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Dimension-8 contributions to light-quark QCD sum rules
Volume 165B, n u m b e r 1,2,3
PHYSICS LETTERS
19 December 1985
DIMENSION-8 CONTRIBUTIONS TO LIGHT-QUARK QCD SUM RULES D.J. B R O A D H U R S T and S.C. G E N E R A L I S 1 The Open University, Milton Keynes, England Received 30 August 1985
Coefficients of dimension-8 quark and gluon condensates in the operator product expansion of light-quark currents are calculated by two methods. In one method mass singularities are explicitly cancelled in four dimensions. In the other 1/~0 poles arising from dimensional regularization of mass singularities in ( 4 - 2 ~0) dimensions are explicitly cancelled by ultraviolet operator renormatization. The results of the two methods agree.
1. Introduction. QCD sum rules [1 ] offer the prospect of relating observable hadronic properties to condensates which reveal the non-trivial structure of the vacuum. Charmonium sum rules involving condensates with dimension d ~< 8 have been analyzed by Nikolaev and Radyushkin [2,3] at Q2 = 0 and by Reinders et al. [4] at Q2 ~ 4m 2" Nikolaev and Radyushkin found unacceptably large d = 8 corrections to their previous d = 6 analysis [5], using a factorization hypothesis to relate gluon condensates of the type (G4) to (G2) 2. Novikov et al. [6] criticised the factorization assumption for gluonic condensates and suggested that the failure of the sum rules at Q2 = 0 might be attributed to overestimation of the {G4) condensates. Reinders et al. [4] on the other hand found fairly insignificant changes to their previous d = 4 analysis [7] at Q2 ~ 4m 2 ' when including (G 3) and (G4) terms estimated by factoHzation. Thi~ findingiseonsistent with ~he suggestion o f Shuryak [8] that cha_maonium states may be too massive to probe the detailed vacuum structure, being sensitive only to the d = 4 condensate (G2). The situation, however, is further complicated by the lattice calculations of Makhaldiani and Miiller-Preussker [9], which suggest that (G4) condensates may be as much as 104 times larger than those assumed by Reinders et al. [4]. Rather than study charmonium, it seems more promising to use the detailed data on e+e - annihilation in the 1 = 1 channel to analyze vacuum structure, via light-quark sum rules, where investigations over a wide range of the Borel parameter M 2 can reveal the presence of d = 6 and d = 8 condensates. Such an analysis was performed by Launer et al. [10], who confirmed the (G 2) value of previous analyses [1,11] and attempted to test the vacuum dominance assumption for the ((-~ff)2) condensate [1 ], which gives the d = 6 non-perturbative corrections. The d = 6 gluonic condensate (G 3) did not enter the analysis, due to the decoupling predicted by Dubovikov and Smilga [12] and confirmed by Hubschmid and MaUik [13] and ourselves [14,15]. Launer et al. [10] also attempted a fit with an unknown d = 8 term, which is presumably dominated by (G4) condensates, whose coefficients were unknown. In this paper we calculate the coefficients of d = 8 quark and gluon condensates contributing to light-quark sum rules in the vector and (pseudo)scalar channels in the limit of zero-quark mass, using both of the methods we have previously developed [14,15] for condensates with dimensions d ~< 6. In the first method [14] (method A) one calculates the quark-loop diagram in a background gluon field, including all effects of the quark mass m. The resuiting mass singularities are then explicitly cancelled by the coefficients of quark operators multiplied by terms singular in m that result from the mixing of coefficients of operators that occurs when one takes matrix elements of the operator product expansion (OPE) in perturbation theory [1,14,16]. After taking the mixing into account we obtain a result which is finite in the limit m ~ 0 and contains gluonic terms of the form (log(Q2//a 2) + con1 Address after 1 October 1985: Department of Physics and Astronomy, University College, London WC1E 6BT, England.
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
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stant) (G4), where
the ultraviolet renormalization scale/a enters via operator renormalization. In the second method [15] (method B) one sets m = 0 ab initio and uses dimensional regularization of mass singularities in n = 4 - 2(9 dimensions to simplify greatly the quark-loop calculation in the background field. Logarithmic mass singularities then appear as 1/co poles, which are cancelled by 1/co one-loop gluonic subtractions relating bare and renormalized quark operators. The price to pay is that one must calculate the tree-graph coefficients of the quark operators in n = 4 - 2co dimensions, retaining terms of O(6o), which, when multiplied by the 1/~ subtractions, modify the finite parts of the dimensionally regularized quark-loop diagrams to give the correct gluonic coefficients. This is necessary because, as shown in ref. [15], mere minimal subtraction of 1/a~ terms resulting from mass singularities does not give the correct result when quark and gluon operators mix under renormalization. The remainder of the paper is organized as follows. In section 2 we indicate, briefly, how we obtained the 105 × 7 matrix,M, which gives the tensor 105 7
To~u~,,,~,,-Tr(f dnx(xIP~PaPuPvPgPhPoPoIO))= i=1 ~
~
1=1
T(i)#vv~hpoMi](O ')
(1)
in terms of 105 ways of forming an eight-rank tensor from the products o f four metric tensors and the 7 gluonic condensates with d = 8. The operator Pz in eq. (1) is the momentum operator in a background field Aau(x), with matrix elements [17]
(x[PulY) = (xliDuly) = [iO/Oxv +gaau(x)Ta ]6(x - y) , where the T a are colour matrices normalized by Tr(TaT b) = ~6ab . In section 3 we use the tensor of eq. (1) in four space-time dimensions to calculate the d = 8 quark-loop contributions for finite quark mass. Method A is then used to cancel mass singularities explicitly. In section 4 we use the tensor o f eq. (1) in n = 4 - 2co space-time dimensions to calculate the dimensionally regularized d = 8 quark loop contributions for zero quark mass. Method B is then used to cancel 1/co poles explicitly. The results o f methods A and B agree and are discussed in section 5.
2. The eighth-rank tensor. To calculate the d = 4 quark-loop contributions one needs only the fourth-rank tensor [15]
Taouv =- Tr t f dnx(xlPo~PaPuPvlO))= (g=ug~v - gaagvv)(G2)/4n(n - 1 ) , a 2 and where G 2 -- (Guv) and vector correlators
i
a Guy =g(OgAu a --OvAua +gfabcAubAv c) .The quark-loop contributions to the scalar
fd, x eiqx(T(J(x)J(O)))=-q2lI s ,
J=- ~k ,
i f d n x eiqx(T(Ju(x)Ju(O))) =- (qt~qv - q2q#v)IIV , Ju -- ~3~utb , are then simply obtained by expanding the quark-loop in a background field to fourth order in Pu [15]. The corresponding sixth-rank tensor, needed to calculate the d = 6 quark-loop contributions, was also given in ref. [15] a and involves the two d = 6 condensates (fabc GauvGbhG~u) and (/a/a), where ]a =_ DuGau. Here we need to expand the quark-loop to eighth order in Pu and use the eighth-rank tensor of eq. (1), given in terms of metric tensors and the vacuum expectation values (O 1 - 7 ) of the 7 gluonic operators of table 1. The first four condensates involve traces of Guy -- GauvT a, but not the source current j a = g2 ~q ~-Ta Taq, which involves a sum over the light-quark fields q = u, d, s. The remaining three operators were neglected by Novikov et al. [6] but retained by Nikolaev and Radyushkin [2], who used a b (fabcGuvGvx D 2 Gxc u) = (8(03 - 0 4 ) - 06
176
205) ,
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PHYSICS LETTERS
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Table 1 The gluonie operators and the L and N coefficients of eqs. (2)-(5). i
o,
1 2 3 4
Tr(Gt~vG~uGa3Go~) Tr(G~vGa(3G~vGa3) Tr(G~,Gv~xGa(3G~) Tr(G#vGa3GuaG3~) a .b Iv -c fabcG~ul~ fabcG~vl~DxG~v l~D'a~.al~
5
6 7
t v
g
•
O.
-3 -1 6 10 5
59•4 91/12 -29/2 -293/6 -41/12 3/2 17/4
0 -3
-10/3 2/3 20/3 4 0 0 -2/3
-19/18 -193/18 -131/9 31 -5/2 -1/3 7/18
in place of our condensate (06). (Note that in ref. [3] Nikolaev and Raduyshkin adopt the opposite sign for (05).) To obtain the 105 × 7 matrix M of eq. (1) in n dimensions is a non-trivial task, involving liberal use of Bianchi identities, integration by parts and equations of motion [5] in the Fock-Schwinger gauge [ 18]. We have succeeded in factorizing it into a 105 × 7 matrix of integers and a 7 × 7 n-dependent matrix, copies of which are available on request. To test it we calculated the matrix element (~ff) for a heavy quark of mass M in a background field, obtaining ( ~ ) d = 8 = (--1901 -- 1602 + 2003 + 7804 -- 2105 -- 206 + 907)/5040rr2M5
•
The coefficients of O 1 - 4 in this heavy-quark expansion agree with the one-loop effective action of Vainshtein et al. [19], which reduces to the Euler-Heisenberg lagrangian [20] in the case of an abelian gauge field. Equipped with the tensor of eq. (1), we computed the d = 8 quark-loop contribution in both methods A and B by routine application of REDUCE3 [21 ].
3. Method A: explicit cancellation of mass singularities. Keeping quark masses in the quark-loop contribution and the tree-graph contributions of method A, we obtain an expression of the form 7
Q8H~h~ -- lira (/=~1 [JiS'VQ4/m4 +KS'VQ2/m2 m
0
10 + 1=1 ~ c/S'V/Q]
+LS'V l°g(Q2/rn2) +MiS'V](oi)/1447r2
"=
7
14
7
- i~l Z{Oi l°gOa2/rn2)/144rr21 +m ~ CiS'v (Q/ - ~ ZiOi/1447r2mI '=
]=11
7 + (Q2BS5V + m2C~I5v)( Q15 -/=~1
\
"=
Z150i]1447r2m2~ + m(Q2BS~V+ m2C~I~V)tQ16
7 - ~
Z160i/144rr2m3)
7
+rn(Q4AS~V + Q2m2BSI~V +m4C~I~V)IQ17 -i~=i Z170i/144rr2m5)) 7 = ~ [Ls'V log(Q2/,u2) i=1
10
+NiS'Vl(ot}/144n 2 +~ 1=1
c/s'v(Q/.)
(2) '
where Q2 = _q2, the coefficientsJ, K and L give the d = 8 quark-loop terms with mass singularities 1/m4, 1/m 2 and log m, respectively, and the coefficients M give the regular quark-loop terms. In addition to the quark-loop contribution there are terms involving vacuum expectation values of combinations of operators whose gluon matrix element vanish to lowest order in perturbation theory. These combinations involve the 17 quark operators Q1-17 of table 2, with dimensions ranging from 3 to 8, from which must be subtracted combinations of the gluonic operators 0 1 - 7 with coefficients given by a mixing matrix Z, whose non.zero elements are given in table 3. The ultraviolet-divergent part of the mixing has been absorbed in the MS renormalization of Q1-1 o' resulting in 177
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Table 2 The quark operators and coefficients of eqs. (2)-(5). P/Z is the momentum operator in a background field. (3,u-rv~p~/a)_and (3'/z3"v')'p)- are totally anti-symmetric tensors constructed from gamma matrices. Square and curly brackets denote commutators and anti-commutators respectively, o/zv -~ 0/2)(~//zq'v - ~rv~/u)•
/
o;
4
Ds
,,?,
qv
,e
~-[ [P, [P/Z,P~,]], [Pp,PoI ] ('rlava'rp"/o)-4 ¢-[ [g, [P#,P~,] ], [Pta,Pv] ]4
~[ [Pu,Pv] ,[Pp, Po] ]Pv(')'/z'ro'Ya)-4 ~-{[P/z,Pp] ,[P/Z, I/] )Po4 {[1/, [P/Z,Pv]] ,[Pp, Pal )('Y/z"/vTp3'o)-4 ~[[Pu,IPv,Pta|],[Po,Po]](')'v3,p,),o)_4 ([Pu, [Pv, P#] ], [Po' P~r] XYv'ro3'a)-4 ~[[P/Z, [P~,,Pu]I,[Pv,[/]]4 ~([Pu, [Pv, Pu]] ,[P~',1/])4 ~[Pu, [Pu ,[Pv,ff/,Pvl ] ] ] 4
8 8 8 8 8 8 8 8 8 8
1/24 -1/12 1/6 2/3 -1/8 1/6 1/3 1 1/3 1/2
14
O-/[P/Z,[Pv,[Pp,Pt,]]][Tp,,),/Z]4 ~[P/z,Pu][Pv,Pp][../~,3,pl4 ~[Pla, Pv] [Pp,Pa] (3'#3'v3'o'ra)-4 ~-[Pta,Pv] [P/Z,Pr,] 4
7 7 7 7
-1/2 -5/6 -3/4 5/3
-
1/9 -1/9 -7/6 -14/9
15
¢[Pt~, [g,P/z] l 4 = -ff
6
-3
-
2/3
16
~P2 4 = m~ ¢4 - ffo/zvG/zv4 /2
5
-6
-
4/3
17
~-4
3
3
-
14/3
6 7 8 9 10 11 12
13
"/a/araq. '
25[288 -13/144 25/72 11/9 -7[96 25/72 17[72 7/12 11/18 3/8
-1136 1/18 -119 4/9 114 -1/9 0 2/9 2/9 1/9
-13/144 17/72 -13/36 2/9 77/144 -13136 -5/36 -17/18 1/9 -5/12
m
-
Table 3 The mixing coefficients of eqs. (2)-(5). J 2 3 3 3 3 3 3 4 4 4
6 1 2 3 4 5 6 1 3 4
-6 6 -6 -12 12 -6 -3 -6 12 12
J
;
8 10 i1 11 11 12 12 12 12 13
5 7 5 6 7 3 4 5 6 1
6 -6 -12 -3 6 -12 12 -6 -3 -18
i
,"
13 13 14 15 15 15 16 16 16 16
2 4 1 5 6 7 1 2 3 4
-18 72 6 27/10 3/20 -6/5 44/35 26/35 -1/35 -111/35
J
;
16 16 16 17 17 17 17 17 17 17
5 6 7 1 2 3 4 5 6 7
4/3 89/280 -3/70 -19135 -16/35 4/7 78/35 -3/5 -2/35 9/35
the appearance of l o g ~ 2 / m 2 ) . (We ignore diagrams involving four-quark matrix elements of the OPE, since they play no role in cancelling mass singularities.) The quark operators of table 2 consist of the operators Q15 - 1 7 encountered in our previous d ~< 6 analyses four d = 7 operators, Q11 - 14, (of which Q11 may be neglected if one ignores the source current j ) and ten d = 8 operators, Q1 - 1 o, (of which Q 6 - 1 0 involve the source current j , and Q5 may be eliminated in four dimensions, using (Tx, (7/z%Tp3'a)_)= 0 and equations of motion). We use the results of ref. [14] to obtain the
[14,15],
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following A and B quark operator coefficients of eq. (2): A1S7=-3,
BlS5=~,
B1S6=2,
BlS7=0,
AV17=2,
BY15=4,
BV6=0,
B1V7=-~.
The remaining quark operator coefficients C of eq. (2) were obtained by expanding tree-graphs to fifth order in Pu" Taking the limit m -+ 0 we obtain the final expression in eq. (2), with the coefficients L and N given in table 1. The cancellation of mass singularities implies the following relations, which have been explicitly verified: 17 10 jS'V=AS~rzilT, KS V = E BS'Vzi, L S ' V = E cS'Vz i . (3a,b,c) j=15 j=l Note that the coefficients M in the regular quark-loop term are by the relation 17 M/S'V =N~/'V + ~ cIS'Vz{. 1=11
not those appearing in the final result, but are given (3d)
4. Method B: explicit cancellation ofl/¢o poles. For method B one uses dimensional regularization of the quark-loop contribution, multiplies by (/12e~/4,r) °a , and obtains 1/~o poles and regular terms. In addition one needs the O(oJ) terms in the coefficients of the bare quark operators, which are related to the renormalized operators Qi by gluonic subtractions involving the mixing coefficients {Z/i [i = 1 ..... 7;j = 1 ..... 10} of table 3. The general result is of the form Q8 IIdS,V = lim ~ {LSl'V[log(Q2/Ia2)- 1/6o] + ArS'V}(oi)/la4rr2 to-+0ki=l 10 7 +j=l ~ (cIS'V + ¢°DS'V)((Q])+ i~1"=Zii(Oi}/a44rr2°°)] 7 = ~ [L/S,V log(Q21/~2) + N S,V] i---1
10
(Oi)/ 144rr2 + G
j=l
CjS'V(Q/)
(4) '
where the coefficients C and Q are given in table 2. Note that the coefficients N of eq. (4), which would be obt~/ined by minimal subtraction of mass singularities, are not those appearing in the final result, but are given by the relation 10
s,v =NS,V _
DjZ,',
(5)
j=l which has been verified explicitly.
5. Discussion. The agreement of methods A and B gives us confidence in the final result, expressed by the coefficients LSl'V_7,NSI'V_ 7 of table 1 and the coefficients cS,_V10 of table 2. A further check on the coefficients L s'_V7 can be obtained by imposing the 5 conditions on (O 1 _7 ) that result in a self-dual background field. These give a vanishing (log (a2/t~2)){G4) contribution, consistent with the observation of Dubovikov and Smilga [12]. The complete vanishing of the 1/Q8 term in [I V, which was predicted by Dubovikov and Smilga, is more difficult to verify, since we have not been able to find a way of generalizing the notion of self-duality to n = 4 - 200 dimensions, which seems to be necessary in both methods, because of ultraviolet-divergent operator mixing. The coefficients N of table 1 clearly depend on the renormalization scheme. It may be that one has to use dimensional reduction rather than dimensional regularization to investigate the complete cancellation implied by self-duality. One may also need to take into account the zero-mode contributions to those quark condensates that one naively expects to vanish asm -+ 0 [12]. 179
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In conclusion, we have completed our program o f computing the coefficients o f quark and gluon operators with d < 8 in the OPE o f light-quark currents. By using the more laborious method A we have also checked, en route, the heavy-quark results of Nikolaev and Radyushkin [2], discovering only minor printing errors. (In the coefficient o f J2043+4 for the vector correlator, 249 should be replaced by 294. In the coefficient o f ~O7 for the scalar correlator, 85 should be replaced by 35. In the coefficient of J6 O 1 - 2 for the pseudoscalar correlator, the plus sign should be replaced b y a minus sign.) By using method B we have demonstrated the failure o f minimal subtraction of mass singularities and the success o f the procedure developed in ref. [15]. The complete d = 8 results are given in tables 1 and 2. I f one neglects quark condensates and gluon condensates involving the source current], one obtains Q8IIS-P8 = ( - 3 0 1 - 0 2 + 603 +
lO04)_log(Q2/la2)/1447r2 + (17701 + 9102 - 1 7 4 0 3 - 58604)/1728zr2
for the scalar (or pseudoscalar) correlator, and v = (-501 Q8IId=8
+ 0 2 + 1003 + 604)
log(Q2/la2)/216zr2 + ( - 1 9 0 1 - 19302 - 26203 + 55804)/25927r2
for the vector correlator. The vector result will allow the phenomenological analysis o f Launer et al. [10] to be continued to d = 8, allowing rival versions o f the hadronic vacuum [2,3,9,22] to be confronted by e+e - data. We thank the Academic Computing Service of the Open University for their assistance. DJB thanks Chris Sachrajda for discussions o f self-dual background fields.
References [1 ] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448. [2] S.N. Nikolaev and A.V. Radyushkin, JINR preprint P2-82-914 (1982). [3] S.N. Nikolaev and A.V. Radyushkin, Phys. Lett. 124B (1983) 243. [4] LJ. Reinders,H.R. Rubinstein and S. Yazaki, Phys. Lett. 138B (1984) 425. [5 ] S.N. Nikolaev and A.V. Radyushkin, Phys. Lett. 110B (1982) 476; 116B (1982) 469(E); Nucl. Phys. B213 (1983) 285. [6] V.A. Novikov, M.A. Shifman, A.I. Vainshtein, M.B. Voloshin and V.I. Zakharov, Nucl. Phys. B237 (1984) 525. [7] LJ. R¢inders, H.R. Rubinstein and S. Yazaki, Nucl. Phys. B186 (1981) 109. [8] E.V. Shuryak, Phys. Lett. 136B (1984) 269. [9] N. Makhaldiani and M. Miiller-Preussker, IINR preprint E2-84-660 (1984). [i0] G. Launer, S. Narison and R. Tarrach, Z. Phys. C26 (1984) 433. [11 ] S.I. Eidelman, L.M. Kurdadze and A.I. Vainshtein, Phys. Lett. 82B (1979) 278. [ 12 ] M.S. Dubovikov and A.V. Smilga, Nucl. Phys. B185 (1981) 109. [13] W. Hubschmid at. ~"S. Mallik, Nucl. Phys. B207 (1982) 29. [ 14 ] S .C. Generalis and D .J. Broadhurst, Phys. Lett. 139B (1984) 85. [15] D.J. Broadhurst and S.C. Genera]is, Phys. Lett. 142B (1984) 75; S.C. Generalis, Analysis of current correlators within the framework of quantum chromodynamics, Ph.D. thesis, Open University report OUT-4102-13 (1984). [16] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. 97B (1980) 257; 100B (1981) 519(E); 103B (1981) 63. [17] E.V. Shuryak and A.I. Vainshtein, Nucl. Phys. B199 (1982) 451; B201 (1982) 141. [18] V.A. Fock, Sow. Phys. 12 (1937) 404; J. Schwinger, Phys. Rev. 82 (1951) 664; A.V. Smilga, Sov. J. Nucl. Phys. 35 (1982) 271. [19] A.I. Vainshtein, V3. Zakharov, V.A. Novikov and M.A. Shifman, Soy. Phys. Usp. 25 (1982) 195. [20] W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714; J. Schwinger, Phys. Rev. 82 (1951) 664. [21] A.C. Heaxn, Reduce User's Manual, Rand Publication CP78 (1984). [22] E. Bag~n, J.I. Latorre, P. Pascual and R. Tarrach, Barcelona preprint (1984).
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DIMENSION-8 CONTRIBUTIONS TO LIGHT-QUARK QCD SUM RULES D.J. B R O A D H U R S T and S.C. G E N E R A L I S 1 The Open University, Milton Keynes, England Received 30 August 1985
Coefficients of dimension-8 quark and gluon condensates in the operator product expansion of light-quark currents are calculated by two methods. In one method mass singularities are explicitly cancelled in four dimensions. In the other 1/~0 poles arising from dimensional regularization of mass singularities in ( 4 - 2 ~0) dimensions are explicitly cancelled by ultraviolet operator renormatization. The results of the two methods agree.
1. Introduction. QCD sum rules [1 ] offer the prospect of relating observable hadronic properties to condensates which reveal the non-trivial structure of the vacuum. Charmonium sum rules involving condensates with dimension d ~< 8 have been analyzed by Nikolaev and Radyushkin [2,3] at Q2 = 0 and by Reinders et al. [4] at Q2 ~ 4m 2" Nikolaev and Radyushkin found unacceptably large d = 8 corrections to their previous d = 6 analysis [5], using a factorization hypothesis to relate gluon condensates of the type (G4) to (G2) 2. Novikov et al. [6] criticised the factorization assumption for gluonic condensates and suggested that the failure of the sum rules at Q2 = 0 might be attributed to overestimation of the {G4) condensates. Reinders et al. [4] on the other hand found fairly insignificant changes to their previous d = 4 analysis [7] at Q2 ~ 4m 2 ' when including (G 3) and (G4) terms estimated by factoHzation. Thi~ findingiseonsistent with ~he suggestion o f Shuryak [8] that cha_maonium states may be too massive to probe the detailed vacuum structure, being sensitive only to the d = 4 condensate (G2). The situation, however, is further complicated by the lattice calculations of Makhaldiani and Miiller-Preussker [9], which suggest that (G4) condensates may be as much as 104 times larger than those assumed by Reinders et al. [4]. Rather than study charmonium, it seems more promising to use the detailed data on e+e - annihilation in the 1 = 1 channel to analyze vacuum structure, via light-quark sum rules, where investigations over a wide range of the Borel parameter M 2 can reveal the presence of d = 6 and d = 8 condensates. Such an analysis was performed by Launer et al. [10], who confirmed the (G 2) value of previous analyses [1,11] and attempted to test the vacuum dominance assumption for the ((-~ff)2) condensate [1 ], which gives the d = 6 non-perturbative corrections. The d = 6 gluonic condensate (G 3) did not enter the analysis, due to the decoupling predicted by Dubovikov and Smilga [12] and confirmed by Hubschmid and MaUik [13] and ourselves [14,15]. Launer et al. [10] also attempted a fit with an unknown d = 8 term, which is presumably dominated by (G4) condensates, whose coefficients were unknown. In this paper we calculate the coefficients of d = 8 quark and gluon condensates contributing to light-quark sum rules in the vector and (pseudo)scalar channels in the limit of zero-quark mass, using both of the methods we have previously developed [14,15] for condensates with dimensions d ~< 6. In the first method [14] (method A) one calculates the quark-loop diagram in a background gluon field, including all effects of the quark mass m. The resuiting mass singularities are then explicitly cancelled by the coefficients of quark operators multiplied by terms singular in m that result from the mixing of coefficients of operators that occurs when one takes matrix elements of the operator product expansion (OPE) in perturbation theory [1,14,16]. After taking the mixing into account we obtain a result which is finite in the limit m ~ 0 and contains gluonic terms of the form (log(Q2//a 2) + con1 Address after 1 October 1985: Department of Physics and Astronomy, University College, London WC1E 6BT, England.
0370-2693/85/$ 03.30 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
175
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stant) (G4), where
the ultraviolet renormalization scale/a enters via operator renormalization. In the second method [15] (method B) one sets m = 0 ab initio and uses dimensional regularization of mass singularities in n = 4 - 2(9 dimensions to simplify greatly the quark-loop calculation in the background field. Logarithmic mass singularities then appear as 1/co poles, which are cancelled by 1/co one-loop gluonic subtractions relating bare and renormalized quark operators. The price to pay is that one must calculate the tree-graph coefficients of the quark operators in n = 4 - 2co dimensions, retaining terms of O(6o), which, when multiplied by the 1/~ subtractions, modify the finite parts of the dimensionally regularized quark-loop diagrams to give the correct gluonic coefficients. This is necessary because, as shown in ref. [15], mere minimal subtraction of 1/a~ terms resulting from mass singularities does not give the correct result when quark and gluon operators mix under renormalization. The remainder of the paper is organized as follows. In section 2 we indicate, briefly, how we obtained the 105 × 7 matrix,M, which gives the tensor 105 7
To~u~,,,~,,-Tr(f dnx(xIP~PaPuPvPgPhPoPoIO))= i=1 ~
~
1=1
T(i)#vv~hpoMi](O ')
(1)
in terms of 105 ways of forming an eight-rank tensor from the products o f four metric tensors and the 7 gluonic condensates with d = 8. The operator Pz in eq. (1) is the momentum operator in a background field Aau(x), with matrix elements [17]
(x[PulY) = (xliDuly) = [iO/Oxv +gaau(x)Ta ]6(x - y) , where the T a are colour matrices normalized by Tr(TaT b) = ~6ab . In section 3 we use the tensor of eq. (1) in four space-time dimensions to calculate the d = 8 quark-loop contributions for finite quark mass. Method A is then used to cancel mass singularities explicitly. In section 4 we use the tensor o f eq. (1) in n = 4 - 2co space-time dimensions to calculate the dimensionally regularized d = 8 quark loop contributions for zero quark mass. Method B is then used to cancel 1/co poles explicitly. The results o f methods A and B agree and are discussed in section 5.
2. The eighth-rank tensor. To calculate the d = 4 quark-loop contributions one needs only the fourth-rank tensor [15]
Taouv =- Tr t f dnx(xlPo~PaPuPvlO))= (g=ug~v - gaagvv)(G2)/4n(n - 1 ) , a 2 and where G 2 -- (Guv) and vector correlators
i
a Guy =g(OgAu a --OvAua +gfabcAubAv c) .The quark-loop contributions to the scalar
fd, x eiqx(T(J(x)J(O)))=-q2lI s ,
J=- ~k ,
i f d n x eiqx(T(Ju(x)Ju(O))) =- (qt~qv - q2q#v)IIV , Ju -- ~3~utb , are then simply obtained by expanding the quark-loop in a background field to fourth order in Pu [15]. The corresponding sixth-rank tensor, needed to calculate the d = 6 quark-loop contributions, was also given in ref. [15] a and involves the two d = 6 condensates (fabc GauvGbhG~u) and (/a/a), where ]a =_ DuGau. Here we need to expand the quark-loop to eighth order in Pu and use the eighth-rank tensor of eq. (1), given in terms of metric tensors and the vacuum expectation values (O 1 - 7 ) of the 7 gluonic operators of table 1. The first four condensates involve traces of Guy -- GauvT a, but not the source current j a = g2 ~q ~-Ta Taq, which involves a sum over the light-quark fields q = u, d, s. The remaining three operators were neglected by Novikov et al. [6] but retained by Nikolaev and Radyushkin [2], who used a b (fabcGuvGvx D 2 Gxc u) = (8(03 - 0 4 ) - 06
176
205) ,
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Table 1 The gluonie operators and the L and N coefficients of eqs. (2)-(5). i
o,
1 2 3 4
Tr(Gt~vG~uGa3Go~) Tr(G~vGa(3G~vGa3) Tr(G~,Gv~xGa(3G~) Tr(G#vGa3GuaG3~) a .b Iv -c fabcG~ul~ fabcG~vl~DxG~v l~D'a~.al~
5
6 7
t v
g
•
O.
-3 -1 6 10 5
59•4 91/12 -29/2 -293/6 -41/12 3/2 17/4
0 -3
-10/3 2/3 20/3 4 0 0 -2/3
-19/18 -193/18 -131/9 31 -5/2 -1/3 7/18
in place of our condensate (06). (Note that in ref. [3] Nikolaev and Raduyshkin adopt the opposite sign for (05).) To obtain the 105 × 7 matrix M of eq. (1) in n dimensions is a non-trivial task, involving liberal use of Bianchi identities, integration by parts and equations of motion [5] in the Fock-Schwinger gauge [ 18]. We have succeeded in factorizing it into a 105 × 7 matrix of integers and a 7 × 7 n-dependent matrix, copies of which are available on request. To test it we calculated the matrix element (~ff) for a heavy quark of mass M in a background field, obtaining ( ~ ) d = 8 = (--1901 -- 1602 + 2003 + 7804 -- 2105 -- 206 + 907)/5040rr2M5
•
The coefficients of O 1 - 4 in this heavy-quark expansion agree with the one-loop effective action of Vainshtein et al. [19], which reduces to the Euler-Heisenberg lagrangian [20] in the case of an abelian gauge field. Equipped with the tensor of eq. (1), we computed the d = 8 quark-loop contribution in both methods A and B by routine application of REDUCE3 [21 ].
3. Method A: explicit cancellation of mass singularities. Keeping quark masses in the quark-loop contribution and the tree-graph contributions of method A, we obtain an expression of the form 7
Q8H~h~ -- lira (/=~1 [JiS'VQ4/m4 +KS'VQ2/m2 m
0
10 + 1=1 ~ c/S'V/Q]
+LS'V l°g(Q2/rn2) +MiS'V](oi)/1447r2
"=
7
14
7
- i~l Z{Oi l°gOa2/rn2)/144rr21 +m ~ CiS'v (Q/ - ~ ZiOi/1447r2mI '=
]=11
7 + (Q2BS5V + m2C~I5v)( Q15 -/=~1
\
"=
Z150i]1447r2m2~ + m(Q2BS~V+ m2C~I~V)tQ16
7 - ~
Z160i/144rr2m3)
7
+rn(Q4AS~V + Q2m2BSI~V +m4C~I~V)IQ17 -i~=i Z170i/144rr2m5)) 7 = ~ [Ls'V log(Q2/,u2) i=1
10
+NiS'Vl(ot}/144n 2 +~ 1=1
c/s'v(Q/.)
(2) '
where Q2 = _q2, the coefficientsJ, K and L give the d = 8 quark-loop terms with mass singularities 1/m4, 1/m 2 and log m, respectively, and the coefficients M give the regular quark-loop terms. In addition to the quark-loop contribution there are terms involving vacuum expectation values of combinations of operators whose gluon matrix element vanish to lowest order in perturbation theory. These combinations involve the 17 quark operators Q1-17 of table 2, with dimensions ranging from 3 to 8, from which must be subtracted combinations of the gluonic operators 0 1 - 7 with coefficients given by a mixing matrix Z, whose non.zero elements are given in table 3. The ultraviolet-divergent part of the mixing has been absorbed in the MS renormalization of Q1-1 o' resulting in 177
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Table 2 The quark operators and coefficients of eqs. (2)-(5). P/Z is the momentum operator in a background field. (3,u-rv~p~/a)_and (3'/z3"v')'p)- are totally anti-symmetric tensors constructed from gamma matrices. Square and curly brackets denote commutators and anti-commutators respectively, o/zv -~ 0/2)(~//zq'v - ~rv~/u)•
/
o;
4
Ds
,,?,
qv
,e
~-[ [P, [P/Z,P~,]], [Pp,PoI ] ('rlava'rp"/o)-4 ¢-[ [g, [P#,P~,] ], [Pta,Pv] ]4
~[ [Pu,Pv] ,[Pp, Po] ]Pv(')'/z'ro'Ya)-4 ~-{[P/z,Pp] ,[P/Z, I/] )Po4 {[1/, [P/Z,Pv]] ,[Pp, Pal )('Y/z"/vTp3'o)-4 ~[[Pu,IPv,Pta|],[Po,Po]](')'v3,p,),o)_4 ([Pu, [Pv, P#] ], [Po' P~r] XYv'ro3'a)-4 ~[[P/Z, [P~,,Pu]I,[Pv,[/]]4 ~([Pu, [Pv, Pu]] ,[P~',1/])4 ~[Pu, [Pu ,[Pv,ff/,Pvl ] ] ] 4
8 8 8 8 8 8 8 8 8 8
1/24 -1/12 1/6 2/3 -1/8 1/6 1/3 1 1/3 1/2
14
O-/[P/Z,[Pv,[Pp,Pt,]]][Tp,,),/Z]4 ~[P/z,Pu][Pv,Pp][../~,3,pl4 ~[Pla, Pv] [Pp,Pa] (3'#3'v3'o'ra)-4 ~-[Pta,Pv] [P/Z,Pr,] 4
7 7 7 7
-1/2 -5/6 -3/4 5/3
-
1/9 -1/9 -7/6 -14/9
15
¢[Pt~, [g,P/z] l 4 = -ff
6
-3
-
2/3
16
~P2 4 = m~ ¢4 - ffo/zvG/zv4 /2
5
-6
-
4/3
17
~-4
3
3
-
14/3
6 7 8 9 10 11 12
13
"/a/araq. '
25[288 -13/144 25/72 11/9 -7[96 25/72 17[72 7/12 11/18 3/8
-1136 1/18 -119 4/9 114 -1/9 0 2/9 2/9 1/9
-13/144 17/72 -13/36 2/9 77/144 -13136 -5/36 -17/18 1/9 -5/12
m
-
Table 3 The mixing coefficients of eqs. (2)-(5). J 2 3 3 3 3 3 3 4 4 4
6 1 2 3 4 5 6 1 3 4
-6 6 -6 -12 12 -6 -3 -6 12 12
J
;
8 10 i1 11 11 12 12 12 12 13
5 7 5 6 7 3 4 5 6 1
6 -6 -12 -3 6 -12 12 -6 -3 -18
i
,"
13 13 14 15 15 15 16 16 16 16
2 4 1 5 6 7 1 2 3 4
-18 72 6 27/10 3/20 -6/5 44/35 26/35 -1/35 -111/35
J
;
16 16 16 17 17 17 17 17 17 17
5 6 7 1 2 3 4 5 6 7
4/3 89/280 -3/70 -19135 -16/35 4/7 78/35 -3/5 -2/35 9/35
the appearance of l o g ~ 2 / m 2 ) . (We ignore diagrams involving four-quark matrix elements of the OPE, since they play no role in cancelling mass singularities.) The quark operators of table 2 consist of the operators Q15 - 1 7 encountered in our previous d ~< 6 analyses four d = 7 operators, Q11 - 14, (of which Q11 may be neglected if one ignores the source current j ) and ten d = 8 operators, Q1 - 1 o, (of which Q 6 - 1 0 involve the source current j , and Q5 may be eliminated in four dimensions, using (Tx, (7/z%Tp3'a)_)= 0 and equations of motion). We use the results of ref. [14] to obtain the
[14,15],
178
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following A and B quark operator coefficients of eq. (2): A1S7=-3,
BlS5=~,
B1S6=2,
BlS7=0,
AV17=2,
BY15=4,
BV6=0,
B1V7=-~.
The remaining quark operator coefficients C of eq. (2) were obtained by expanding tree-graphs to fifth order in Pu" Taking the limit m -+ 0 we obtain the final expression in eq. (2), with the coefficients L and N given in table 1. The cancellation of mass singularities implies the following relations, which have been explicitly verified: 17 10 jS'V=AS~rzilT, KS V = E BS'Vzi, L S ' V = E cS'Vz i . (3a,b,c) j=15 j=l Note that the coefficients M in the regular quark-loop term are by the relation 17 M/S'V =N~/'V + ~ cIS'Vz{. 1=11
not those appearing in the final result, but are given (3d)
4. Method B: explicit cancellation ofl/¢o poles. For method B one uses dimensional regularization of the quark-loop contribution, multiplies by (/12e~/4,r) °a , and obtains 1/~o poles and regular terms. In addition one needs the O(oJ) terms in the coefficients of the bare quark operators, which are related to the renormalized operators Qi by gluonic subtractions involving the mixing coefficients {Z/i [i = 1 ..... 7;j = 1 ..... 10} of table 3. The general result is of the form Q8 IIdS,V = lim ~ {LSl'V[log(Q2/Ia2)- 1/6o] + ArS'V}(oi)/la4rr2 to-+0ki=l 10 7 +j=l ~ (cIS'V + ¢°DS'V)((Q])+ i~1"=Zii(Oi}/a44rr2°°)] 7 = ~ [L/S,V log(Q21/~2) + N S,V] i---1
10
(Oi)/ 144rr2 + G
j=l
CjS'V(Q/)
(4) '
where the coefficients C and Q are given in table 2. Note that the coefficients N of eq. (4), which would be obt~/ined by minimal subtraction of mass singularities, are not those appearing in the final result, but are given by the relation 10
s,v =NS,V _
DjZ,',
(5)
j=l which has been verified explicitly.
5. Discussion. The agreement of methods A and B gives us confidence in the final result, expressed by the coefficients LSl'V_7,NSI'V_ 7 of table 1 and the coefficients cS,_V10 of table 2. A further check on the coefficients L s'_V7 can be obtained by imposing the 5 conditions on (O 1 _7 ) that result in a self-dual background field. These give a vanishing (log (a2/t~2)){G4) contribution, consistent with the observation of Dubovikov and Smilga [12]. The complete vanishing of the 1/Q8 term in [I V, which was predicted by Dubovikov and Smilga, is more difficult to verify, since we have not been able to find a way of generalizing the notion of self-duality to n = 4 - 200 dimensions, which seems to be necessary in both methods, because of ultraviolet-divergent operator mixing. The coefficients N of table 1 clearly depend on the renormalization scheme. It may be that one has to use dimensional reduction rather than dimensional regularization to investigate the complete cancellation implied by self-duality. One may also need to take into account the zero-mode contributions to those quark condensates that one naively expects to vanish asm -+ 0 [12]. 179
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In conclusion, we have completed our program o f computing the coefficients o f quark and gluon operators with d < 8 in the OPE o f light-quark currents. By using the more laborious method A we have also checked, en route, the heavy-quark results of Nikolaev and Radyushkin [2], discovering only minor printing errors. (In the coefficient o f J2043+4 for the vector correlator, 249 should be replaced by 294. In the coefficient o f ~O7 for the scalar correlator, 85 should be replaced by 35. In the coefficient of J6 O 1 - 2 for the pseudoscalar correlator, the plus sign should be replaced b y a minus sign.) By using method B we have demonstrated the failure o f minimal subtraction of mass singularities and the success o f the procedure developed in ref. [15]. The complete d = 8 results are given in tables 1 and 2. I f one neglects quark condensates and gluon condensates involving the source current], one obtains Q8IIS-P8 = ( - 3 0 1 - 0 2 + 603 +
lO04)_log(Q2/la2)/1447r2 + (17701 + 9102 - 1 7 4 0 3 - 58604)/1728zr2
for the scalar (or pseudoscalar) correlator, and v = (-501 Q8IId=8
+ 0 2 + 1003 + 604)
log(Q2/la2)/216zr2 + ( - 1 9 0 1 - 19302 - 26203 + 55804)/25927r2
for the vector correlator. The vector result will allow the phenomenological analysis o f Launer et al. [10] to be continued to d = 8, allowing rival versions o f the hadronic vacuum [2,3,9,22] to be confronted by e+e - data. We thank the Academic Computing Service of the Open University for their assistance. DJB thanks Chris Sachrajda for discussions o f self-dual background fields.
References [1 ] M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Nucl. Phys. B147 (1979) 385,448. [2] S.N. Nikolaev and A.V. Radyushkin, JINR preprint P2-82-914 (1982). [3] S.N. Nikolaev and A.V. Radyushkin, Phys. Lett. 124B (1983) 243. [4] LJ. Reinders,H.R. Rubinstein and S. Yazaki, Phys. Lett. 138B (1984) 425. [5 ] S.N. Nikolaev and A.V. Radyushkin, Phys. Lett. 110B (1982) 476; 116B (1982) 469(E); Nucl. Phys. B213 (1983) 285. [6] V.A. Novikov, M.A. Shifman, A.I. Vainshtein, M.B. Voloshin and V.I. Zakharov, Nucl. Phys. B237 (1984) 525. [7] LJ. R¢inders, H.R. Rubinstein and S. Yazaki, Nucl. Phys. B186 (1981) 109. [8] E.V. Shuryak, Phys. Lett. 136B (1984) 269. [9] N. Makhaldiani and M. Miiller-Preussker, IINR preprint E2-84-660 (1984). [i0] G. Launer, S. Narison and R. Tarrach, Z. Phys. C26 (1984) 433. [11 ] S.I. Eidelman, L.M. Kurdadze and A.I. Vainshtein, Phys. Lett. 82B (1979) 278. [ 12 ] M.S. Dubovikov and A.V. Smilga, Nucl. Phys. B185 (1981) 109. [13] W. Hubschmid at. ~"S. Mallik, Nucl. Phys. B207 (1982) 29. [ 14 ] S .C. Generalis and D .J. Broadhurst, Phys. Lett. 139B (1984) 85. [15] D.J. Broadhurst and S.C. Genera]is, Phys. Lett. 142B (1984) 75; S.C. Generalis, Analysis of current correlators within the framework of quantum chromodynamics, Ph.D. thesis, Open University report OUT-4102-13 (1984). [16] L.J. Reinders, H.R. Rubinstein and S. Yazaki, Phys. Lett. 97B (1980) 257; 100B (1981) 519(E); 103B (1981) 63. [17] E.V. Shuryak and A.I. Vainshtein, Nucl. Phys. B199 (1982) 451; B201 (1982) 141. [18] V.A. Fock, Sow. Phys. 12 (1937) 404; J. Schwinger, Phys. Rev. 82 (1951) 664; A.V. Smilga, Sov. J. Nucl. Phys. 35 (1982) 271. [19] A.I. Vainshtein, V3. Zakharov, V.A. Novikov and M.A. Shifman, Soy. Phys. Usp. 25 (1982) 195. [20] W. Heisenberg and H. Euler, Z. Phys. 98 (1936) 714; J. Schwinger, Phys. Rev. 82 (1951) 664. [21] A.C. Heaxn, Reduce User's Manual, Rand Publication CP78 (1984). [22] E. Bag~n, J.I. Latorre, P. Pascual and R. Tarrach, Barcelona preprint (1984).
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