Isospin-violating quark distributions in the nucleon

Physics Letters B 274 (1992)433-438 North-Holland

PHYSICS LETTERS B

Isospin-violating quark distributions in the nucleon Eric S a t h e r Centerfor TheoreticalPhysics, Laboratoryfor Nuclear Science and Department of Physics, MassachusettsInstitute of Technology, Cambridge, MA 02139, USA

Received 30 September 1991

The quark distributions in the nucleon which measure isospin breaking are estimated in quark models and then evolved to experimental energies. The difference dP(x) - u" (x) is roughly (2-3)% ofdP(x). Surprisingly, uP(x) - d" (x) is only half as large. These distributions produce an error of 0.002 in the value of sin2 0wextracted from neutrino scattering unless they are taken into account.

1. Introduction Strong-interaction phenomena respect isospin symmetry to a high degree. For this reason theorists routinely assume the isospin symmetry of parton distributions, as in the derivation of the GrossLlewellyn Smith sum rule. Yet electromagnetism and the u p - d o w n quark mass difference must break the isospin symmetry of parton distributions, removing the degeneracy of u p (x) with d" (x) for example. This paper describes a quark-model estimate of the isospin-violating quark distributions in the nucleon. Quark models have a history o f successfully explaining the properties of quark distributions. For example, using an approach based on the bag model ~, Close and Thomas [2] reproduced the strong deviation of the ratio u P ( x ) / d ~ ' ( x ) from 2 at large x which signals the breaking o f SU (6) symmetry. A related approach employed here predicts the breaking of isospin symmetry, albeit on a much smaller scale. The sources ofisospin violation suggest an isospin asymmetry of the parton distributions on the order o f aEM or (md--mu)/AQcD. The calculations described here accordingly reveal a minorityThis work is supported in part by funds provided by the US Department of Energy (DOE) under contract #DE-AC0276ER03069. ~ This method of calculating quark distributions, developed by Signal and Thomas [ 1], informs the analysis of section 2.

quark difference, 8d(x) -=dP(x) - u" (x), that is a few percent of the isospin-averaged minority distribution, d ( x ) . One might expect the majority-quark difference, 8 u ( x ) - u P ( x ) - d " ( x ) , to be twice as large as the minority difference. Strikingly, it is only half as large. These isospin-violating quark distributions must be understood in order to extract a high-precision value of sin 2 0w from deep-inelastic neutrino scattering [ 3 ]. When eventually measured, they will expose the roles of electromagnetism and quark mass in nucleon structure.

2. Isospin violation in quark models Because the nucleon transforms as an isodoublet, with the neutron related to the proton by an isospin rotation, isospin asymmetry in the nucleon can only be observed by comparing matrix elements of operators in the proton with their isospin-rotated counterparts in the neutron. Hence the parton distributions which measure isospin breaking are the majority- and minority-quark differences, the corresponding antiquark differences, heavy-quark differences, and the gluon difference. O f these the light-quark differences are the easiest to measure as well as to model, and are the subject of this investigation. The quark distribution in a target can be obtained

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from the forward, virtual quark-target scattering amplitude

A(k,p) = J- d a x e x p ( - i k x )

(Pl T ( ~ ( x ) q / ( 0 ) ) [ P ) c , (2.1)

by contracting with 7 + and integrating over all quark momenta such that the quark carries a fraction x of the total +-momentum: q ( x ) = 51 f

d4k (-~)4c~(k+-xP +) rr[y+A(k,p)] • (2.2)

We will use quark-model ideas to describe a nucleon in its rest frame, assuming that it consists of three constituent quarks, each in an energy eigenstate, as, for example, in the bag model. Then we can write

= 2 ,,/2 M ( 2 g ) 4 ~(k°-E).P

(k) ,

(2.3)

where the factor of the target mass M comes from the covariant normalization of the target state in the scattering amplitude, eq. (2.1). The expression for the quark distribution, eq. (2.2), now simplifies to q(x) -----fd3k~(x-(E+k_-)/M).~(k).

(2.4)

In the bag model .~(k)=2lu+(k)12, where u(k) is the quark wave function, u+=P+u, and P + = ½( ~ + c d ) is a light-cone projector [4]. In virtual quark-nucleon scattering, a quark is removed from the nucleon, leaving an intermediate diquark state containing the two remaining quarks, and later a quark is reinserted (see eq. (2.1) ). Therefore the energy E of the released quark is the difference in energy between the nucleon and the diquark, and so includes the quark-diquark binding energy B in addition to the energy EQ of the quark mode initially occupied by the quark; i.e.,

E=M-ED

=EQ + B ,

(2.5)

where ED is the diquark energy. Now consider the isospin-violating quark distributions produced by electromagnetism and the quarkmass difference. The effect ofisospin breaking on the 434

quark distribution in eq. (2.4) can be separated into two contributions. One is the result of modifying (k). The other is purely kinematic: M a n d E inside the delta function shift. We can compare the two contributions produced by the quark-mass difference in the bag model. There M, E, and the quark wave function are functions of quark mass. Numerical calculations show that the simple kinematic effect is far more important: in the region where the total mass shift effect is significant, x < 0.7, the variation of the wave function contributes only (10-20)% of the total effect, with the higher percentage at low x. We will therefore retain just the kinematic contribution that arises from varying the argument of the delta function. This allows us to relate a variation of q(x) due to a small perturbation of M or E to d q ( x ) / d x , the derivative o f q ( x ) with respect to x at fixed M a n d E. For a general variation,

5 q ( x ) = l"C ~, M~d [axx q (vx ) ] - Si E d q ( x ) )

Tr['/+A(k,P) ]

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(2.6)

Above we assumed that each quark occupies an energy eigenstate. We could try to make a more realistic model by smearing out this energy. The energy shift at each value of E would be the same ~SE, however, and eq. (2.6) would not change. One virtue of this approximate formula is that experimentally-measured quark distributions can be substituted for the model distributions. Many of the defects of the quarkmodel distributions are thereby avoided, and features such as SU(6) breaking are automatically incorporated. Moreover, the calculation of the isospinviolating distributions becomes remarkably transparent. To obtain the isospin-violating quark distributions we need only determine the target mass and quark (or diquark) energy shifts caused by the quark-mass difference and electromagnetism. The contribution of the mass of a quark to the mass of a constituent quark or hadron containing the quark is by definition its current quark mass [5]. To include electromagnetism, we ignore the small quark self-interactions and assume that all the electromagnetic energy resides in (partly) spin-dependent "bonds" among the quarks. Then to first order in isospin violation, the electromagnetic binding energy of a nucleon is the sum of the bond energies:

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BEM = 2Bmin-maj emin emaj q- Bmaj-maje 2maj •

(

2.7 )

In each nucleon two u - d bonds connect the minority quark to the two majority quarks and so the quarkdiquark binding energies are identical for the minority quarks: i.e., 5B = 0. Hence for the minority-quark difference, 5 E = ~EQ = rod-- m, ~ 4.3 MeV. Of course, 8 M = r n p - m , = - 1.3 MeV. The minority difference is therefore

8d(x)=-l(l~Ml +(md-m.)

d[xdv(x)]

~dv(x)

)

,

(2.8)

where dv(x) is the isospin-averaged, minority valence distribution. In the case of the majority difference the intermediate diquark in each nucleon consists of an up and a down quark, one a majority and the other a minority quark. So the diquark energies are the same. Hence 5E= 8M and

M(d

M \dx[XUv(X)l-duv(X)) "

(2.9)

Here u~(x) is the average majority valence distribution. The factor 5M/M indicates the small size of the majority-quark difference. Without electromagnetism each 5M in eqs. (2.8) and ( 2.9 ) would have been just - ( m d - m . ) . The two distributions would then have been roughly the same size due to cancellation between d (xu~)/dx and duv/dx. Electromagnetism diminishes both distributions but clearly affects the majority difference more profoundly.

3. Isospin violation at experimental energies

These results do not hold at all energy scales. Taking moments of both sides of the formula for the variation in the quark distribution, eq. (2.6), we find for j>~2

5qJ= ( 1--j) (~MqJ-SEq J-~ ) /M ,

(3.1)

where qJ is thejth moment o f q ( x ) : 1

qJ= ~ dxx j-~ q(x) . 0

(3.2)

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The jth moment of the variation ~q evolves with a different anomalous dimension than the ( j - 1 )st moment of the original quark distribution, so the formula cannot scale. This is not surprising: the analysis of section 2 treated a nucleon as three constituent quarks, presumably a valid description at some lowenergy scale. Therefore before substituting experimental values for the moments of the quark distributions into the equation for 5qj, they must first be evolved down to the constituent quark scale where the formula is valid. Then, to make useful predictions, the moments of the isospin-violating distributions must be evolved up to the energies of deep-inelastic scattering. By assumption there are no sea quarks at the constituent quark scale. Hence the isospin-violating distributions calculated at that scale constitute the initial conditions for the evolution of both singlet and non-singlet distributions. First consider the QCD evolution ofnon-singlet, i.e. valence, distributions. To first order in as, they evolve according to

AJqJ(t),

~ qJ(t) =

(3.3)

where Aj is thejth non-singlet anomalous dimension [6] and t=log(Q2/it~) is the logarithmic energy scale. Integrating, we find llOgkqt

)/

=

1

dt'o~s(t')-d.

~

(3.4)

o

Though a first-order result, this formula accommodates all higher-order running of as and so is less sensitive to higher-order corrections [ 7 ]. We will use the moments of bag model distributions to determine the constituent quark scale. The evolution equation (3.4) can be used consistently to relate bag model results to experimental data ~2: substituting the CCFR Tevatron data [ 8 ] for qJ(t) and bag moments for qJ(0), d falls in the narrow interval between 0.48 and 0.56 for moments 2-8, with an average value of ( d ) =0.50 ( d = 0 indicates no evolu~2 To compare moments of the measured distributions with bag model moments, which are SU(6) symmetric, we compare dS~from the bag with the combination ~( ½u~ + 2d~) of experimental moments in which, to first order, the SU ( 6 ) violation [2] (coming from one-gluonexchange) cancels. 435

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d 6d{(t) = as(t) A' 6d~(t)

0.006

dt

0.004

3

+--(e'-8TC

d

2 j djr ( t ) . e~)aEMA

(3.6)

0.002

Here, working (as always) to first order in isospin breaking, we can neglect the QED evolution of d~ ( t ), and therefore the contribution to 6d(. (t) from QED evolution is just -azmA,,tdJv(t)/Sn. The appearance of an explicit factor of t requires us to finally specify the energy of the constituent quark scale. The scale parameter t is given implicitly by the defining equation for d, eq. (3.4). For second-order running of~s and with a s = 0 . 2 5 at the scale of the experimental data, Q 2 = 1 2 . 6 GeV 2, the average value ( d ) = 0.50 corresponds to t = 4.2, and thus a constituent quark scale of 0.45 GeV. The numerical results for m o m e n t s of the valence distributions at the experimental scale are shown in table 1. The growth of the percentage asymmetry of the minority-quark distributions with i n c r e a s i n g j is analogous to the more dramatic growth of u~/d{. (The smaller increases in the majority-quark asymmetry are due to QED evolution. ) Singlet evolution is complicated by mixing. For example, 6 ( u + lT+d+d) mixes under Q C D evolution with the gluon difference, G P - G ". But the analysis follows the same course as for non-singlet evolution, and uses the same value of d, only the evolution equations become matrix equations. The initial glue distributions, both isoscalar and isospin-violating, were taken to be zero ,3. The results are shown in table 2.

0.000

-0.002 0

0.3

0.4

0.6

0.8

X

Fig. 1. Simple parameterizations of the isospin-violatingvalence distributions. They are weighted by x, as they are when measured, and so vanish at x=0. When the factor ofx is removed, these distributions diverge at x=0. The integral of each of these curves weighted by 1/x must be zero in order to preserve the normalization of the valence distributions. tion). Using the average value we can invert this equation: q/(0) = qJ(t) exp( ( d ) A j) .

(3.5)

We then use this formula to project the data down to the constituent quark scale, plug the resulting moments into eq. (3.1) (with 6M and 6E taken from section 2 ) to find the m o m e n t s of the isospin-violating distributions, and finally use this formula again to bring the isospin-violating distributions up to the scale of the data. When we include the QED c o n t r i b u t i o n to evolution, isoscalar and non-isoscalar distributions mix, introducing additional isospin asymmetry. For example, the evolution equation for the minority-quark difference becomes

~3 The glue then carries 47% of the target momentum at the experimental scale, not significantly different from the measured 44%.

Table 1 Moments of the isospin-averagedand isospin-violatingvalence distributions.

436

Moment

Uv

fiUv

%

dv

6d,.

%

2 3 4 5

0.27 0.080 0.033 0.016

- 1.4X 10 3 -5.0)<10 4

-0.51 -0.63 -0.72 -0.79

0.11 0027 0.0096 0.0042

2.3)<10-3 7.6)<10 4

2.1 2.8 3.6 4.3

-2.4X

10 -4

- 1.3)< 10-4

3.4)< l0 -4

1.8X 10 4

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Table 2 Moments of singlet distributions. Moment

u+a

8(u+a)

%

d+d

8(d+d)

%

2 3 4 5

0.30 0.082 0.033 0.016

-- 1 . 8 X 10 - 3

-0.59 -0.65 -0.73 --0.80

0.14 0.029 0.0099 0.0042

2.9X 10-3 8.1 X 10 - 4

2.1 2.8 3.5 4.2

-5.3X 10-4 -2.4X

10 - 4

-- 1.3X 10-4

The measured isospin-averaged distributions were fit to the form x a ( 1 - x ) ~ [8]. We can now fit the non-isosinglet distributions to this form as well. Suppose, for example, that the exponents t2 and fi characterize the distribution d(x). W h e n isospin violation splits d(x) into dP(x) and u n ( x ) , ~ and fisplit at first order into a _+ 8 a and fi+ 8ft. F o r valence distributions, whose normalizations are fixed by sum rules, the perturbations fia and 8fl can be calculated from the second and third m o m e n t s o f the isospinviolating valence distributions. The minority and majority valence distributions are then differences o f distributions o f this form, and are plotted in fig. 1. 4. Application to v scattering Isospin-violating quark distributions complicate the extraction o f sin: 0w from deep-inelastic neutrino scattering. In their absence, the Paschos-Wolfenstein relations [9] equate ratios o f neutral- and chargedcurrent cross sections, R + - a ~ c + aNC ~ a ~ C -+ a c9 c

(4.1) '

to binomials in sin 20w. These relations can be used to d e t e r m i n e sin20w to high accuracy only after the cross sections have had the contributions from the isospinviolating distributions removed. F o r a target composed o f equal numbers o f protons and neutrons, the distributions 8 ( (u + a ) - ( d + d ) ) contribute to the sum and difference of charged-current and, to a lesser extent, neutral-current cross sections that a p p e a r in R +-. The total cross sections measure the second moments o f these distributions with respect to a universal, Bjorken scaling variable ~4, hence the contribution o f target mass shifts found earlier must he

3 . 5 X 10 - 4

1.8X 10-4

dropped. F o r the valence distributions (occurring in R - ) this m o m e n t is - 3.8 × 10 -3, and for the sum o f quarks and antiquarks (in R ÷) it is - 4 . 9 × 10 -3. Failure to correct for these isospin-violating moments in R ÷ (R - ) would cause an overestimate o f sin20w by 1.4× 10 -3 ( 2 . 4 × 10-3). Hence in o r d e r to obtain a high-precision m e a s u r e m e n t o f sin 2 0w from neutrino-scattering experiments and thereby probe the high-energy behavior o f the standard model, we need to i m p r o v e our understanding o f nucleon structure, in particular its response to the breaking o f isospin. In the differences o f the form 8 u - 8d that a p p e a r above, the m o m e n t s o f the majority- and minoritydifferences have opposite signs and reinforce one another. This is because 8 u - S d equals (uP+un)(do+dn), the difference between the up-quark and the down-quark distributions s u m m e d over the two nucleons; relative to the down quark, the up radiates more strongly during evolution and, more important, has less mass, shifting it to lower x in each nucleon. (Recall that electromagnetic interactions among the quarks reduce but do not reverse the effect o f quark mass.) F o r a target with unequal numbers o f protons and neutrons, the sum flu + 8d also contributes to R +. Here the opposite signs o f 8u and 8d lead to a near total cancellation, and further this sum appears weighted by the neutron excess, ( N - Z ) ~ ( N + Z ) . Therefore it can be completely ignored and the corrections to sin 2 0w are the same as for a target with no neutron excess. ~4 In deep-inelastic scattering, the momentum of the struck quark appears in the cross section; the momentum of the nucleon containing the quark is not relevant. Hence we use a Bjorken scaling variable, which is defined with respect to a target-independent mass, e.g., the average nucleon mass.

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5. Conclusion T h e size o f the i s o s p i n - v i o l a t i n g q u a r k distributions puts t h e m at the f r o n t i e r o f e x p e r i m e n t a l accessibility. As a t t e m p t s are m a d e to o b t a i n f u n d a m e n t a l p a r a m e t e r s a n d s t r u c t u r e f u n c t i o n s f r o m deep-inelastic scattering to e v e r - i n c r e a s i n g accuracy, these dist r i b u t i o n s will n e e d to be u n d e r s t o o d a n d t a k e n into account. T h i s w o r k r e p r e s e n t s an initial step in that direction.

Acknowledgement I t h a n k Peter R o w s o n and c o l l a b o r a t o r s for m o t i v a t i n g this w o r k a n d R.L. Jaffe for essential g u i d a n c e t h r o u g h o u t . T h a n k s are also o w e d to J i m Olness w h o c o r r o b o r a t e d s o m e o f the early results a n d X i a n g -

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d o n g Ji w h o e x p l a i n e d an i m p o r t a n t point.

References [ l ] A.I. Signal and A.W. Thomas, Phys. Lett. B 211 ( 1988 ) 48 l; Phys. Rev. D 38 (1989) 2832. [2] F.E. Close and A.W. Thomas, Phys. Lett. B 212 (1988) 229. [3] R.H. Bernstein et al., Precision measurements of neutrino neutral current interactions using a sing-selected beam, FNAL Proposal P-815 (1990). [4 ] R.E. Jaffe, Phys. Rev. D 11 ( 1975 ) 1953. [5] S. Weinberg, Trans. NY Acad. Sci., Ser. 2, 38 (1977) 185. [6] H. Georgi and H.D. Politzer, Phys. Rev. D 9 (1974) 416; D.J. Gross and F. Wilczek, Phys. Rev. D 9 (1974) 980. [7] R.L. Jaffe and G.G. Ross, Phys. Len. B 93 (1980) 313. [8 ] M. Shaevitz, Fits to the CCFR Tevatron structure functions measured in neutrino scattering on iron, talk Division of Particles and Fields of the APS ( 1991 ), to be published. [9 ] E.A. Paschos and L. Wolfenstein, Phys. Rev. D 7 ( 1973 ) 91.