Extracting the pion-nucleon sigma-term from data

Volume 213, number 1 PHYSICS LETTERS B 13 October 1988

EXTRACTING THE PION-NUCLEON SIGMA-TERM FROM DATA ~

J. GASSER, H. LEUTWYLER Institut fiir theoretische Physik, Universitdt Bern, Sidlerstrafle 5, CH-3012 Bern, Switzerland

M.P. LOCHER and M.E. SAINIO Paul Scherrer Institut, ~ CH-5234 Villigen, Switzerland

Received 7 July 1988

Exploiting unitarity and analyticity we derive a set of integral equations which determine the behaviour of the S- and P-wave phase shifts near threshold in terms of two subtraction constants. These constants also fix the value of the -Y-term.

In the low-energy region, the behaviour of the nN scattering amplitude is governed by chiral symmetry [1]. The leading contribution stems from the tree graphs generated by the chiral effective lagrangian [2] ~'

Atree =a~ -v = ~ ,

B~ee = Bffv + 1 '

+ 2( 1 1 ) ga ( ~ ) B p v = g m---T~_s T m---TT-~_u - ~m2 , (1)

where F , = 93.3 MeV is the pion decay constant. The strength of the pion-nucleon coupling is also pre- dicted by chiral symmetry: g~--gAm/F,:. We do not make use of this approximate relation, however, but work with the phenomenological value g2/4n= 14.28 [31.

To simplify the notation, we remove the pseudo- vector Born term contribution from all of the amplitudes under consideration

¢r Work supported in part by Schweizerischer Nationalfonds. ~ Formerly Schweizerisches Institut f'tir Nuklearforschung (SIN) ~ We use the notation of ref. [ 3 ]. The pion mass is denoted by/1

and m is the mass of the proton. The pion LAB energy m = # + T~ is related to the CMS m o m e n t u m q by q2= (092_ It 2) / ( 1 + x 2 + 2xm / /l ) with x = / l / re . The variable v is defined by v= ( s - u ) / 4 m .

F - F - F p v .

Chiral symmetry then relates the value of the isospin even amplitudeb + =A ÷ + u/~ + at the Cheng-Dashen point

X = F ] I 5 + Is=m2, t=2,u2 (2)

to the nucleon matrix element of the quark mass term [4,51

m u + m d a= 4---m- ( p l o u + a l p ) . (3)

The analogous low-energy theorem for the isospin odd amplitude relates the value o f / ~ - / v at the Cheng-Dashen point to the nucleon matrix element of the vector current (Adler-Weisberger relation) [1,51.

The comparison of these predictions with experimental data meets with two problems,

(i) The predictions are exact only if the mass of the u and d quarks is set equal to zero. To compare the low-energy theorems of chiral symmetry with data, one needs to estimate the size of the corrections generated by the quark masses.

(ii) Since the Cheng-Dashen point is outside the physical region, one needs to extrapolate the available experimental information to this point. The re- markable work of the Karlsruhe group [ 3 ] shows that analyticity and unitarity allow one to perform this

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85


extrapolation in a meaningful manner. The result for the Z-term is X = 64 + 8 MeV [6 ].

A S-term of this size calls for rather drastic changes in the standard pictures [ 7-10 ], as it implies that the matrix element (p l gs lp) is large and that half o f the nucleon mass is generated by the mass of the strange quark ~2. We therefore consider it necessary to reex- amine the low-energy structure of the 7tN amplitude, both experimentally and theoretically [14]. Con- cerning point (i), the situation has recently improved considerably. A complete evaluation o f the quark mass corrections at the one-loop level of chiral perturbation theory has been achieved within chiral SU (2) X SU (2) [ 15 ]. This analysis allows one, in particular, to estimate the size of the off-shell corrections which occur in the low-energy theorem for the isospin even amplitude. As discussed in refs. [ 9,10,15,16 ], the difference between a and S is of order (mquark) 3/2. Numerically, the one-loop correc- tion amounts to S - a = 5 MeV [ 15 ]. The main contribution and the main uncertainty stems from the t- dependence of the scalar form factor ( p ' I Qu+ dd I P ) . This problem is under study.

In the present paper, we address point (ii). The main problem here is that the S-term is an inherently small effect which measures the chiral asymmetry generated by the u and d quark masses. Since the data analysis underlying the value of X quoted above is very complex, it is difficult to analyze error propagation and to reliably relate the uncertainties in the value of S to the uncertainties in the experimental information. [ In this connection, we mention that the prediction of chiral symmetry concerning the isospin odd amplitude (one-loop improved version of the Adler-Weisberger relation) is in excellent agreement with the Karlsruhe analysis [ 10 ]. In contrast to the case of the S-term, this test of the theory does however not concern an inherently small effect. ] In the following, we propose an extrapolation method which allows one to analyze error propagation in a more di- rect manner.

At low energies r~N scattering is dominated by the

~2 In ref. [ 8 ], the implications of a large S-term were analyzed within chiral perturbation theory to one loop. The related question of the valence quark contribution to the spin of the nucleon has recently attracted interest [ 11 ], in particular in connection with the dark matter searches [ 12 ] and with axion couplings [ 13 ].

six partial waves S~ ~, $3 ~, P~ ~, P31, P~ 3 and P33- (No te that chiral symmetry suppresses the isospin even S- wave; in D + the P-waves are comparable to the S- waves already at LAB momenta of order 30 MeV/c! ) The experimentally well explored region, where both the differential and the total cross sections are measured, begins at CMS momenta qo of the order of the pion mass (qo =/z corresponds to kLA B-.~ 170 MeV/c, T~-~ 80 MeV). We therefore need to extrapolate the six phase shifts listed above from the experimental region q> qo down to threshold. It is well known that this cannot be done on the basis of the effective range formula [ 3 ]

Re f + = (sin 2~l+ )/2q=q2t(al+ +q2bl+ +. . . ) , (4)

because this formula fails before the experimental region is reached. ( I f the tree graph contributions are removed, the isospin odd S- and P-waves are very well represented by (4) for a considerable range o f momenta. For the isospin even partial waves, this is however not the case. )

We shall use forward dispersion relations to discuss the energy dependence of the invariant amplitudes. The standard equations for D and B are

Re/3+ (0)) = / ) + (#)

+ 2 ( 0 ) : - / z 2 ) i r d0)'0)' ] [ I m D + ( 0 ) ' ) ] L0)'2-0)2J L 0)'2-Iz2 '

,u

g20) 209 ~ d0)' R e / ) - ( 0 ) ) = ~Sm 2 + --~r J 0 ) '2 -0 ) ~ - - - - - - 5 I m D - (0) ' ) '

/1

ReB+(0 ) ) = 20) i ,do)' imB+(0) , ) CO 2 - - 0 ) 2

/z

2m 2 i d0)'0)' R e B - (0)) = + - _ 0 ) ~ I m B - (0) ' ) . 7~ 0 ) , 2

/a

(5)

These relations impose four constraints on the six S- and P- wave phase shifts. To fully determine the energy dependence of these partial waves, we introduce two further invariant amplitudes E -+ (0)), defined by

0 +0)B) ,-o E ( 0 ) ) = ~ ( A _ , (6)

where the partial derivative is taken at fixed s. (Note

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Volume 213, n u m b e r 1 PHYSICS LETTERS B 13 October 1988

that the functions E +- (o9) are not symmetric under c o - , - co. ) The dispersion relations obeyed by E(o)) are easily obtained from the fixed-t dispersion relations for A (s, t ) and B (s, t):

Re/~+ (co) =/~+ (/z)

+ i [-. dco: co' 1 co' )] ~r Lco'2-cozJ L

,u

- i d w ' [h'2(co', col I m D + (co ' ) /t

-/~, (co', co) I m B + (co') ] , QC~

R e / ~ - ( c o ) = 2co~ do)' _ co , 5-,--_ co ~ I m E - (co) /z

+ i do)' [h2(co', co) I m D - ( c o ' )

- h i (co', co) I m B - ( c o ' )] ,

hn(co', co) = (co' +co) -" /27rm,

fin(co', co) =h,(co ' , co) - h , ( c o ' , St) . (7)

At low energies, the main contribution to the dispersion integrals stems from the region of the A-res- onance, which generates a rapidly growing P33 phase shift. The effective range formula describes the behaviour of this partial wave quite well, once the tree graph contribution is removed. The corresponding contribution to the invariant forward amplitudes can be read off from the partial wave decomposition

D(co) - 47rx~ss ~ [ (l+ l ) f+(q )+l f_ (q ) ] , m l=o

B( co) = ( 47rm/q 2)

X ~ [ ( l + l ) ( e - l - 1 ) f + ( q ) + l ( e + l ) f _ ( q ) ] , l=O

E(co) = t~qSq2 t~ol(l+ l )(l+ Z ) [f+ (q) + fu+l)_ (q) ] ,

(8)

where e stands for ( 1 + qZ/m 2 ) l/2. In the case of Re B and of Re E, the contribution generated by the P33 wave is approximately linear in q2. Indeed, plotting Re(B+-B+ee) a n d R e ( E Z - E + ¢ e ) v e r s u s q2 from

ref. [ 17 ], one essentially obtains a straight line up to q ~/t. In the D amplitudes, the P-wave contributions are however suppressed by a kinematic zero at q=0.The kinematic suppression factor ~q2 is re- sponsible for the strong curvature seen in Re D +. The essential point here is that these features reflect prop- erties of the absorptive parts in a region where the amplitudes are essentially measured directly and where the error bars are under good control.

The six dispersion relations (5), (7) involve two subtractions, which stem from the subtraction function occuring in the fixed-t dispersion relation for the amplitude A + (s, t). In the Karlsruhe analysis, the size of the subtraction function is obtained on the basis of an iteration involving discrepancy function tech- niques. We shall adopt these results only for the amplitudes in the experimentally well determined region q> q0, but we do not make use of these values in the extrapolation to the low-energy region. In our opinion, the subtraction constants are the main source of uncertainty in the extrapolation and we therefore treat them as parameters to be determined on the basis of low-energy precision measurements. The subtraction constants are related to the scattering lengths a++ and a++,

g2x3 15+ (Iz)=4n( l + x)a~+ +

~t ( 4 _ X 2 ) '

g2x2 E + (/z) = 6re( 1 +x)a ++ (9)

f13(2__X)2 "

The two scattering lengths a J-+ and a ++ are the two parameters of our extrapolation procedure. Given a value of these two scattering lengths, the dispersion relations (5), (7) fully determine the behaviour of the S- and P-wave phase shifts in the low-energy region in terms of information derived from the prop- erties of the scattering amplitude at higher energies. The extrapolation procedure involves the following steps:

Choose a value for a~-+ and ai~+. Choose a value for qo and calculate the absorptive

parts I m D +, I m B +, I m E + in the region q>qo, using one of the existing phase shift analyses [ 18,19,20 ]. (Note that, in the isospin basis, Im D and Im E are positive. )

In the lowzenergy region q< qo, the D, F, ... waves only generate small corrections. The uncertainties in

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their extrapolation from the experimental region to threshold do not strongly affect the invariant amplitudes. We take the partial waves with l>~ 2 from ref. [17] .

With this input, the six dispersion relations listed above amount to a set o f non-linear integral equations for the six S- and P-wave phase shifts, which can be solved by iteration starting with zero S- and P-waves in the interval q< qo. In practice this procedure converges after three to four iterations. Since we are close to threshold the phases are small and the solution is unique. Using the amplitudes KH.80 [ 18 ] for kLAB > ko = 172 MeV/c and with the subtraction constants a~-+, a ~-+ from the same source we have reproduced all the other low-energy parameters aft+, b~+, a ?_ and a i-+ within one standard deviation (table 2.4.7.2. ofref. [3] ).

We now discuss the implications of our analysis for the S-term. To determine Z we need to extrapolate the ampli tude/3 + ( u, t) in the variable t to t = 2fl 2. In terms of the coefficients which occur in the Taylor expansion o f / j ÷ in powers of u 2 and t

1 ~ + + + 2 + =d~o +dloU +do, t+d~ov4 +d+lv2t+d~2t2 +...

the X-term can be written in the form

X=F 2 (d6~o +2/t2d6~) + Z , , (10)

where we have isolated the leading contributions o f order mqua~k. The remainder X~ is generated by the curvature o f / J + (0, t).

At one loop, chiral perturbation theory predicts X~ = 6.1 MeV. The Karlsruhe analysis, on the other hand, leads to a value which is substantially larger: X~ ,,, 12 MeV [ 21 ]. The origin of this difference was identified in ref. [ 11 ]. The curvature in the t-dependence is mainly generated by the nn cut at t > 4/12. The nn phase shift underlying the Karlsruhe analysis corresponds to an S-wave scattering length o f a ° = 0.26. As shown by Koch [ 6 ], the curvature is rather sensi- tive to the value o f the scattering length used, i f a ° is increased from the current algebra prediction 0.16 to 0.28 the L-term increases by about 8 MeV. Actually, a ° is known very accurately from chiral perturbation theory. The one-loop corrections to current algebra were worked out in ref. [22] with the result a ° = 0 .20+0.01. Accordingly, the value Xl -~12MeV quoted above, which corresponds to ao ° =0.26, is reduced, roughly by the fraction 6 . ( 2 8 - 1 6 ) of the

variation considered by Koch, i.e. by about 4 MeV. We conclude that a t-channel dispersion analysis based on nn phases which are consistent with chiral symmetry leads to Z~ = 8 MeV.

In the language of chiral perturbation theory, the difference between a ° = 0 . 1 6 and a ° = 0 . 2 0 corresponds to a two-loop contribution to the nN amplitude which enhances the value of Z1. There are however also other effects manifesting themselves at the same order. In particular, the A generates a contribution of opposite sign (with g~/4n= 13.5 GeV-2 [ 3], the corresponding curvature amounts to Z~ = - 3 MeV). In the following, we use

X 1 = 8 "~ 2 MeV

In our opinion, the error bar generously covers the uncertainties associated with the extrapolation in the variable t. The point here is that the uncertainties in the curvature are small; the main issue in a determination of the S-term, therefore, is the value o f the constants d+o and d~] which occur in eq. (10). In fact, chiral perturbation theory controls the behaviour of the amplitudes 6 -+ in the entire region I v I ~< #, 0 ~< t ~< 2/~ 2 within small uncertainties, once the coefficients d~-o, d~o and dJ-j of the Taylor expansion are known [ 10]. Using the chiral representation of the amplitude, one can express the S-term directly in terms of the threshold parameters, e.g. as #3

X=nF~[ (4+2x+x2)a~+

- 4/-t2b6V+ + 12x#2a ++ ] + ~ o . ( 11 )

To one loop, chiral perturbation theory predicts 2 7 o = - 12.6 MeV, including effects generated by the A. In this case, it is more difficult to estimate the size o f the higher order effects, because the origin of the contributions o f the type v2t and v 4 to the amplitude 15 + is not easily identified. We do therefore not at- tempt to quantify the uncertainties associated with eq. (11).

Inserting the central values of the coefficients d~-o=-l .46+_0.10, d~]=1.14_+0.02 from ref. [3] , the formula (10) gives S = 59_+ 2 MeV. [Using the

#3 Relations of this type were proposed earlier [ 23 ]. In the notation of ref. [ 10] eq. ( 11 ) corresponds to r=0. The virtue of this choice is that the value of the constant 2:o is fixed by the chiral representation of the amplitude (chiral perturbation theory to one loop).

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central values of the threshold parameters given in the same references, formula (11) leads to 27= 60 MeV ].

The coefficients d~-0 and d+j coincide with the forward amplitudes at o~= 0

d+o = /5+ (0), d~-~ = E + ( 0 ) .

The dispersion relations (5) and (7) therefore deter- b~+, aF_, ai-+ as mine the threshold parameters ar-+, + +

well as the coefficients drY0, drF~ as functions o f our two basic parameters a~-+ and a ++. Since the permis- sible variations in these parameters are small, the result is well represented by a set of linear relations. Using the partial waves [18] for kLAn> 172 MeV/c we obtain

bo~+ = - 0 . 0 4 3 - 0 . 1 7 A o - 2 . 8 A l ,

a~-_ =-O.056-O.O05Ao+O.95A~,

dffo = - 1.492+ 14.6Ao-0.4Aj ,

d~-~ = 1.138+0.003Ao+20.8A~,

Ao=a~-+ +0.010, Al=a~-+ - 0 . 1 3 3 ,

in units of pion mass. Fig. 1 shows the dependence o f 27 on the input pa-

rameters, using eq. (10). The rectangle on the left represents the Karlsruhe values with errors quoted. The band on the right corresponds to the mesic a tom

0.137 . . . . . . . . . . . . . . . . . . . . . . i ..... ', ..... ! .....

o, °V / / , , / t o,~F ° / / i ', /~

o133[-..A2l / ! / ~ i3°A o.-2V ~ / / , /

°-°E / ! / i t -0.01 -0.02 -0.03 - 0.04.

aO * (p-1)

Fig. 1. The value of the X-term as a function of the parameters a++ anda++ using eq. (10) and -rl=8MeV. The rectangle on the left shows the range in the scattering lengths quoted by the Karlsruhe group [ 3 ]. The error band on the right reflects the result of the 7t-p atomic measurement a~+ o =a~-+ +ar-+ =0.059 _+ 0.006/~ - ' [24]. The crosses with numbers refer to the curves in fig. 2.

result [ 24 ]. It is well known that the mesic a tom data are not consistent with the Karlsruhe value for a~-+ [25].

The central value of the rectangle now occurs at 27= 56 + 2 MeV, lower than the value 64 + 8 MeV [ 6 ] by one standard deviation. Half of the shift stems from the reduction o f the 7tn scattering length, the remainder from the fact that, if aft+ and a ~-+ are kept fixed, the forward dispersion relations generate small changes in d+o and in d~-~ which both tend to reduce the value of 27. [Using eq. ( 1 1 ) one instead obtains 27= 59 MeV, a value which, within the uncertainties associated with this equation, is perfectly consistent with the above result. ]

We briefly discuss the uncertainties connected with the determination of S;

Variations of the amplitudes for kLA B > 172 MeV/c. We have replaced the amplitudes KH.80 [18] by KA.84 [ 19] which differ mostly for the high partial waves. The corresponding threshold parameters are reproduced with a precision similar to the KH.80 case. The CMU-LBL phases which are available [ 3 ] for 0.4<kLAB<2.5 GeV/c are in between ref. [ 18] and ref. [ 19 ]. The effect on the S-term is small. For the whole range of a~-+, a++ shown in fig. 1 the change of S never exceeds 0.7 MeV.

In general we have found little sensitivity of S to the iso-antisymmetric scattering amplitudes. Scaling Im B - or Im E - by 2% remains invisible in fig. 1.

The high-energy tails (kLAB> 10 GeV/c ) for the relevant isosymmetric dispersion integrals are com- pletely negligible.

The D-waves are input for our iteration scheme. A 30% variation leads to AS<0 .1 MeV. F-waves are even less important.

In fig. 2 we compare three solutions, marked with numbers 1-3 in fig. 1, for the n+p differential cross section at the lowest available energy (kLAB=79 MeV/c ) for the purpose o f illustration.

We conclude that the overall uncertainty o f S in this analysis is less than 2 MeV. This uncertainty in- cludes effects generated by the differences between the various phase shift analyses mentioned above, but does not account for experimental errors otherwise. The main problem in the determination of the S-term is thus reduced to a precise determination of the two basic parameters a~+ + and a ~ +. A detailed analysis of the current experimental information about the range

89


1.4

1.2

1.0

~1~, 0.8

~1~ 06

0.4

0.2

I \ I ' [ , i z i , i , i , i ,

\ \

i I i L L I i I I I L i , i , i

2 0 /~0 60 80 100 120 140 160 180

8cM {degrees)

Fig. 2. The differential cross section for n+p scattering at kLAB=79MeV/c(T,~=20.8MeV). The solid line gives the Karlsruhe solution (KH.80) (number 1 in fig. 1), the dashed line corresponds to the cross 2 in fig. 1 and the dash-dotted line to the cross 3. The experimental points are from ref. [26].

o f these p a r a m e t e r s will be g iven elsewhere . N e w low-

ene rgy p r e c i s i o n e x p e r i m e n t s [27 ] will a l low one to

n a r r o w th i s r ange fu r the r .

T h i s s t u d y w o u l d n o t h a v e b e e n poss ib l e w i t h o u t

t he g e n e r o u s he lp o f G. H 6 h l e r , R. K o c h a n d H . M .

S t a u d e n m a i e r w h o m a d e t he resu l t s o f t he K a r l s r u h e

ana lys i s a v a i l a b l e for us, a n d in n u m e r o u s d iscus-

s ions c lar i f ied deta i l s o f t he i r results . P a r t o f th i s work

was d o n e whi l e o n e o f us ( J . G . ) was v i s i t i ng t he

C e n t r e de P h y s i q u e T h 6 o r i q u e in L u m i n y , Marse i l l e .

H e w o u l d l ike to t h a n k t he C N R S a n d the U n i v e r s i t y

o f A ix -Marse i l l e II for f i n a n c i a l s u p p o r t a n d t he

m e m b e r s o f t he I n s t i t u t e in L u m i n y for t he w a r m

h o s p i t a l i t y e x t e n d e d to h i m .

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Extracting the pion-nucleon sigma-term from data