QCD sum rules are used to obtain the isovector and isoscalar'' axial vector coupling constants, g{sub A} and g{sub A}{sup S}. We find g{sub A} = 1.26 {plus minus} 0.08, g{sub A}{sup S} = 0.13 {plus minus} 0.08 with sum results for g{sub A}{minus}1 and g{sub A} + g{sub A}{sup S}. These sum rules also show that in the limit of chiral symmetry restoration, g{sub A} {yields} 1 and g{sub A}{sup S} {yields} {minus}1

Henley, E.M. (Washington Univ., Seattle, WA (United States))

Hwang, W.Y.P. (National Taiwan Univ., Taipei (Taiwan, Province of China). Dept. of Physics)

Kisslinger, L.S. (Carnegie-Mellon Univ., Pittsburgh, PA (United States). Dept. of Physics)

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ilt DOE/ER/40427--35-N91-/DE92 014227NUCLEON AXIAL COUPLING CONSTANTS AND QCD SUM RULESErnest M. Henley 1, "_-Y.P. Hwang 2 and L.S. Kisslinger a1Department of Physics, FM-15University of Washington, Seattle, Washington 981952Department of Physics, National Taiwan UniversityTaipei, Taiwan 10764, R.O.C.aDepartment of Physics, Carnegie-Mellon UniversityPittsburgh, Pennsylvania 15213Symposium at Drexel UniversityOctober 31 and November 1, 1991PREPARED FOR THE U.S. DEPARTMENT OF ENERGYThis report was prepared as an account of work sponsored by the United StatesGovernment. Neither the United States nor the United States Department of En-ergy, nor any of their employees, nor any of t;heir contractors, subcontractors, ortheir employees, makes any warranty, express or implied, or assumes any legal li-ability or responsibility for the product or process disclosed, or represents that itsuse would not infringe privately-owned rights.By acceptance of this article, the publisher and/or recipient acknowledges the U.S.Government's right to retain a nonexchmive, royalty-free license in and to any copy-right covering this paper.DISTI::tlBUTIONOF THIS DOOUMENT IS UNLIMI'_gNUCLEON AXIAL COUPLING CONSTANTS AND QCD SUM RULES*Ernest M. Henley I*, W-Y.P. Hwang 2 and LIS. Kisslinger 3I Department of Physics, FM-15 and Institute for Nuclear Theory, HN-12University of Washington, Seattle, Washington 981952 Department of Physics, National Taiwan UniversityTaipei, Taiwan 10764, R.O.C.3 Department of Physics, Carnegie-Mellon UniversityPittsburgh, Pennsylvania 15213AbstractQCD sum rules are used to obtain the isovector and "isoscalar" axialvector coupling constants, gA and gAs. We find gA = 1.26 =t=0.08, gAs =0.13-t-0.08 with sum rules for gA-- 1 and gA +gSA. These sum rules also showthat in the limit of chiral symmetry restoration, gA ""* 1 and gSA ---+-1.I. IntroductionLet me first say that it is a pleasure to be here to help celebrate this happyoccasion of a colleague and friend. Although my talk is not directly in the field tb thas been of primary concern to Abe Klein, I hope he will accept it as the tribute itis intended to convey.One of the major questions for nuclear physics remains how to incorporate thebasic hadronic theory QCD into its framework. To overcome the nonperturbativeaspects of the theory, numerous methods have been proposed and used: lattice QCD,light-cone techniques, bag models, solitons, string and related flux-tube models,Nambu-Jona-Lasinio model, Q('D sum rules, and many others. Each of them hasbeen found useful in a limited realm, but none of them has so far been found tobe widely applicable and successful without requiring added features. But many ofthe methods are still being developed, and their applicability continues to be testedand extended.The method of QCD sum rules is, I believe, particularly useful for describinghadcuic structures. It has been applied to both mesons and baryons and found tobe remarkably successful. The met ho,i is to replace the non-perturbative long-rangeaspects by several (hopefully) universal vacuum expectation values (VEV's). Theshort-range aspects on the other hand, are calculated perturbatively. The methodwas developed by physicists in the USSR 1, who also extended it to computinghadronic properties in the presence of an external (e.g., electromagnetic) field. 2,aIn this talk I want to discuss the calculation of gvA = gA and gS, the isovectorand "isoscalar" axial vector coupling constants of the nucleon by means of QCDsum rules, a,4 I will also show that gA _ 1 and gS _ -1 in the limit of chiralsymmetry restoration at high nuclear densities or temperatures.We are particularly interested in the response of nucleons to electroweak fields.These responses have been considered in the past. 3,4 One of the critical tests of amodel of nucleons is its ability to generate the correct axial vector coupling constant,gA = 1.254 :k .006. Non-relativistic quark models predict 5/3 and relativistic onespredict 0.75 < gA _ 1.5. QCD sum rules have been shown 3 to give gA = 1.25,but this result has been obtained with non-standard values of the condensate. Byincluding terms consistently up to dimension d = 8, we use a QCD sum rule forgA-- 1 to show that gA = 1.26 with accepted values of these condensates. In additionwe calculate gS and compare it to recent EMC measurements.II. MethodThe method begins with a correlation operator Tr(q) in an external axial fieldZtLTr(q)= i J d4x e < 01T(q(x)fi(0))10>, (1)where the current 7/ has the quantum number of the hadron in question. For anucleon a useful form has been found to be 2-4rl(x ) = e_bc {u,,(x)T C %, ub(x)} ")'_")'SdC(x) , (2)where C is the charge conjugation operator and a...c are color indices. If vN(p) is anucleon spinor (normalized so that 9v = 2MN, with MN the nucleon mass) we canwrite< Olq(O)lN(p) >= ,kg vN(p), (3)where _N is a constant.The correlator can be written in an operator product expansion=c, + Z °", (4)72where I is the unit operator, the Cz, C, are Wilson coefficients and O, is an oper-ator; the On's can be ordered by dimension and the Wilson coefficients fall off bycorresponding powers of Iq21. _r is calculated in a region of Iq21 --_ 1 GeV 2, wherethe VEV's take into account nonperturbative QCD aspects, not included in theperturbative part.The correlator rr has various structure functions (labeled by j), each of whichsatisfies a dispersion relation1 _0_ ImrrJ(:_)ds- + subtraction constants (5)rrj(q2) _ rr s -- q2The subtraction terms are eliminated by assuring convergence via a Borel trans-formB[F(q_)]- lira 7i.( -g_ F(q_)--, - d_r._ _.(_)_-_/M___,-__ rrq2/n-.M_ ,finite(6)The correlator rr in the presence of Z, is evaluated first for the quarks in thenucleon', it is equated to the nucleon-based dispersion calculation,1 1rr(p) = -[AN[ 2 gA_75 + excited states and continuum. (7)13- MN 1_- :X.INBy comparing coefficients, several sum rules can be obtained for the coefficients ofP' Z/iT 5, _: 7_, and ia"v Z,p,@. Only the first of these is used to obtain gA because(i) it contains gA alone, and (ii) the excited and continuum state contributionsare less important than for the other terms, because they would cancel for paritydoublets, aWe begin with the quark propagator in configuration space in the presence ofthe external constant axial-vector field Z,. If we keep terms up to second order inan expansion of < 01q(x)_(0)[0>, we obtainS ab __ _ab i(2rr)4x4(i_-gx" Zi_7_) + 32_:2x 2 gc G_v(¢ a"" +a"V¢){ 1 ( 1 x2m02)+ 1 1+ _b < _q> -_ 1+ _ _ gx_75+ 3-6gx_Z_C'_'y5+_i-_ g_ _ _-*' ""'The various terms of this propagator are shown in Fig. 1. The constants g arethe coupling of the quarks to the external field. In the standard model they aregu = --gd = 1 (9a)for the up and down quarks; these are the coupling constants for the isovector gA.The "isoscalar" coupling gS is defined bygu = gd = 1. (95)a b cd eXf g hQuark propagator diagrams.lt should be noted that this is not truly the weak isoscalar axial coupling constant.The latter is given by the sum of gA for the proton and neutron, and vanishes inthe absence of"strange quarks. 5The susceptibilities h: and X and the mass m_ in Eq. (8) are defined by< 01_g_. aqlO>= -m_o< _q>,< 014g_G,_'Y_ql0> = gxZ_,< Ftq>,<0 q'YuT_q0>= gxZ_, < Clq> . (10)The sum rule for gA is then obtained from the coefficients of p. Z/rTs in Tr(p).After the application of a Borel transformation, we obtainM_E2 M_ 2G 2 M_ 5 a2L4/98L4/9 f 32L4/9 '_ gc > Eo - 18L68/sl _a E0 +1 2G 2-_ 288L4/9 xa < g_ >/J_(gA + AMg)exp(- 2 2=: 3,4N/Ms) , (11)Where a = -(21r) 2 < qq >, the length L is L = 0.6211n(10MB) corresponding toAQCD = 0.1 GeV, with the Borel mass Mu given in GeV; the quantity _v -=(27r)4,\_v/4. In Eq. (11) we have kept anomalous dimensions. The functions E0, El,and E2 are given byE0= 1-e -xBi = 1-(l+x)e -xE2 = 1 -(1 + x + lx2)e-" , (12)with x = W 2/M_. These functions account for contributions of excited states upto a mass W. We choose W 2 _ 2.3 GeV 2. The constant A represents residualcontributions of excited and continuum states. The terms which contribute to theleft-hand side of Eq. (11) are shown in Fig. 2.Similarly, for the "isoscalar" gAs we findi MiM_E2 M_ < g2G2 > Eo + xaM_E1 - 18_-S--/si _aZo8L4/9 + 32L4/9 6L4/91 L4/9 1 Xa < g2G 2 >- 1--_a_ + 288L4/92 s 'AS 7_,I_ exp(-JPf_/M_ ,=/3N(g A + ) ) (13)In order to obtain a sum rule for gA -- 1, we use the Belyaev-Ioffe 6 sum rule for themass MNM_ E2 + M_ < g2G 2> Eo -t- la2Lt/9 1 a2m2°8L4/9 32L4/9 6 24M_/_ exp( 2 2= -MNMB). (14)After subtraction of Eq. (14) from Eq. (11), we obtain1 a2L4/9._ 1 a2m 2 1 _aM_ 1 2G 224 M_ 18 Z6S/sl E0-{ 288L4/9 xa <gc >= /32N{(gA- 1)+AM_} exp(-M_/M_). (15)One advantage of the sum rule, Eq. (15) for gA- 1 is that it has a smaller variationin powers of M_ than that for gA. This occurs because the first two terms ofEqs. (11) and (14) are equal to each other. From Eq. (15), we immediately see thatif a _ 0, the left hand side = 0 and gA -- 1 -" O.The constants required for the evaluation of the sum rules are obtained fromother experiments. They are no_Atadjusted. These constants area _ 0.55 GeV a , baa ,_ 0.140 GeV 4 ,_a _ 0.70 GeV 2 .< g2cG2> _ 0.47 GeV 4 ,m02_ 0.8 GeV 2 (16)a b cz"J,d-1 d-2e-1 e-2f-1 f-2 f-3/g hsDiagrams included in the evaluation of the sum rules for gA and gA'Of these values, _ is the least well known; we estimate the error in the sum rule byevaluating gA -- 1 for Ra = -0.140 GeV 4 as well as for the value given in Eq. (16).The value of _ is obtained from the mass sum rule and the mass of the nucleon,/3_v = 0.26 GeV 6.A reduction in the range of powers in M_ can also be obtained for gas througha sum rule for g_ + gA,J116L4/9 <gcG > Eo + 6L4/--------_ xa M_E1 9L68/s 1 _aEo2 a2L4/9 1 2 2+-_ + 144L4/gM _ xa < gcG >=/32N(gA + gS + A'M_)exp(-M_/M_) . (17)Again, we note that if a ---, 0 and < G 2 >--, 0, the left hand side vanishes andgS = --gA - --1.III. Results and DiscussionIn the sum rule for gA -- 1, the quark condensate < _q > or a dominates the con-tribution on the left-hand side of Eq. (14) and the susceptibility terms proportionalto n and X are relatively unimportant.Solutions to the sum rules are found numerically; they axe stable for Ms >_1.8GeV. Although this is a relatively large Borel mass, we find that the continuum con-tributions from A and A s are very small. A smaller mass is desirable to emphasizethe contribution of the nucleon and deemphasize excited states. We findgA -- 1.26 4- 0.08gS __0.13 :t: 0.08. (18)The central value of gA is clearly consistent with experiment. The value of gAs differsfrom that found by Gupta, Murthy, and Pasupathy 4 who obtain g_ = 0.35 with avalue of < _q > smaller by 20% than the "standard" one we use. This result showsthe sensitivity of gAs to the susceptibilities and quark condensate.From the EMC data and baryon decays, one obtains 7gS _ Au + Ad _ 0.31 :i=0.08. (19)Our value is not quite consistent with experiment if A_7- Ad = 0. If _ = X =O, gA _ 1.3 and gAs = --0.56. This again shows the sensitivity to g_ to thesusceptibilities. If a = 0, we obtain gA = 1.00 :J=0.02 and g_ = -0.68; if < GG >= 0 in addition, gS = -1.0. The latter result is, indeed, a strange one. Togetherwith gA "- 1, it implies Au = 0, Ad = -1 for the proton when chiral symmetry isrestored.Can we use these results to predict what happens in nuclei? Although we expecta partial restoration of chiral symmetry in nuclei, we cannot simply interpolatebetween the free nucleon (gA = 1.26) and chiral restoration (gA = 1) limits. Forinstance, other condensates such as the vector < qtq > can occur in nuclei. However,it is likely that 1 <gA < 1.26.References* Talk given by EMHt Work supported in part by the U.S. Department of Energy and NSF grant PHY-9023586.1. M.A. Shifman, A.J. Vainstein, and V.I. Zakharov, Nucl. Phys. B147 (1979)385,448.2. B.L. Ioffe and A.V. Smilga, Nucl. Phys. B232 (1984) 109.3. V.M. Belyaev and Ya. I. Kogan, Pis'ma Zh. Eksp. Teor. Fiz. 37, (1983) 611[JETP Lett. 37 (1983) 730]; C.B. Chiu, J. Pasupathy, and S.J. Wilson, Phys.Rev. D 32 (1985) 1786.4. V.M. Belyaev, B.L. Ioffe, and Ya. I. Kogan, Phys. Lett. 151B, (1985) 290; S.Gupta, M.V.N. Murthy, and J. Pasupathy, Phys. Rev. D 39, (1989) 2547.5. E.M. Henley, G. Krein, S.J. Pollock and A.G. Williams, Phys. Left. B269(1991) 31.6. V.M. Belyaev and B.L. Ioffe, Zh. Eksp. Tor. Fiz. 83 (1982) 876 [Sov. Phys.JETP 56 (1982) 493].7. J. Ashman et al., Phys. Lett, B206 (1988) 364; Nucl. Phys. B328 (1989) 1./r

Nucleon axial coupling constants and QCD sum rules

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