Volume 233, number 3,4 PHYSICS LETTERS B 28 December 1989

p - M E S O N - N U C L E O N T E N S O R C O U P L I N G IN RELATIVISTIC QUARK M O D E L S

H.J. WEBER t Jesse W. Beams Laboratory and Institute of Nuclear and Particle Physics, University of Virginia, C71arlottesville, VA 22901, USA

Received 26 September 1989

The pNN tensor-to-vector coupling constant ratio T~ V is estimated to be ~ 3.7 in the MIT bag model including a pion cloud contribution of ~ 1.5. In a consistent relativistic many-body framework of the constituent quark model on the light cone, T~ V increases by more than 50% to ~ 5 and comes closer to the empirical value 6.6 _+ 0.6.

The p-meson-nuc leon coupling of the form ~

1~-- ' lpNN Fp=fl'NOo'½TNUN, up=gpNNY -vl 2~nN aU~ Qv,

(1)

with Dirac-nucleon spinors UN=u(P) , U h = u ( P ' ) and momen tum transfer Q = P ' - P , is usually con- structed from symmetry principles in meson field theory [2,3]. F rom NN potent ials based on meson exchanges below the pion product ion threshold [4 ], a vector coupling constant gpNN ( q 2 = 0 ) may be ex- t racted in the range 2<~g~NN/4n<~3 in good agree- ment with the vector-meson dominance model ( V D M ) [2] . The empir ical p -nuc leon tensor-to- vector coupling constant ratio T/V----toNN/ gpNN = 6.6 --+ 0.6 from an analysis [ 5 ] o f n N scattering and nn-~Nfq in NN scattering is now used in most phenomenological NN potentials, in contrast to the significantly lower nucleon isovector magnetic mo- ment/~v = 3.7 predic ted by the V D M for the pNN ten-

sor coupling. In quark models, however, there seems to be no

need for in termedia te vector mesons to explain the nucleon magnetic moments [6] , thus avoiding the VDM concept. The hidden S U ( 2 ) v × U ( 1 ) symme- try [ 7 ], though, has revised the idea of the massive, composi te vector mesons as effective gauge bosons in the QCD phenomenology at distances that are inter-

Supported in part by the US National Science Foundation. ~1 The metric conventions, units and notation are those of ref.

[1].

media te between quark-g luon dynamics at short dis- tances and nonlinear chiral or Skyrme models of pion dynamics at long distances. In ref. [8] it is argued that the VDM be responsible for t he / t v= 3.7 part of the empir ical pNN tensor- to-vector coupling ratio, ~ 6.6, and that quark-g luon dynamics at r~<0.5 fm

should provide the remainder . Here we provide est imates for the p -nuc leon ten-

sor coupling from the relativistic const i tuent quark model to examine relativistic effects. While the bag model and the nonrelat ivis t ic quark model (NQM, with relativist ic correct ions) underes t imate T~ V, we find in the consistent relativist ic many-body light- cone formal ism that relativist ic effects, while signifi- cant, are not large enough to explain the empir ical

value 6 -7 . In QCD, meson -qua rk couplings arise ~2 in the bo-

sonizat ion of its generating functional in the path in- tegral formulat ion, when the gluon and quark fields are formally integrated [ 12 ]. After a Fierz rearrange- ment of the quart ic quark action it can be writ ten in quadrat ic form and then formally integrated upon in- t roducing bilocal boson fields and expanding them in terms of local meson fluctuations about the vacuum

configuration. The resulting p -meson-quark transi t ion opera tor

~2 This has been recognized independently and used for calculat- ing meson-baryon coupling constants in quark confinement models such as the MIT bag and constituent quark models [9], and for the construction of NN and N-hyperon potentials [ 10]. The NN potential has recently been used to search for possible quark effects in the triton [ 11 ].

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Volume 233, number 3,4 PHYSICS LETTERS B 28 December 1989

GiT~'½r is of purely vector type in the long wave- length expansion that is appropriate at Q2= 0 and for vertex form factors at low Q2.

For crude estimates we use first the MIT bag color confinement model [ 13,14 ]. This yields

f d3r (N ' I//7~'~qlN) exp(iQ.r)

= 1 - uNO ' ~ ' N / d N , (2)

where

O U = ( F ( + ) - 3mN5Q F , ) 7 ~

R

=4zr I (g2 +f2)J° (Qr)r2 dr , F~+) 0

R

FI = 4to _1 fgJl ( Qr) r 2 dr , ( 3 ) 0

to lowest order in P and P' , where Q=IQI and Q2 = _ 0 2 , and g ( f ) is the large (small) radial Dirac- quark wavefunction. For a quark mass [13] mq =

0.108 G e V / c 2, eq. (2) yields T / V = 2 . 5 , and 2.2 for Olq=:0 . These values are too small when compared with the empirical 77 V~ 6.6.

However, the p-nucleon tensor coupling may be enhanced by pion cloud contributions of the p me- son. In a two-pion triangle mechanism with inter- mediate nucleon and A(1232) states (shown in fig. 1, cf. eqs. ( 3.14)- ( 3.18 ) in ref. [ 14 ] for details) T~ Vincreases from 2.2-2.5 by ~ 1.5, altogether to 3.7-

P I I I

2 / \

/ \

N - - d . . . . . . N > N N~A

Fig. I. Two-pion exchange triangle diagrams with intermediate nucleon and delta states contribute ~ 1.5 to the pNN tensor-to- vector coupling ratio.

4, which is closer to the VDM value but still less than -~ of the empirical 6.6 + 0.6. Moreover, translation and Lorentz invariance are violated along with the exclu- sion principle for quarks in the MIT bag model (and chiral bags), where only the interacting quark (in the impulse approximation) is treated as a Dirac parti- cle. These serious problems necessitate rather model and observable dependent recoil and center-of-mass corrections. They are avoided in consistent relativis- tic many-body theories such as the light-cone formal- ism [ 15 ] that is adopted next for constituent quarks, for which there is growing evidence in QCD.

In the relativistic constituent quark model on the light cone the relevant 9oPP vertex is given as

3f Jgo = -~Gj21 d F ~'~'a~iT~r3u~uP , (4) ' = X j

where

~ ( ) d2q3Td2Q3T d F = f i 6 l - ~ x i )2 j=l (167r3

Here the A)=p + / P + are the invariant momentum fractions, P= _~pj and P' the total momentum of the initial and final proton, and q3, Q3, the relative three- quark momentum variables with q ]= - -q~r , Q2= -Q~T, so that for the transverse (T) and+ components (P+ =Po+P~> 0)

p3=Q3+x3 P, pl,2=++_q 3 - x l ' ~ Q 3 + x l . 2 P (5) 1 - x 3

holds. For space-like momentum transfer Q, it is con- venient to choose Q+ =0, so that x~ =.r> q~ = q3 and Q ; = Q3+ ( 1 - x 3 ) Q . We have assumed that the nu- cleon is dominated by a three-valence quark config- uration with constituent quark masses mj= mq~ ~my so that Zj mj ~ ,tl N.

The totally symmetric nucleon momentum distri- bution is taken as a common relativistic gaussian,

q)(x, q3, Q3)=Nexp(-~I~/6c~2) , ~P=0VZP, (6)

where M] is the covariant three-quark mass squared [15]. The size parameter a=0 .32 GeV~ ~mN and mq=0.38 G e V / c 2 are determined by the nucleon magnetic moments and its axial form factor, for which we obtain/~v=2.68 n.m.,/Zn= -- 1.63 n.m., and gA= 1.25, respectively.

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Volume 233, number 3,4 PHYSICS LETTERS B 28 December 1989

The relativistic spin wave function ZP of the proton is constructed from the conventional Clebsch- Gordan prescription for free quarks, but with their total energy taken to be mN. The quark instant (equal time) and light-cone (equal r = t + z ) states are re- lated by the Melosh transformation. In the weak binding limit the resulting nucleon spin wave func- tion is given by

Zp(1, 2, 3) =Ip(13, 2) +Ip(23, 1) ,

Ip(12, 3)=a~,(7"P+mN)75V~2"t2~3uN(P) • (7)

The normalization N in eq. (6) is obtained from the proton charge form factor at Q2= 0. A third vector- spin invariant (11 in refs. [ 15,16 ] ) does not occur in eq. (7). This consequence of the Melosh construc- tion [ 17 ] is consistent with the fact that Ii generates a second-class axial vector current that is not seen in experiments [ 16 ].

Substituting eqs. (6), (7) into eq. (4) and per- forming the spin sums given for the 9o-proton vertex

• 2 -, fdFCPq~P J~o =41mNGUN X3

× { - (Pl) Tr[ (P ' ) (p'3)y'~(p3) (P) (P2) ]

-- (P l ) ( P ) (P3)YX(P3) ( P ' ) (P2)

+ (P2) Tr[ (P ' ) (Pl) (P) (P3)Y'~(P;) ]

4- (P2) ( P) (P, ) ( P' ) (P'3 ) 72(P3)

+ (P~)Y~(P3) (P) (Pl) (P') (P2)

+(p'3)7~(p3) Tr[(P' ) (pl ) (P)(p2)]}uN, (8)

with the notation

(Pi) - 7"pi+mq ,..., ( p , ) _ 7"P'-FmN 2mq 2mN

The total symmetry of the momentum distribution OP has been used to relabel the quarks so that quark 3 is interacting in each term of eq. (8). When eq. (8) is cast in the form of eq. ( 1 ) the vector and tensor coupling may be extracted. The analysis and evalua- tion ofeq. (8) were carried out by means of symbolic and numerical computer codes.

The result for T~ V= 3.3 for pure vector p-quark coupling shows that relativistic effects are signifi- cant. The dependence on the momentum transfer Q2

of the vector coupling and the ratio T~ V of vector to

tensor couplings are shown in fig. 2. The vector ver- tex form factor has a RMS value of 0.65 fm and may be compared with the dipole shape of the electromag- netic form factor of the proton. T~ Vvaries slowly with the quark model parameters a ~ m q , which is dis- played in table 1. (The p-quark vector coupling con- stant G is determined so as to reproduce the empiri- cal vector coupling g2op p /4~~ 2.89. )

The pion cloud will increase T/V by ~ 1.5. More- over, a small amount of p-quark tensor coupling could bring T/ V in agreement with the empirical value. This is an aspect of the relativistic q~l structure of the p-meson wave function, which contains a vec- tor and tensor component [ 18 ], viz.

1.0 3.5

3.O ,, ~ ~ ~ ~ T / v

2D

0 .2

0 0 ,L2 i i i

0 . 4 0 G 0 .8 LO

- O 2 ( (GeV/c )2 )

Fig. 2. The pNN vector form factor (solid line, V, with left hand scale) and the ratio T/V of tensor to vector couplings (dashed line, T/V, with right hand scale) of the relativistic constituent quark model for transverse size a=0.32 GeV and quark mass mq=0.38 GeV/c 2 compared with the electromagnetic dipole form factor of the proton (dot-dashed line).

Table 1 The ratio T~ Vof pNN tensor-to-vector coupling constants versus transverse momentum size ~e and constituent quark mass mq.

[GeV] mq [GcV/c 2 ]

0.25 0.30 0.35 0.40

0.25 4.22 3.87 3.58 3.34 0.30 4.07 3.75 3.49 3.26 0.35 3 90 3.62 3.37 3.16 0.40 3.73 3.46 3.25 3.06

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Volume 233, number 3,4 PHYSICS LETTERS B 28 December 1989

Table 2 The ratio T~ V of pNN tensor-to-vector coupling constants for ~x=0.32 GeV and m q = 0 . 3 8 GeV/c z versus p-quark tensor cou- pling strength G.

ro T~ V

1 6.87 0.6 5.44 0.2 4.02

-0.2 2.59 -0.6 1.17 - l -0.26

~'0 =0,zTx2W'E(mo - rpy.Q )ux, . (9)

G e V / c 2, which is the correct re la t ion expected f rom the SU (3) f lavor symmetry .

In conclus ion , then, relat ivist ic effects f rom a con- s is tent relat ivis t ic m a n y - b o d y fo rmal i sm are large enough for the 9NN tensor coupl ing to come close to exp la in ing the empir ica l value 6 . 6 + 0 . 6 o f T / V .

Moreover , q u a r k - g l u o n b i n d i n g may generate a ten- sor c o m p o n e n t in the qcl wave func t ion of the p me- son, which is shown to have the potent ia l o f inf luenc- ing strongly the pNN tensor coupl ing.

It is a pleasure to t hank W. K o n e n of the U n i v e r - sity of M a i n z for his co l labora t ion on the electromag- net ic and axial form factors of the nucleon.

Here e denotes the po la r iza t ion vec tor of the p me- son, while 0o and rp are m o m e n t u m dis t r ibut ions. This b o u n d qq s t ructure of the p m e s o n suggests augmen t - ing the pure vector q u a r k - m e s o n t r ans i t ion opera tor

½GOTurq by the tensor part - ½G(ro/mp)CITa~,,Q~q. Inc lud ing it in the bag mode l es t imate (wi th cons t an t

r o for s impl ic i ty ) increases toNN by

R

10m N r o f AtoNN = 3mp 4~ J d r r 2 ( g 2 + S f l 2) , (10)

o

and T / V b y 4.8r o for quark mass [13] m q : 0 . 1 0 8 G e V / c 2, or 4.65r~ tbr m q = 0 . For r p =0 .5 4 ( a n d m q = 0 . 1 0 8 G e V / c 2 ) , or r~=0 .63 ( a n d m q = 0 ) , the

sum of the es t imates in eqs. (3 ) , (5 ) an d the p ion cloud c o n t r i b u t i o n reproduces the empir ica l va lue of T~ V.

The co r re spond ing results for the relat ivis t ic con- s t i tuent quark mode l are exhib i ted in table 2. The de- pendence of T / V on the p - q u a r k tensor coupl ing strength r o resembles that o f the bag model .

Final ly, it is in te res t ing to no te that the corre- spond ing results for the m me s o n [eq. (4) wi thout

1 3 the isospin opera tor ~zj, i.e. eq. (8 ) wi thou t the mi- nus signs ] yield a vanish ing tensor coupl ing at Q2 = 0, which grows to 0.6 at Q 2 = - 0 . 6 4 G e V / c 2 a nd then falls off. Fur thermore , the m vector coupl ing at Q 2 = 0 is 6 . 5 2 ~ ~ x 2 . 8 9 for c~=0.28 GeV an d m q = 0 . 2 7

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