The axial vector coupling and magnetic moment of the quark

Physics Letters B 284 (1992) 384-389 North-Holland

PHYSICS LETTERS B

The axial vector coupling and magnetic moment of the quark D u a n e A. D i c u s Center for Particle Physics, Department of Physics, University of Texas, Austin, TX 78712, USA

Djordje Minic, Ubirajara van Kolck 1 Theory Group, Department of Physics, University o['Texas, Austin, TX 78712. USA

and R o b e r t o Vega Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309, USA

Received 12 April 1992

Details of an estimate of the corrections of order 1/N to the axial vector coupling of the constituent quark, where N is the number of colors, are presented. Analogous estimate is made for the magnetic moment of the constituent quark.

Some time ago Weinberg [1 ] argued that in the leading order in 1/N, where N is the n u m b e r o f colors, the axial vector coupling o f the constituent quark is equal to one and its a n o m a l o u s magnetic m o m e n t is zero. This justifies the usual treatments o f the constituent quark and bag models, where the quark is treated as a bare Dirac particle, p r o v i d e d the corrections in 1 / N are shown to be small, especially for the magnetic moments. More recently Weinberg [2] (see also ref. [ 3 ] ) has given an estimate o f the corrections of order 1 / N to the axial vector coupling o f the constituent quark. His calculation was done using the chiral quark model lagrangian [4] in the chiral limit and the limit of large n u m b e r o f colors. The essential input was the analogue o f the famous A d l e r - W e i s berger sum rule for p i o n - q u a r k scattering. First order corrections in 1 / N to the leading result were shown to come from tree-level p i o n - q u a r k scattering and q u a r k - a n t i q u a r k pair p r o d u c t i o n diagrams. The latter contribution turned out to be logarithmically divergent but relatively small even for the values o f a c u t o f f a s large as 5 GeV. Fellow of CNPq, Brazil. 3 84

It is our aim in this paper to elaborate on this last result, already quoted in ref. [ 2 ]. Following the same pattern of reasoning we also analyze the magnetic m o m e n t o f the constituent quark using the same chiral lagrangian and the analogue o f the D r e l l - H e a r n G e r a s i m o v sum rule for p h o t o n - q u a r k scattering. Our results seem to indicate that the chiral quark model with coefficients o b t a i n e d by sum rules works well. The lowest order terms in the chiral lagrangian density in which the relevant degrees o f freedom are the constituent u- and d-quarks, treated as massive particles subject to color force only at large separations, and pions treated as p s e u d o - G o l d s t o n e bosons, have the following form ~ '

~ Here we neglect isospin breaking due either to a quark mass difference in the QCD lagrangian or to electromagnetic interactions. Accordingly, we do not include purely electromagnetic contributions to the sum rules. Elsevier Science Publishers B.V.

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PHYSICS LETTERS B

p i o n - q u a r k scattering in the form used by Weinberg [2] in the chiral limit ( m ~ = 0 ) :

1

£#=

2(1

+n-/F~)1

gA=l-- ~

2 "~ D/~Ir-

2(l+u2/F~)

-(~{~+m+~ 2igAF~

[a_ (O))--ff+(O))] ,

(2)

0

2i

1

F~ 1 + ~r2/F~ +

25 June 1992

1

where a+ and a_ stand for the total cross-sections for scattering o f n + and ~ - respectively on a constituent u-quark at rest. The incoming energy of the pion is denoted by ~o. In the large N limit F~ goes as x/N. Taking this fact into account it is easy to conclude that the only relevant processes making contributions of order 1/N z in the total cross-sections, or equivalently of order 1 / N in gA, are tree-level p i o n quark elastic scattering (fig. 1 ) and the quark-antiquark pair production (fig. 2 ) [ 2 ]. (In the latter case the extra 1/xflN in the amplitude, coming from the coupling to the produced pairs, is canceled in the expression for the total rate, by the sum over the colors of the produced pair.) Using the lagrangian density

t- (~rx ~ )

)

l+~r2/F]Y~t.~u g/,

(1)

where u denotes the pion field, g/is the quark field, m~ ( ~ 135 MeV) is the pion mass, m stands for the mass of the constituent quark ( ~ 360 MeV), F~ is the pion decay constant ( ~ 190 MeV), gA is the axial vector coupling in the leading order in 1/N, hence is set to one, and t = l a where a are the usual Pauli matrices in isospin space. In order to estimate corrections to gA tO first order in 1 / N w e use the Adler-Weisberger [ 5 ] sum rule for p~

P~"

-p

-

-

>p.

P~

p-

P,~

-

-p

>p

-p

(a)

(b)

P~ > p"

(c)

Fig. 1. Feynman diagrams for pion-quark scattering that contribute to the Adler-Weisberger sum rule.

p~ "'-.

P1

7.

".., "

-

P~

~

.....

"" ' -

~

-P2

"'~

_p.

~< ,' >

p-

(a)

p

(b)

P (c)

P1

"'i

p)

'

(d)

Pl

<

)p.

"'" )-p

~P2 s

~.

-"

(e)

"

>p.

/'2

i

)p

'

:~ p -

(0

Fig. 2. Quark-antiquark pair production diagrams that contribute to the Adler-Weisberger sum rule to the same order in 1/N as those of fig. 1. 385

Volume 284, number 3,4

PHYSICS LETTERS B

given by ( 1 ) one finds that the only relevant contribution for p i o n - q u a r k scattering comes from x - u--*x°d (contributions coming from x + u ~ x + u and x - u - , x - u cancel). The corresponding differential cross section is [2 ] (do) d-~ . . . .

= m2 u2(1--COS0) 2 Od 27r2F4 [ l + u ( l _ c o s 0 ) ] 3 ,

(3)

where 0 is the angle between p;~ and p= (see fig. 1 ) and u = ~ / m. The situation is a bit more involved for quark-antiquark pair production. Three processes contribute: x - u - , d d a , x + u - , u u d and x - u - , d u 0 . The last process can be broken into two non-interfering parts (to leading order in l / N ) . The first comes from diagrams o f fig. 2 where the p r o d u c e d pair with mom e n t a p, and P2 consists of a d-quark and a u-quark. The contribution from these diagrams to the sum rule ( 2 ) is exactly canceled by the corresponding contribution from the process x + u ~ u u a . The second part o f the process x - u--* duff comes from diagrams of fig. 2 where the produced pair is uf~. The corresponding contribution to g2 is exactly equal to that o f u - ~ d d a . Because o f the complication coming from the three particle phase space we only quote the expression for the a m p l i t u d e squared for x - u ~ d d a : [M12=29

m6[ ( p - p ' ) . p = ] 2 F 6 (pl +p2)2(p'p~) (p"p,~)

+ 2 9 /H4(pl -~-p2) 2 F6(p, p) 2 .

(4)

Doing the three particle phase space integrals and making use of the A d l e r - W e i s b e r g e r sum rule ( 2 ) we find a logarithmically divergent contribution to gA. The final expression for g2 up to first order in 1 / N is I'H 2

g~=l

2~z2F~

Table 1 A (MeV)

.¢

2000 2800 5 000 10000 20000

0.022 0.135 0.634 1.76 3.42

integral over ~o in (2) is d o m i n a t e d by energies less than or o f the order o f a typical N - i n d e p e n d e n t Q C D energy scale, such as the mass of the p-meson ( m o ~ 770 M e V ) . This condition is satisfied for the tree-level p i o n - q u a r k scattering. On the other hand the contribution for the pair production process is logarithmically divergent and it obviously does not meet the above condition ~2. In particular the threshold for the pair production process is at 4 m ~ 1400 MeV which is not small c o m p a r e d to the above scale. Thus, although this process is formally o f the same order in 1 / N a s the tree-level p i o n - q u a r k elastic scattering, one could think that its inclusion would be problematic even if it did not diverge. In any case its contribution turns out to be small for any reasonable cutoff. We have also checked how the second term in (5) changes if the mass o f the pion is taken into account. N u m e r i c a l calculation shows that the change in the final result is only three percent even if the mass o f the pion is taken to be half o f the constituent quark mass. Also, because the energy threshold is large the pion mass can be neglected in calculating ,¢. Following the same logic we calculate the corrections o f order 1 / N to the magnetic m o m e n t o f the constituent quark. Here we have to take into account

In"

2

,J.

(5)

The second term in ( 5 ) comes from p i o n - q u a r k elastic scattering and the last term comes from q u a r k antiquark pair production. J denotes a numerical integral evaluated from 4m to a cutoffA. In table l we s u m m a r i z e the d e p e n d e n c e o f , J on different values o f the cutoffA. As p o i n t e d out by Weinberg [ 2 ], the validity o f the described estimate o f the axial vector coupling of the constituent quark is based on the a s s u m p t i o n that the 386

25 June 1992

,2 Because we work only to leading order in 1/N certain interference terms have been neglected in (5). An example is the diagrams of fig. 2 for x-u~dda multiplied by those with the identical d-quarks interchanged. To check the numerical validity of the 1/N approximation, and to check that the neglect of these terms was not the cause of the divergence of the sum rule integral, we have calculated them for x-u~dda. They change the values of.¢ given in the previous table by only about then percent and do not improve the convergence of the integral.

Volume 284, number 3,4

PHYS1CS LETTERS B

the lowest order terms in the chiral lagrangian density describing the electromagnetic interaction of quarks and pions. They are given by the following expression:

X

mo~

i

+zd ~n - 3 m - o )

--In O)

( I + ¢t2/F~ ) 2 +

[eA~'(nXO~,Tt)~+ ~e l 2A,,A,, (n 2 -~3)1 2

{+,_ [

+ -too) -B

2 1 + F,~2 1A-Tt2/F2~ ( g 2 t 3 - t ' n g 3 )

+ (Z+Zd)2+Zd(Z+Zd)

-

1

m(m+2o)) 2 (6)

where A~, denotes the photon field, - e stands for the charge of the electron and z~e and zde stand for the charges of the up and clown quark respectively. Again we set gA = 1 in the leading order in 1/N. In order to estimate the anomalous magnetic moment of the constituent quark, we use the analogue of the Drell-Hearn-Gerasimov sum rule [6] for photon-quark interaction, given by the following formula: ~o

K:=

m:

"[do)[a,(o))-a~(o))]

2;g20~Z 2 J

(7)

o)

t

where ae and aA represent the cross-sections for parallel and antiparallel photon and quark spins and o) denotes the incoming energy of the photon in the frame where the target quark is at rest. Also, ze denotes the charge of the target, c~=e2/4zr, t is the threshold for the relevant process and x stands for the anomalous magnetic moment of the constituent quark. Counting the powers of N is done as before. Only two types of processes make'contributions of order 1/N to the cross-sections O'p and o-A and correspondingly of the same order to x 2. These are: pion-photoproduction (fig. 3) and quark-antiquark pair photoproduction (fig. 4). The relevant calculations have to be performed while keeping the pion mass finite and then letting it to zero. Taking this fact into account, we obtain the following expression for the cross-sections coming from pion-photoproduction diagrams

( w i t h A a = O'p - a.~ ):

(

2o) za z m + z d 2 m -

m

--Z~ ( m + o ) ) ( m Z + m o ) - ½ m ~ ) } )

)

F, l + g 2 / F 2 Ys(tXTt)3 tff,

_

z2m~ A + - m 2 + m ~ In too) A_ - m 2 + m ~

+ ieA,,~7*'( [ ( ½z, + z,j) + ( zo-- zd)t3 ]

2gA

m-o))

rim2

/

1

ASo=

25 June 1992

,

(8)

where z = l ( z = 0 ) for the case where the charged(neutral) pion is in the final state (the up quark charge has been eliminated by zu = z + za). Also A + _ = m ( m + o ) ) - l m ~ + _ B with B = [ ( m o ) - ½ m 2 ) 2 - m ; ~ m - ~] I / ' ~-. The pion mass is important only in the second term which contributes to the sum rule as

2az2m~ 7 do) F~

2o)

J ~ln--

as m, goes to zero. This term, which is lost if one naively sets m, to zero before calculating Aa, gives a large contribution to ~c2 but is exactly canceled by the other terms ~3. Numerical evaluation of the integral in (7) gives ~c2=0 for both the charged and neutral pion, independent of the value Of Zd. Again, the contribution coming from quark-antiquark pair photoproduction is slightly more complicated, but in this case the sum rule integral is finite. The corresponding expressions for the magnetic moments of the constituent u- and d-quarks are .~

K~,= ~ N

t/7-

,4 (zu, Za),

(9)

, 2 [ m2 ~2 #ca = ~ N t ~ ) ,Y2(zu, Zd) ,

(10)

where .Y~,,Y2stand for numerical integrals whose values ~4 are less than 0.001. Because the energy thresh~3 We thank S. Drell for suggesting this to us. ~4 In fact as the number of sampling points in our Monte Carlo routine is increased the values of the above numerical integrals seem to converge to zero.

387

Volume 284, number 3.4

PHYSICS LETTERS B

P~

P~

P~

25 June 1992

P~

~

P~

. oS

-p

-

!

-p-

-p

-

(a)

•p"

. p

- p-

(b)

•p

'

(c)

• p "

(d)

Fig. 3. Feynman diagrams for pion-photoproduction that contribute to the Drell-Hearn-Gerasimov sum rule.

Px

~p

-

-p

~

.

rio

-

(a)

. -p

" P

P~

~ P2

,: p

(d)

I

•

p"

(c)

P~

P

•

p

(b)

P~ ," ~, ' ~

.

P1

Pr

"

P~ A/X/~

P2 P

(¢)

,' P

~

P,

~-

P2 P

(0

P~

i i

P

(g)

P

Fig. 4. Quark-antiquark pair photoproduction diagrams that contribute to the Drell-Hearn-Gerasimov sum rule to the same order in I / N a s those of fig. 3. old is large the pion mass can be neglected in calculating ,4 and .~. O f course the same objections as before can be raised to contributions from the pair production processes which obviously do not come from low energies but again, their contributions turn out to be numerically negligible ,5 Thus the 1/Ncorrections to Ic2 are essentially zero. Just by dimensional analysis we expect contributions to Ic2 of O ( 1 / N 2) to be of the following form: 1 (m2"~ K~,d~ ~ \ 2 ~ 1

2 .

(11)

(5), (9) and (10) we argue that for consistency of our calculation the N = 3 limit should be taken after the axial vector coupling and the magnetic m o m e n t s are obtained for the nucleons in the framework of the large N constituent quark model ~6. ( i n the large N limit of the naive quark model [ 8 ] the proton is built from k + 1 up and k down quarks and analogously the neutron consists o f k up and k + 1 down quarks, where N = 2 k + 1.) We also note that an extension of the usual power counting argument [ 7 ] indicates that the d o m i n a n t contribution to both p i o n - n u c l e o n and p h o t o n - n u c l e o n interactions is the impulse approxi-

In order to discuss the m e a n i n g of the equations .5 The same comment would also apply to diagrams with possible four-quark vertices [ 7 ] which we have not considered. 388

~6 If this limit is taken first the values for the magneticmoments of the nucleonsare identical to the ones that follow. The value of the axial vector coupling,on the other hand, decreases a bit.

Volume 284, number 3,4

PHYSICS LETTERS B

mation, in which the pion or photon interacts independently with each quark in the nucleon. All other processes are of higher order in the chiral expansion. On the other hand, all these processes are of the same order, N, in the 1/Nexpansion. In the case of the axial vector coupling of the nucleon, we find in the non-relativistic limit

(gA).uc,eon=½(N+ 2)gA = ] [ N + 2 + N ( g A - 1)] + O ( 1 / N ) .

(12)

Taking A= 3500 MeV for definiteness, which corresponds to the momenta of the incoming particles taken as mp in the center of mass, gA=0.87 and (gA),,deo,= 1.54, which should be compared to the experimental value 1.25. Analogously, we find the following expressions for the magnetic moments of the proton and neutron: U P = ~ [ u u ( N + 5 ) - / Z d ( N - I) ] ,

(13)

/-iN = ~ [pd(N+ 5)-/,Z~ ( N - 1 )] ,

(14)

where/*,,d = ( 1 + Ku,d)Zu,de/2rn are the magnetic moments of the constituent up and down quarks. Using the usual quark charges in ( 13 ) and (14) we obtain pp = ~

e

[ N + 3 + ~N(Zxu +ted) ] + O ( 1/xflN), (15)

e [N+l+]N(2~cu+~cd)]+O(1/xfN). /iN = -- 12---m {16) Since ~cuand ~Cdare consistent with being zero the values for the magnetic moments are/~p = 2.6 e/2mN and Ily=--l.7e/2mN, where mN is the nucleon mass. These values should be compared with the experimental results, ltp=2.79e/2mh and / I N = - - l . 9 1 e / 2mh. Alternatively one could use the values of the quark charges obtained from the quark model under the requirement that the nucleons have their usual

25 June 1992

charges for any N. Explicitly, zu= (N+ 1 )/2N and Zd= ( 1 - - N ) / 2 N which in the large N limit goes to zu= ½ and z a = - ½. Then the corresponding values would be/ip = 2.2 e/2mN and/iN = --/tp. The value obtained for the axial vector coupling agrees within 20% with the experimental value. Perhaps the agreement could be improvement by taking into account relativistic corrections [ 9 ]. Concerning the anomalous magnetic moment, we have shown explicitly that the potentially dangerous O ( 1 / x / N ) corrections are negligible. We consider these results to be evidence that the constituent quark model can be understood in the large N limit in terms of chiral symmetry (in the form of the chiral quark model) and reasonable assumptions on the high-energy behavior of amplitudes (embodied in sum rules). We thank S. Weinberg for helpful conversations and encouragement and S. Brodsky, S. Drell, M. Peskin and J. Polchinski for discussions. We acknowledge participation ofC. Ord6fiez in the initial phase of this collaboration. This work is supported in part by the Robert A. Welch Foundation, NSF Grant PHY 9009850, US Department of Energy Grant DE-FG0585ER40200 and US Department of Energy Grant DEAC03-76SF0015. References [ l ] S. Weinberg, Phys. Rev. Len. 65 (1990) 1181. [2] S. Weinberg, Phys. Rev. Lett. 67 ( 1991 ) 3473. [3] S. Peris, Phys. Lett. B 268 ( 1991 ) 415; Ohio State University preprint OSU/92-DOE/ER/01545-569. [ 4] A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984) 189. [5] S. Adler, Phys. Rev. 140 (1965) B736; W.I. Weisberger, Phys. Rev. 143 (1966) 1302. [ 6 ] S.D. Drell and A. Hearn, Phys. Rev. Lett. 16 (1966) 908; S.B. Gerasimov, Yad. Fiz. 2 (1965) 598 [Sov. J. Nucl. Phys.

2 (1966) 430]. [71S. Weinberg, Nucl. Phys. B 363 ( 1991 ) 3. [8] G. Karl and E. Paton, Phys. Rev. D 30 (1984) 238. [9] C. Hayne and N. Isgur, Phys. Rev. D 25 (1982) 1944.

389