authors, 655
Volume 245, number 3, 4
PHYSICS LETTERS B
e.g. ref. [18], and simply associate a mean squared radius
(r~)=~q~
(81
to each constituent quark q. The numerical value o f the coefficient ~ d e p e n d s on the effective quark masses mq. For our choice of masses we need ~ = 0.9 in order to make experimental and calculated values for the proton radius to agree exactly. Therefore, a quantity A has to be a d d e d to the mean squared radius obtained from a wave function calculation within the non-relativistic quark model, where 1
A~t = -
Y~ n q=l --m2q,
Ah - L eq ~ , q=l mq
(9)
for strong interaction and charge radii, respectively. Here eq are the charges o f the constituent quarks in units of the elementary charge e. A d d i n g the terms eq. (9) to the contribution from the wave function one reproduces the experimental radii well in absolute magnitude and flavour d e p e n d e n c e (table 1). The value for the pion is an exception. One would obtain a perfect fit if the radius from the non-relativistic quark model is set equal to zero. Strong spin forces which account for the large w-p mass difference and which are not taken into account in this calculation may be resposible for the shrinking of the pion wave function as c o m p a r e d to the model calculation based on eq. (7) (cf. also the values for w and p calculated in the relativistic bag model). A similar reasoning as for the w may reduce the discrepancy for the radii o f the kaon. We draw the attention to the fact that our values A, which account for the a p p a r e n t size of the constituent quarks, nearly equal the sole contribution o f the vector d o m i n a n c e model to the hadronic radius. This numerical coincidence does not seem accidental to us but rather points to some common physics. What could be the origin of the apparent radius ~-l/mq o f the constituent quark? We list various effects, parts of them have already been discussed in the literature. We use Q E D as a guide line: (1) Darwin term: The non-relativistic reduction of the Dirac equation for an electron moving in the C o u l o m b potential a i r leads to the a p p e a r a n c e of various terms, among them the Darwin term (see, for instance, ref. [21]): The virtual creation o f particleantiparticle pair in the C o u l o m b field leads to a 656
16 August 1990
fluctuation ("Zitterbewegung") in the position coordinate of the electron, which appears in the nonrelativistic Schr/Sdinger picture as an a p p a r e n t size o f the moving electron, this size being 6 (r2)~-----8m~'
(10)
This effect always appears when going from a relativistic to a non-relativistic position coordinate and is calculated by Hayne and Isgur [18] for the nonrelativistic quark model. (2) Lamb shift: For an electron b o u n d in an electromagnetic potential terms in addition to those created by the Dirac equation appear. For instance, the emission of a p h o t o n before the interaction of the electron with the proton leads to a displacement of the electron whose effect on the Lamb shift can be simulated by attributing an apparent radius (rZ)eL to the electron r ?e=_Z~_2 c + l n me
,
(11)
where c is a numerical constant and too a cut-off frequency. Effects of this type have been calculated by Poggio et al. [19] for the non-relativistic quark model. (3) Condensate: An ab initio calculation of a h a d r o n within Q C D has to start with current quarks whose masses are very small. A not yet well understood mechanism (chiral symmetry breaking h la N a m b u - J o n a - L a s i n i o [22] to form a condensate inside the hadron?) allows to pass from point-like and massless u, d quarks and gluons to a largely non-relativistic problem where constituent quarks with sizable masses interact via effective forces. The forces o f this model problem, among them linear confinement, are relatively weak. It is very likely that the mechanism which leads from the nearly zero mass and pointlike current quarks to the constituent quarks is not only responsible for the effective masses mq but also for an effective size o f the constituent quarks. After the completion o f this work we received a preprint by Klimt et al. [23] in which they calculate the radius of constituent quarks within the N a m b u Jona-Lasinio model. Translated into the language of eq. (8) they find ~:= 0.3. In this p a p e r we have presented strong interaction radii d e d u c e d from total cross sections for most
Volume 245, number 3, 4
PHYSICS LETTERS B
known stable hadrons. Wherever the comparison is possible, the strong interaction radii are very similar to the charge radii. Therefore, we are confident about the physical significance of the strong interaction radii. The most important result from this new body of data is the dependence of the radii on strangeness. It seems beyond doubt that hadrons become smaller with increasing n u m b e r of strange valence quarks. Within quark models such a dependence is expected since a larger mass of the quarks reduces the kinetic energy and allows for a smaller system. However, calculations show that this expected dependence in the relativistic and non-relativistic quark models is too weak compared to experiment. In the non-relativistic models this deficiency can be overcome by attributing an apparent radius to each constituent quark. We find ,2\1/2 r /u,d
0.6fm,
(r2}~/2=0.4fm.
(12)
We note that the apparent radius is sizable. Its contribution to the hadronic radius is comparable or even larger than that obtained from the wave function so that the constituent quarks overlap considerably. Adding the contributions from the wave function and the apparent size together quantitatively accounts for the absolute values of the radii as well as their dependence on strangeness. The ideas presented in this paper support the following very qualitative picture o f a hadron: The interactions among the partons inside a hadron may be seen in a hierarchy: (i) The very strong interaction of a quark with its (gluon) surrounding gives rise to a constituent quark with effective mass mq and an effective radius (r2}qt/2• 1/mq. (ii) The fairly weak residual interaction among the constituent quarks is responsible for hadronic spectra. While the first process is a genuine relativistic and possibly nonperturbative effect, the final system of interacting constituent quarks (gluons being absent!) seems to be treatable approximately as a non-relativistic problem. An important part of the non-perturbative effects namely the formation of resonant intermediate states
16 August 1990
seems correctly described in the vector dominance model, though in a language of the pre-quark era. Clearly our interpretation of the hadronic radii in terms of apparent quark radii is very speculative. However, the existence of a new set of data for radii which we have presented in this paper may provide helpful constraints on future theoretical work. We gratefully acknowledge many stimulating discussions in particular with S. Brodsky, D. Gromes, O. N a c h t m a n n , H.J. Pirner, and M. Strikman. The work has been supported in part by the Bundesminister fiir Forschung u n d Technologie under contract 06 HD710.
References [1] T.T. Wu and C.N. Yang, Phys. Rev. B 137 (1965) 708. [2] A. Donnachie and P.V. Landshott, Nucl. Phys. B 267 (1986) 690. [3] B. Povh and J. Hfifner, Phys. Rev. Lett. 58 (1987) 1612. [4] J.F. Gunion and D.E. Soper, Phys. Rev. D 15 (1977) 2617. [5] F.E. Low, Phys. Rev. D 12 (1975) 163. [6] D.S. Ayres et al., Phys. Rev. D 15 (1977) 3105. [7] S. Gjesdal et al., Phys. Lett. B 40 (1972) 152. [8] S.F. Biagi et al., Z. Phys. C 17 (1983) 113. [9] T.H. Bauer, R.D. Spital, D.R. Yennie and F.M. Pipkin, Rev. Mod. Phys. 50 (1978) 261. [10] J.J. Aubert et al., Nucl. Phys. B 213 (1983) 1. [11] M. Bozzo et al., Phys. Lett. B 147 385 (1984) 392. [12] F. Borkowski et al., Nucl. Phys. B 93 (1975) 461. [13] S.R. Amendolia et al., Nucl. Phys. B 277 (1986) 168. [14] S.R. Amendolia et al., Phys. Lett. B 178 (1986) 435. [15] W.R. Molzon et al., Phys. Rev. Lett. 41 (1978) 1213. [16] T. DeGrand, R.L. Jaffe, K. Johson and J. Kiskis, Phys. Rev. D 12 (1975) 2060. [17] S. Godfrey and N. Isgur, Phys. Rev. D 32 (1985) 189. [18] C. Hayne and N. Isgur, Phys. Rev. D 25 (1982) 1944. [19] E.C. Poggio and H.J. Schnitzer, Phys. Rev. D 20 (1979) 1175. [20] H. Krasemann, Phys. Lett. B 96 (1980) 397. [21] J.D. Bjorken and S.D. Drell, Relativistic quantum mechanics (McGraw-Hill, New York, 1964). [22] Y. Nambu and G. Jona-Lasinio, Phys. Rev. 122 (1961) 345; 124 (1961) 246. [23] S. Klimt, M. Lutz, U. Vogl and W. Weise,preprint TPR-8938.
657
PHYSICS LETTERS B
16 August 1990
Systematics of strong interaction radii for hadrons B. P o v h a n d J. H i i f n e r Max-Planck-lnstimt fiir Kernphysik Heidelberg, D-6900 Heidelberg, FRG and lnstitut fiir Theoretische Physik der Universitiit, D-6900 Heidelberg, FRG Received 21 May 1990
Hadron-proton total cross sections and slope parameters can be used to determine strong interaction radii of hadrons. Those radii agree with charge radii, where available. We present values for the RMS radii of A, E, E, p, ~band J/0. A quark model with constituent quarks accounts for the absolute values as well as the flavour dependence of the radii if one attributes a mean squared radius (r~) = m( 2to the constituent quarks q with effectivemass mq. We discuss possible origins of this relation.
The radius of a hadron is not an unambiguously defined observable but depends on the probe used in the experiment. The interaction of electrons with a hadron h yields the mean squared charge radius (r2¢h)h. We shall show how a strong interaction mean squared radius (r~)h 2 can be extracted from h a d r o n - p r o t o n (hp) collisions. The charge radius of a spin zero hadron is obtained from the form factor Fh(q) for elastic electron-hadron collisions for small momentum transfers q2 IG(q)l 2 = 1 - ~ 1q 2 2( r c h ) h + O ( q ) 4.
(1)
Because of the nature of the strong interaction there exists no generally agreed procedure for the definition of a strong interaction radius. Wu and Yang [ 1] have proposed that high-energy elastic hadron proton cross sections dtr~tp/dq -~ are related to the charge form factors of projectile and target hadron via d (_el Ihp
= const. If.(q)12lFp(q)[ 2.
(2)
This relation, if true, implies that the slope parameter bhp, which determines the q2 dependence of do-~l/dq 2 ~ exp(-bhpq 2) at small q: is related to the RMS charge radii via _1 2 2 bhp--~((rch)hq-(rch)p).
(3)
We define a mean squared strong interaction radius
by the relation 2
__
1
(rst)h - 3[bhp- ~b~,],
(4)
where b,;o, is a mean value for the slope parameter from pp and ~p scattering. The hypothesis by Wu and Yang [1] requires that strong interaction radii are equal to the charge radii. This equality can only hold approximately since (Gt), 2 is different for a hadron and antihadron and also depends on the CM scattering energy x/~. Both effects are absent in eq. (2), but are included in more refined versions of theories for high-energy elastic collisions, like the Pommeron exchange model [2]. We have compared charge radii with strong interaction radii deduced at v~ = 16 GeV. We have formed averages of the strong interaction radii for h and h. The result is displayed in fig. l a for p, ~r and k, for which charge radii as well as slope parameters are available. The comparison shows close agreement between (r~,)h 2 and (rch)h 2 and supports the hypothesis by Wu and Yang. In the foreseeable future there is little hope to have data on charge radii for other hadrons like ~ , w or p±. For any information on their sizes as well as on sizes of neutral particles like A, ~b, J/0, we have to rely on strong interaction results, though their significance is less clear. Unfortunately, eq. (4) is not a very convenient relation to determine radii for two reasons: (i) Since (r~,)h is proportional to a difference of two quantities, the
0370-2693/905 03.50 © 1990 - Elsevier Science Publishers B.V. (North-Holland)
653
Volume 245, number 3, 4
1.0
PHYSICS LETTERS B I
(a)
/ / /
E
/
,,...-,
2 2
0.5 V / /
0
i
i
i
0
I
i
i
i
I
1.0
I
50
i
0.5 /
I
(b)
4O
..¢,- p / _--
E
3o
b
20
~
-Vs = 16 GeV
/ ,,
K+:~ K-
'~-
/
10 +?-// / 0
i
0
,
I
0.5
,
I
i
i
]
1.o
2]
Fig. l(a). Relation between the strong interaction hadronic radii defined in eq. (4) and the charge radii for proton, pion and kaon. (b) Relation between the hadron-proton total cross section and the strong interaction mean squared radius of a hadron, defined by eq. (4).
errors can be sizable and (ii) experimental values of slope parameters are not available for all hadrons. Total cross section data are more abundant. Recently we have discovered [3] an empirical linear relation between the total cross section ~r~p and the quantity 2 (r~,)h both quantities taken at the same value of s. For small hadrons, a linear relation between ~r~hpand (r~)h is derived by G u n i o n and Soper [4] within the Low model [5] of two gluon exchange. We shall use the empirical systematics, fig. lb, to determine strong interaction radii from total cross section measurements via the relation (rst)h
=
~ ~ t(o ts ) (r~,)p ~ ( ~ )
2 3 (r~,)p - ~b~,I (s).
(5)
The values of the strong interaction radii together 654
16 August 1990
with electromagnetic radii (where available) are listed in table 1. It is for the first time that such an extended list for hadronic radii is available. Although the notion of a strong interaction radius is less well founded than that of the charge radius the data presented in table 1 may provide important insight into the systematics of hadronic sizes. We discuss them in detail. (i) The values for electromagnetic and strong interaction radii are very similar (as displayed already in fig. la), except of course for neutral particles like the K ° for which the hadronic radius is expected to be as large as the one for K ±. (ii) The strangeness content of a hadron has a clear effect on its radius. Hadrons become smaller when they contain more strange valence quarks. More quantitatively: The values of (r~t)h decrease by nearly constant steps 6(r 2) = (0.08 + 0.03) fm 2 when the number of strange valence quarks increases by one unit. This is so for the J=½+ baryons in the chain p ~ (A, Z ) ~ 7=, for the 0- mesons ~r ~ K and for the 1mesons p--, ~b, where in the latter case the difference corresponds to two steps since the + is predominantly an sg system. (iii) The comparison of the radii for 7r and p mesons displays a strong spin dependence. As expected on general grounds, the pion which more b o u n d has a smaller radius. How can the absolute values and the systematics of the data be understood? We briefly report on the vector d o m i n a n c e model and quote results from a relativistic bag model but have a more detailed discussion within the non-relativistic quark model. In the vector dominance model the charge radii of the baryons and mesons are related to the masses of the vector mesons p, to and + into which the photon converts before interacting with the hadron. In its most naive version (point like hadron and single pole dominance) one has 2
---- (r~h)p__
6
~
= 1 { 6 + 6 '~ 2
2
(6) "
It fits well the ~r± and K ± and K ° data for charge radii but fails for the baryons. In strong interactions other mesons should play the role of p, to, +. Therefore in the vector dominance model it is not obvious that strong interaction and charge radii should be similar. The relativistic bag model [16] starting with massless
Volume 245, number 3, 4
PHYSICS LETTERS B
16 August 1990
Table 1 Experimental values for the mean squared charge and strong interaction radii (r 2) in fm 2 of hadrons and predictions from various models. The experimental values for the strong interaction radii have been deduced from the cross section measurements of refs. [6-11] using eq. (5). The errors are those from the cross section measurements only and not from the possibility that eq. (5) may hold only approximately. The charge radii are from refs. [12-15]. The values of (r 2) for the vector dominance model are calculated from eq. (6) The predictions for the relativistic bag model have been evaluated from the formulae and Table 111 of ref. [16]. The values from the non-relativistic mode contain a contribution from the wave function calculated as described in the text and the contribution A, eq. (9), from the apparent size of the constituent quarks. Hadron
Experiment strong interaction
p A, 52 f~ ~r p, to K÷ K° cb
3/0
Theory charge
vector dominance model
0.67 ± 0.02 0.58 ± 0.02 0.50 ± 0.02
0.67 ± 0.02
0.40 0.34 0.29
0.41 ± 0.02 0.52 ~0.05 0.35 ± 0.02
0.44 ± 0.01
0.40
0.21 ±0.02 0.04± 0.02
0.34 ± 0.05 -0.054+0.026
0.53 0.49 0.46 0.51 0.24 0.47 0.21 -0.011 0.38
0.32 -0.09 0.23
light and massive strange quarks r e p r o d u c e s m e a n s q u a r e d radii and their flavour d e p e n d e n c e semiq u a n t i t a t i v e l y i.e. on the level o f 20% to 40%. In most cases the p r e d i c t e d v a l u e is smaller than the experim e n t a l one. The d e p e n d e n c e on the strangeness content o f the h a d r o n s has the right trend (except for the ~b) but is too weak. The non-relativistic q u a r k m o d e l [17] is r e m a r k a b l y successful in the p r e d i c t i o n o f h a d r o n i c spectra. O n e starts f r o m a non-relativistic h a m i l t o n i a n ( t h o u g h velocities are fairly high) in which constituent quarks with effective masses m, interact via effective forces. The c o n s t i t u e n t quarks are a s s u m e d point-like. F e w c a l c u l a t e d values for h a d r o n i c radii are published. In o r d e r to be able to c o m p a r e with the i n c r e a s e d b o d y o f d a t a in table 1, we h a v e c a l c u l a t e d w a v e functions a n d their radii by the v a r i a t i o n a l m e t h o d . We h a v e n e g l e c t e d all spin d e p e n d e n c e s . T h e n the h a m i l t o n i a n for a m e s o n is 2 1 /4 = P- - ~ s - + K r , 2tz r
relativistic bag model
(7)
w h e r e / z is the r e d u c e d mass. A similar h a m i l t o n i a n is a s s u m e d for the baryons. We have used masses o f
non-relativistic quark model wave function
A
(r 2)
0.27 0.24 0.21 0.18 0.14 0.14 0.13 -0.019 0.11 0.04
0.40 0.31 0.23 0.14 0.40 0.40 0.27 -0.085 0.14 0.02
0.67 0.55 0.44 0.32 0.54 0.54 0.40 -0.10 0.25 0.06
0.3 G e V and 0.5 G e V for the light and strange constituent quarks, respectively. F u r t h e r m o r e a ~ = 0 . 5 and K = 1 G e V / f m where b o t h values are a s s u m e d i n d e p e n d e n t o f strangeness. The calculated values o f (r 2) are given in table 1 in the c o l u m n " w a v e funct i o n " . The values are too small by m o r e than a factor two. This result is not c h a n g e d by m o r e s o p h i s t i c a t e d calculations [17]. F u r t h e r m o r e , a l t h o u g h the predicted radii b e c o m e smaller with increasing strangeness content, this t r e n d is too w e a k c o m p a r e d to experiment. This deficiency o f the non-relativistic q u a r k m o d e l has b e e n k n o w n b e f o r e [18], but b e c o m e s even m o r e visible with the i n c r e a s e d b o d y o f data presented in this paper. It has b e e n s u s p e c t e d for s o m e time that the a s s u m p t i o n that c o n s t i t u e n t quarks are point-like might be r e s p o n s i b l e for the d i s p r e p a n c y b e t w e e n theory and data. For instance in the c a l c u l a t i o n o f widths for w e a k decays o f m e s o n s it has b e e n o b s e r v e d that one s h o u l d not use 14,(0)12, the nonrelativistic w a v e f u n c t i o n o f the relative m o t i o n o f the two q u a r k s at r = 0 but rather at r = 1/rnq. This p h e n o m e n o l o g i c a l p r o c e d u r e is s u p p o r t e d in theoretical w o r k by Poggio et al. [19] and K r a s e m a n n [20]. We f o l l o w the suggestion o f the earlier authors, 655
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e.g. ref. [18], and simply associate a mean squared radius
(r~)=~q~
(81
to each constituent quark q. The numerical value o f the coefficient ~ d e p e n d s on the effective quark masses mq. For our choice of masses we need ~ = 0.9 in order to make experimental and calculated values for the proton radius to agree exactly. Therefore, a quantity A has to be a d d e d to the mean squared radius obtained from a wave function calculation within the non-relativistic quark model, where 1
A~t = -
Y~ n q=l --m2q,
Ah - L eq ~ , q=l mq
(9)
for strong interaction and charge radii, respectively. Here eq are the charges o f the constituent quarks in units of the elementary charge e. A d d i n g the terms eq. (9) to the contribution from the wave function one reproduces the experimental radii well in absolute magnitude and flavour d e p e n d e n c e (table 1). The value for the pion is an exception. One would obtain a perfect fit if the radius from the non-relativistic quark model is set equal to zero. Strong spin forces which account for the large w-p mass difference and which are not taken into account in this calculation may be resposible for the shrinking of the pion wave function as c o m p a r e d to the model calculation based on eq. (7) (cf. also the values for w and p calculated in the relativistic bag model). A similar reasoning as for the w may reduce the discrepancy for the radii o f the kaon. We draw the attention to the fact that our values A, which account for the a p p a r e n t size of the constituent quarks, nearly equal the sole contribution o f the vector d o m i n a n c e model to the hadronic radius. This numerical coincidence does not seem accidental to us but rather points to some common physics. What could be the origin of the apparent radius ~-l/mq o f the constituent quark? We list various effects, parts of them have already been discussed in the literature. We use Q E D as a guide line: (1) Darwin term: The non-relativistic reduction of the Dirac equation for an electron moving in the C o u l o m b potential a i r leads to the a p p e a r a n c e of various terms, among them the Darwin term (see, for instance, ref. [21]): The virtual creation o f particleantiparticle pair in the C o u l o m b field leads to a 656
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fluctuation ("Zitterbewegung") in the position coordinate of the electron, which appears in the nonrelativistic Schr/Sdinger picture as an a p p a r e n t size o f the moving electron, this size being 6 (r2)~-----8m~'
(10)
This effect always appears when going from a relativistic to a non-relativistic position coordinate and is calculated by Hayne and Isgur [18] for the nonrelativistic quark model. (2) Lamb shift: For an electron b o u n d in an electromagnetic potential terms in addition to those created by the Dirac equation appear. For instance, the emission of a p h o t o n before the interaction of the electron with the proton leads to a displacement of the electron whose effect on the Lamb shift can be simulated by attributing an apparent radius (rZ)eL to the electron r ?e=_Z~_2 c + l n me
,
(11)
where c is a numerical constant and too a cut-off frequency. Effects of this type have been calculated by Poggio et al. [19] for the non-relativistic quark model. (3) Condensate: An ab initio calculation of a h a d r o n within Q C D has to start with current quarks whose masses are very small. A not yet well understood mechanism (chiral symmetry breaking h la N a m b u - J o n a - L a s i n i o [22] to form a condensate inside the hadron?) allows to pass from point-like and massless u, d quarks and gluons to a largely non-relativistic problem where constituent quarks with sizable masses interact via effective forces. The forces o f this model problem, among them linear confinement, are relatively weak. It is very likely that the mechanism which leads from the nearly zero mass and pointlike current quarks to the constituent quarks is not only responsible for the effective masses mq but also for an effective size o f the constituent quarks. After the completion o f this work we received a preprint by Klimt et al. [23] in which they calculate the radius of constituent quarks within the N a m b u Jona-Lasinio model. Translated into the language of eq. (8) they find ~:= 0.3. In this p a p e r we have presented strong interaction radii d e d u c e d from total cross sections for most
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known stable hadrons. Wherever the comparison is possible, the strong interaction radii are very similar to the charge radii. Therefore, we are confident about the physical significance of the strong interaction radii. The most important result from this new body of data is the dependence of the radii on strangeness. It seems beyond doubt that hadrons become smaller with increasing n u m b e r of strange valence quarks. Within quark models such a dependence is expected since a larger mass of the quarks reduces the kinetic energy and allows for a smaller system. However, calculations show that this expected dependence in the relativistic and non-relativistic quark models is too weak compared to experiment. In the non-relativistic models this deficiency can be overcome by attributing an apparent radius to each constituent quark. We find ,2\1/2 r /u,d
0.6fm,
(r2}~/2=0.4fm.
(12)
We note that the apparent radius is sizable. Its contribution to the hadronic radius is comparable or even larger than that obtained from the wave function so that the constituent quarks overlap considerably. Adding the contributions from the wave function and the apparent size together quantitatively accounts for the absolute values of the radii as well as their dependence on strangeness. The ideas presented in this paper support the following very qualitative picture o f a hadron: The interactions among the partons inside a hadron may be seen in a hierarchy: (i) The very strong interaction of a quark with its (gluon) surrounding gives rise to a constituent quark with effective mass mq and an effective radius (r2}qt/2• 1/mq. (ii) The fairly weak residual interaction among the constituent quarks is responsible for hadronic spectra. While the first process is a genuine relativistic and possibly nonperturbative effect, the final system of interacting constituent quarks (gluons being absent!) seems to be treatable approximately as a non-relativistic problem. An important part of the non-perturbative effects namely the formation of resonant intermediate states
16 August 1990
seems correctly described in the vector dominance model, though in a language of the pre-quark era. Clearly our interpretation of the hadronic radii in terms of apparent quark radii is very speculative. However, the existence of a new set of data for radii which we have presented in this paper may provide helpful constraints on future theoretical work. We gratefully acknowledge many stimulating discussions in particular with S. Brodsky, D. Gromes, O. N a c h t m a n n , H.J. Pirner, and M. Strikman. The work has been supported in part by the Bundesminister fiir Forschung u n d Technologie under contract 06 HD710.
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