Form factor of the σ-term

Volume 253, number 1,2 PHYSICS LETTERS B 3 January 1991

Form factor of the a-term

J. Gasser , H. Leu twyle r Institute for Theoretical Physics, University of Bern, Sidlerstrape 5, CH-3012 Bern, Switzerland

and

M.E. Sainio Research Institute for Theoretical Physics, University of Helsinki, Siltavuorenpenger 20C, SF-O0170 Helsinki, Finland

Received 24 September 1990

We use dispersion relations to calculate the nucleon form factor of the scalar operator au +dd and show that the corresponding mean square radius is remarkably large.

1. The preceding paper [ 1 ] updates the information about the magnitude of the a-term in nN scattering. In this context, the form factor

1 a( t )a(p ' ) u (p )= 2mm (p' lrh(au+dd) IP) ,

r h = ½ ( m u + m d ) , t = ( p ' - - p ) z, (1)

and the t-dependence of the scattering amplitude play an important role. The present note exhibits the analysis underlying the corresponding numbers quoted in ref. [ 1 ]. In particular, we wish to show that the slope o f the form factor a(t) is unexpectedly large, a result which affects previous determinations of the a-term from nN scattering by a correction of roughly - l0 MeV.

2. Since the proton is composite, the form factor a(t) tends to zero as t ~ o o and the once-subtracted dispersion relation

t o~ dt ' a ( t ) = a ( O ) + J t , ( t , _ t _ i e ) I m a ( t ' ) (2)

4 / t 2

should be dominated by low-energy intermediate

~r Work supported in part by Schweizerischer Nationalfonds.

states. In the elastic region 4/22< t < 16122, the imaginary part is given by

3 " o ( F ~ ( t ) f + ( t ) 1 - . (3) Im a ( t ) = 2 4 m 2 - t

Here, F ~ ( t ) = ( n ° ( p ' ) l r h ( a u + d d ) l n ° ( p ) ) is the pion a-term a n d f o (t) is the I = J = 0 nN partial wave in the t-channel. The form factor F~(t) was determined recently [2 ] by solving the Muskhelishvili- Omnbs integral equations for the nn/KI?, system, using the phase-shift analysis o f refs. [ 3,4 ] and fixing the subtraction constants with chiral symmetry. The partial wave f o (t) was analyzed by HShler and col- laborators [ 5,6]. We heavily rely on their results in the space-like region where it is controlled by the nN data, but modify the nn phase shift used in the continuation to the fimelike region. The continuation is performed as follows [ 5,6 ].

3. Introduce the Omn6s function t l

t f dt ' r ° ( t ' ) Ao( t )= -n a t' t ' - t - i e ' (4) 4,u 2

where ri°(t) is the I = J = 0 nn phase shift. An appre- ciable inelasticity only sets in at KI( threshold. (For a quantification of this statement in the framework

260 0370-2693/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. ( North-Holland )


ofchiral perturbation theory see ref. [ 7 ]. ) In the following we choose t~ = (0.9 GeV) 2. The function

f ( t ) =exp[ -Ao( t ) ] f ° ( t ) / ( 4 m 2 - t ) (5)

contains a left-hand cut extending from tnc=4# 2 -- # 4 / m 2 t o -- oo; since the exponential removes most of the nn cut, a substantial right-hand discontinuity only occurs for t> tt. We exploit the fact that the imaginary part has a zero at t = 2 # 2 and write the dispersion relation in the form

/tic

f ( t ) - t - 2 # 2 -J dt ' I m f ( t ' ) + R y ( t ) . (6) n t ' - t - i E t ' - -2# 2

12

If the cutoff 12 is taken sufficiently far to the left, the remainder Rf (t) only represents distant singularities, such that, for small values of t, it can be approx- imated by Ry(t) =a+bt. In the region 0</</no, the imaginary part o f f o is given by the Born term,

I m f ( t ) =exp[ - 3 o ( t ) ] ~g2m(2#2- t )

X ( 4 # 2 - t ) - l / 2 ( 4 m 2 - t ) -3/2 , (7)

where g,-- 13.4 is the nN coupling constant. In the space-like domain t2< t<0, w e use the values for f o (t) given in ref. [5]. As a test of the machinery, we insert the nn phase shift used in that reference and compare the left-hand side of eq. (6) at small space- like momentum transfers with the value of the integral. We find that Rf ( t ) is indeed linear to a good approximation with a = 0.61 #, b = 0.30#3, t2 = - 70# 2 and the representation (6) reproduces the continuation given in ref. [ 5 ].

4. The nn phase shift underlying our analysis is shown in fig. 1. Chiral symmetry provides a strong constraint near threshold as it predicts both the scattering length and the effective range rather accu- rately: a°=0 .20+0.01 , b °=0 .24+0 .02# -2 [8]. The data on the other hand consistently require the phase to slowly rise through ½ n before the KI( threshold is reached. We interpolate between these two regimes with a phenomenological formula of the type

t g ~ ° = q ( f 2 + q 2 ) - l / 2 ( O t + f l q 2 + y q 4 ) 1-- ~00 '

where q is the center of mass momentum. The shaded region shown in fig. 1 results if a and fl are adjusted

1 O0 °

8 0 °

6 0 °

4 0 °

2 0 °

0 o

2 5 O

6°(~)

/ /

/ /

3 5 0 4 5 0 5 5 0 6 5 0 7 5 0 B 5 0

V ~ (MeV)

Fig. 1. nn phase-shift as a function of energy. The behaviour near threshold is controlled by chiral symmetry which predicts ao ° = 0.20_+ 0.01, b ° = (0.24 + 0.02 )# -2 [ 8 ]. The data at x / s > 600 MeV are from ref. [ 3 ]. We determine the sensitivity of our results to the uncertainties in the nn phase by varying 6o o (s) in the range indicated by the shaded area. The dashed line depicts the phase-shift used in ref. [ 5 ] which is based on the solution of the Roy equations obtained by Froggatt and Petersen [ 9 ].

to the chiral predictions, while ~, and mo are varied within the range permitted by the data of ref. [ 3 ]. For comparison we show the phase shift used in ref. [ 5 ], which is based on the solution of the Roy equations obtained by Froggatt and Petersen [ 9 ]. (For al- ternative parametrizations of 6 ° see ref. [ 10 ]. )

5. The evaluation of the form factor a(t) is now straightforward. The representation ofF~ (t) given in ref. [ 2 ] leads to the results shown in fig. 2, and fig. 3 gives the corresponding values of the dispersion integral occurring in eq. (2). The uncertainties indicated arise from the ambiguities in the nn phase shift mentioned above [when evaluating a(t) - a(O) for a given representation of the phase shift, we have used the same representation also in the solution of the integral equations for F~(t)]. Numerically, the result for z % - a ( 2 # 2) - a ( 0 ) is

/%= 15.2+0.4 MeV, (8)

261


2 5 o - I r~(t) I /~ ~ /

: j /

/ / 2 . 0 0 :"

/ /

1.50

1 . Q O \...X

\ \

0 . 5 0

0 . 0 0 , , , , , , . , . . . . . . . . 0.0 0.2 0.4 0.6 0.8

,/7 (CeV)

Fig. 2. Solution of the integral equations for the a-term of the pion [2]. The solid lines correspond to the solutions using the phase-shifts shown in fig. 1 as input. The dash-dotted curves de- pict the Omn~s representation F~(t) =/z2( 1 +bt) exp,do(t) with b = 0.38 fm 2. The uncertainties indicated by the shaded regions arise from the ambiguities in the ~ phase shift. The one-loop result o f chiral perturbation theory (shown as a dashed line) in- cludes all contributions of order p4. The dotted line depicts the tree approximation F~ (t) = / t 2.

where the error bar only shows how the result de- pends on the uncertainty of the nTt phase shift representation indicated in fig. 1.

The result (8) is remarkable as it implies that the slope o f the scalar nucleon form factor, a ( t ) = t7( 1 + ~ ( r 2 ) s t +... ) is larger than hitherto assumed. For t r=45 MeV we find a mean square radius o f (r2)SN ~ 1.6 f m 2, about twice as large as the mean square radius ( r E ) ~,i rn" "~ 0.82 fm 2 o f the electric (iso- vector) charge distribution GV(t)! ~. This shows that the scalar current au + d d is much more sensitive to the pion halo surrounding the nucleon than the electromagnetic current. The effect is also seen in the pionic matrix elements o f these currents, but it is less

~t As can be seen in fig. 3, the anomalous threshold associated with ~n intermediate states not only generates a large slope but it in addition produces a significant curvature which also contributes to the value of Aogiven in eq. (8).

MeV

4 0

30-

20

10-

O-

-10

~(t) - ~(o)

......"'" .- /.,-

. /" .-

- 2

/ / j /

J t "

t l~, 2 - 1 0 1 2 3 4

Fig. 3. Result of the dispersive analysis of the form factor a( t ) . The shaded region shows the difference tr(t)-a(O). For comparison we also indicate the one-loop result of chiral perturbation theory (dashed line) and the dipole formula for the scaled electric form factor, a(O)[2GV(t) - 1], with a(0)=45 MeV (dash-dotted line). The figure shows that the scalar form factor rises more steeply with t than the electromagnetic one. The strong curvature generated by the anomalous threshold is visualized with the tangent at t=0 (dotted line).

pronounced there: the mean square radius of the scalar form factor F~(t) is ( r 2 ) S = 0 . 6 0 fm 2 [2] , also larger than the mean square radius of the charge distribution, ( r2)en "m" ---0.44 fm 2 [ 11 ].

The mean square radius ( r 2 ) s is determined by

the integral f d t l m a ( t ) / t z and A~ represents the analogous integral with t 2_. t ( t - 2/z 2 ). The function Im tr(t) l ta/t 2 is shown in fig. 4. It is evident from this figure that the main contribution to the integral stems from the low energy region (x/~~ 300-600 MeV), where we expect our representation of the partial wave f o+ (t) to be reliable, because there the curvature generated by the states above 4 / ~ = 50/12 should be negligible. For comparison we also show the imaginary part of the scaled electric form factor 2tr(0) GV(t) l t4 / t 2 [note that G V ( 0 ) = ½ ]. Clearly in that case the contribution to the integral arising from the low-energy region is much smaller, comparable to the one from the p-resonance. This is why the scalar radius of the nucleon is so large.

262

MeV

2.0-

1.5

1.0

0.5

o.o t / l~ ~ 0

Ima(t ) u4/t 2

/ f ~ ' \ '~ X / \ '~')_, / \\ • ""-" .L-~ - ~ j ' 1 "\

i

10 20 30 40


Fig. 4. Imaginary part of the scalar form factor. The shaded region shows the behaviour of the function Im a( t )Fta/t 2, the dashed line depicts the chiral representation at one loop. For comparison the dash-dotted line shows the imaginary part o f the scaled electric form factor, 2a (0) Im GV(t)l~a/t 2, with a (0 ) =45 MeV (taken from ref. [5], fig. 5.2.1 ).

6. Let us compare the result (8) with chiral perturbation theory where the t-dependence of the form factors starts showing up at one-loop order. The leading nonanalytic contribution to the quantity J~ is given by 3g2/13/64nF 2 -~ 7 MeV and the full one-loop result is J~= 4.6 MeV [ 12] - this is the number pre- viously used in determinations of the a-term. The calculation described above shows that this number underestimates the actual value by about 10 MeV. The origin of the difference is the following. In the one- loop calculation, the two factors F*~ ( t ) f ° ( t) in eq. (3) are replaced by the corresponding Born approx- imations. In the case o f f ° (t), the Born term van- ishes in the vicinity of t=/12, and the data indeed confirm the occurrence of a zero there, but the Born approximation miserably fails if t is not in the vicinity of this point, because the I = 0 nn S-wave interaction is attractive and strongly grows with energy. For l> 4/12, this interaction amplifies the Born approximation f o r f o+ (t) by about a factor of two and a similar, though less pronounced effect also occurs in F~(t) . In chiral perturbation theory, the amplification generated by the nn interaction also shows up,

but only at higher order. In fact, for t> 4/12, the one- loop contribution to f o+ (t) is of the same order of magnitude and of the same sign as the tree graph; the one-loop contribution to F~(t) also clearly exhibits the amplification due to the nn interaction (see fig. 2 and refs. [ 10,7 ] where higher order contributions to F~ are discussed in detail). Since the chiral perturbation theory estimate for A, is, however, based on the Born approximation of the two factors occurring in eq. (3), it does not account for these effects. Iron- ically, the first attempt [ 13 ] at evaluating the leading nonanalytic contribution to A~ contained an alge- braic e r ro r - a factor of two - and gave A,= 2 × 7 MeV, remarkably close to the correct answer.

7. The continuation of the scattering amplitude proceeds along similar lines (for notation, see ref. [ 1 ] ). The amplitude/9÷ (t) contains a left-hand cut at t < -4m/1 ,-- -27/12 and a right-hand cut for t > 4/12. The main contribution to the curvature at small t stems from the lower end of the right-hand cut where the imaginary part is dominated by the S-wave contribution,

i m / $ + ( t ) - 16n t l m f O ( t ) + . . . 4m 2 _ (9)

Accordingly, we write the dispersion relation in the form

II dt' Imf° + ( t ' )

/ ) + ( t ) =16t2 t,2(t, t _ i ~ ) 4 m Z - t ' 4l/2

+RD(t ) • (10)

RD(t ) contains the singularities associated with the left-hand cut as well as what remains of the right-hand discontinuity if the S-wave contribution is removed. The D-wave term is readily estimated with the results given in ref. [ 5 ] - for t= 2/12, this contribution to the twice subtracted dispersion relation (10) is entirely negligible. The experience with the coupled channel integral equations for F, (t) indicates that the discon- tinuities above t=4/12 do not generate significant curvature at low t, either. Furthermore, if t is small, the left-hand cut can be replaced by the contribution generated by the A-resonance which occurs at ta - - 63/12 and which produces a small negative cur-

=doo + d o ~ t - c a t with vature, R o ( t ) + + 2

263


Ca = 0 .012/ t -5 . ( 1 1 )

We have checked that in the range - 4 / ~ 2 < t < 4/~ 2, the representation (10), ( 11 ) indeed agrees with the values o f / ) ÷ (t) given in ref. [ 5 ] if the partial wave f o (t) is calculated with the nn phase shift used there. In this case the numerical value o f the curvature term

- - 2 - - + 2 + AD=F,~[D ( 2 / t ) - d o o - 2 d ~ # 2] is AD=13 MeV, also in agreement with ref. [ 5 ].

8. It is then a simple matter to work out the change in this number which occurs if the nrc phase shift is modified. Since the values of d~-o and d~-i are irrele- vant here and since the remainder in the quanti ty R I stems from the left-hand cut and is not affected by the modification o f the nn phase, if suffices to evalu- ate the change occurring in the dispersion integral over I m f o with the result

A D = 1 1 .9+0.6 M e V , (12)

where the error bar again only indicates the sensitivity o f the result to the uncertainties in the nn phase shift shown in fig. I. Comparison with the number quoted above shows that if the scattering length is lowered from a ° =0 .26 to a ° =0.20, the value Of AD only decreases by 1 MeV. At first sight this appears to contradict Koch's result [14] who found that a smaller 7tn phase shift reduces the value o f the amplitude at the Cheng-Dashen point considerably. The contradiction is only an apparent one, however: in the discrepancy function technique used in ref. [ 14 ], a change in the phase shift also generates a modification o f the coefficients dd-o, dd-i and the change in the value of the amplitude at t = 2 l? is, therefore, not the same as the change in AD.

9. The main conclusion to be drawn from this analysis is that the singularity associated with nn intermediate states strongly affects the t-dependence both o f the scattering amplitude and of the a-term form factor. In the determination of the a-term, however, only the difference AD--A~ matters where the two effects almost compensate one another:

~D--A~= - -3 .3+0 .2 M e V . (13)

The cancellation also implies that the result is not sensitive to the uncertainties in the ;rTr phase shift (this is what the error bar stands for - at this level of accuracy, the curvature generated by the discontinu- ities at t> 4/12 may well show up, too) . The fact that the difference '~D-- ZI~ is substantially smaller than the individual terms can be understood on the basis o f chiral perturbation theory, where the leading contributions to both dD and A~ are o f order/z 3. The leading terms are A D = 2 3 a + .... A~= 18a+.. . with a=lt3g~ /384ztF~ (the remainder is o f order ]~4 log #). The similarity of the two coefficients explains why the two terms (Ao--- 9 MeV, A~= 7 MeV) almost cancel one another.

We greatly profited from the cooperation with Pro- fessor G. H6hler; in particular we are indebted to him for detailed information concerning the behaviour of the nN amplitude in the unphysical region.

References

[ 1 ] J. Gasser, H, Leutwyler and M.E. Sainio, Phys. Lett. B 253 (1991) 252.

[2] J.F. Donoghue, J. Gasser and H. Leutwyler, Nucl. Phys. B 343 (1990) 341.

[3] B. Hyams et al., Nucl. Phys. B 64 (1973) 134; W. Ochs, Thesis, University of Munich (1974).

[4] K.L. Au, D. Morgan and M.R. Pennington, Phys. Rev. D 35 (1987) 1633.

[ 5 ] G. H6hler, in: Landolt-B6rnstein, Vol. 9 b2, ed. H. Schopper (Springer, Berlin, 1983 ).

[6] R. Koch, Ph.D. thesis, University of Karlsruhe ( 1978); E. Pietarinen, A calculation of 7r~--,Nlq amplitudes in the pseudophysical region, University of Helsinki report HU- TFT-I 7-77 (1977).

[ 7 ] J. Gasser and U.-G. MeiBner, Chiral expansion of pion form factors to two loops, preprint BUTP-90/20.

[ 8 ] J. Gasser and H. Leutwyler, Phys. Lett. B 125 ( 1983 ) 321, 325;Ann. Phys. (NY) 158 (1984) 142.

[9] C.D. Froggatt and J.L. Petersen, Nucl. Phys. B 129 (1977) 89.

[10] T.N. Truong, Phys. Rev. Lett. 61 (1988) 2526. [ 11 ] S.R. Amendolia et al., Nucl. Phys. B 277 (1986) 168. [12]J. Gasser, M.E. Sainio and A. gvarc, Nucl. Phys. B 307

(1988) 779. [ 13] H. Pagels and W.J. Pardee, Phys. Rev. D 4 ( 1971 ) 3335. [ 14] R. Koch, Z. Phys. C 15 (1982) 161.

264

Documents

Form factor of the σ-term