On the low energy structure of QCD

Volume 125B, number 4 PHYSICS LETTERS 2 June 1983

ON THE LOW ENERGY STRUCTURE OF QCD ~

J. GASSER and H. LEUTWYLER Institut fu'r theoretische Physik, Sidlerstrasse 5, 3012 Bern, Switzerland

Received 21 March 1983

We analyze the low energy structure of off-shell amplitudes in QCD. The method used automatically keeps track of the Ward identities associated with chiral symmetry. The infrared singularities generated by the Goldstone bosons are shown to lead to substantial deviations from the soft pion results.

Since the masses of the u and d quarks are very small, the divergence of the isovector axial current nearly vanishes *a. The lagrangian of QCD therefore exhibits an almost exact chiral symmetry; the corre- sponding low energy theorems (that would be exact if the masses m u and m d were zero) should hold to a very good approximation in the real world. To have a rough order of magnitude estimate of the deviations from the soft pion theorems to be expected one may compare the size of the terms in the QCD hamiltonian which break SU(2) X SU(2) and SU(3) respectively. The term which breaks SU(2) × SU(2) is smaller than the piece which breaks SU(3) by the factor (mu+md) / ( m s - mu) ~ ~z • Since the SU(3) relations generally hold at the 20% level one might expect the SU(2) × SU(2) relations to have an accuracy of the order of 20%: 12 -~ 2%.

As pointed out by Li and Pagels [2] this crude order of magnitude estimate may however be quite mis- leading. The soft pion theorems are statements about the behaviour of particles which in the symmetry limit (m u = m d = 0) are massless. Chiral perturbat ion theory [3] which treats the quark mass term as a perturba- t ion of the massless theory contains infrared singularities. These singularities may enhance the size of the perturbat ion and lead to deviations from the soft pion results that are substantially larger than what is indi- cated by the above crude estimate.

As an example, consider Weinberg's prediction [4]

Work supported in part by Schweizerischer Nationalfonds. , I For a review of the information about quark masses, see

ref. [1].

for the S-wave mr-scattering length: a 0 = 7M~r/32rcF~r2 2 where F~r = 93.3 -+ 0.1 MeV is the pion decay constant. The infrared singularities characteristic of chiral perturbat ion theory imply that the expansion o f a 0 in powers of the quark mass rh 1 =-~(m u + md) (or, equiv- alently, in powers o f M 2 ~ rh) is not a simple Taylor series, but contains nonanalytic terms of the type M2 logM2:

+ 0 (M2)I. (11

The coefficient of the nonanalytic term happens to be large in this case: if the scale/~ is taken at 1 GeV the correction amounts to 25% rather than to 2% as estimated above. This example shows that the nonanalytic terms in general produce sizeable corrections to the low energy theorems of current algebra +2.

The low energy structure o f the mr-scattering amplitude in the limit of massless pions was analyzed by Lehmann and his collaborators [5]. Weinberg [6] has given a general analysis of the low energy behaviour of on-shell matrix elements in the presence of a quark mass term. To work out the quark mass expansion of the theory we have to extend this framework and study the low energy behaviour of off-shell amplitudes.

¢2 T.N. Truong told us some time ago that he found large "unitarity corrections" to some soft pion theorems. Our analysis clearly confirms these results: in some cases the infrared singularities of chiral perturbation theory enhance the SU(2) X SU(2) asymmetries by one order of magnitude.

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(The literature contains a number of papers which analyze "unitari ty corrections" to the Trot-scattering amplitude ,3. These corrections do contribute to the nonanalytic terms we are looking for, but are not the only contributions.)

We consider the Green's functions of the vector, axialvector, scalar and pseudoscalar currents. These Green's functions are generated by the vacuum-to-vacuum amplitude associated with the lagrangian

./~= ./~ QCD + ? q 7 u ( v u + 7 5 a , ) q _ F q ( s _ i 3 " s p ) q , mquark = 0

exp {iZ [v, a, s, p] } = (Oout[Oin)o,a,s,p, (2)

where the external field v r (x), au (x), s ( x ) , p (x ) are hermitean, colour neutral matrices in flavour space. Note that s ( x ) includes the quark mass term. To sim- plify the discussion we restrict ourselves to two quark flavours u, d. Furthermore, to avoid anomalies [8], we disregard the flavour singlet vector and axial currents, i.e. take tr o u = tr a u = 0. In this case the lagrangian (2) is invariant under independent SU(2) transformations of the right- and left-handed component of q:

1 VL(X) ½ (1 q(x) [VR(X)~ (1 +3,5)+ --3'5)] q(x),

(3)

provided the external fields are transformed accordingly:

+ <,,<' : vR(o,, +<,,,) + ivp.o,,v .,

, , = + iVLOuV{ , v u - a u VL(V u - a u ) V L +

s' + i n ' = V R (s + iv) V~. (4)

The generating functional Z [ v , a, s, p] is invariant under these transformations. This property guarantees that the Green's functions generated by Z obey the appropriate Ward identities. To analyze the structure of Z at low energies (external fields of long wave- length) we first put s ( x ) = p ( x ) = 0, i.e. look at the Green's functions of the vector and axial currents in the chiral limit. In this limit the spectrum of the theory contains massless pions, which, if the matr ix ele- ment (0 IA u 17r) 0 does not vanish, produce poles which dominate the low energy- behaviour of the Green's

4:3 Unitarity corrections are reviewed in ref. [7], a survey of the more recent literature will be given in a more complete account of this work

functions. A technique which allows one to determine the leading low energy contributions is well known [9]. One considers the action of a suitable classical effective lagrangian. In the present context the non-linear o-model coupled to external vector and axial vector fields provides us with such a lagrangian [10]. More specifically, we denote by U A (x) a four-component real 0 ( 4 ) vector field of unit length, u T u = 1, and define its covariant derivative as

v. = a. v ° + (x) v i,

u;-- a . U + d 1o (x)Vl- u °,

_ 1 7.ioi, au _ 1 i i v u --~ - ~ r a u . (5)

The effective action reads

Z 1 = F 2 f d x ~ V u u T v u u . (6)

The field U A (x) is determined by the external fields v u (x) , au(x) through the classical equations of motion which follow from the requirement that Z 1 be an extremum. To get a unique solution we impose Feynman boundary conditions and specify the direc- t ion of the condensate in flavour space as the limit rh ~ + 0 of the massive theory: U A (x) -+ 6 A as t -+ + ~ . This determines Z 1 as a functional of the external fields - one easily checks that this functional is gauge invariant and reproduces the low energy behav- tour e.g. of the two-point function (0 IT [ A u A v] 10): in the chiral limit the leading low energy behaviour of the vector and axial vector Green's functions is determined by a single constant F . The low energy structure of the scalar and pseudoscalar densities on the other hand is not determined by the pion decay constant alone, but involves a second low energy constant B which measures the vacuum expectation value

(0 i f iu l0) o = (0l~ldl0) 0 = - F Z B (7)

of the scalar densities in the chiral limit. The expansion of Z in powers of the external field s ( x ) contains a linear term:

z = - f o x (0ifiul0)0 tr s(x) + . . . .

which, by itself, is not gauge invariant, because tr s ( x ) transforms like a component of a chiral four vector. Wri t ings(x) = sO1 + s i ' ; i ; p ( x ) = p0"l + pi 'ci the trans- formation law (3) shows that (s O , p i ) and (p0, _s i) transform as independent four vectors. The linear term

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in Z is proportional to s 0. To reproduce this term in a gauge invariant manner it suffices to add the contribu- tion 2F2B(sOU 0 + p i u i ) to the effective lagrangian (6) which then becomes

Z 1 = F2f dx(½ V~t U T V" U + x T u ) , (8)

where we have absorbed the constant B in the external fields, X A (x)= 2B(s0(x), pi(x)). The equation of motion for U A (x) reads

V la Via U A - U A (U T V Ia Via U) = X A - U A (UTx).

(9) Solving this equation to first order in U i one finds

that the quark mass shifts the pion pole to M 2 = 2rhB: the pion mass is determined by rh =½ (m u + m d ) , by the vacuum expectation value (0 Ifiu [0) 0 and by the pion decay constant [11].

As shown by Weinberg [6] there are two classes of contributions at the next order in the low energy expansion. The first category is related to unitary; since S-matrix elements are not unitary in the tree graph approximation (which is generated by the functional Z 1 [v, a, ×] ) one has to include graphs with loops. The one-loop graphs are suppressed by one power o fp 2 in comparison with the tree graphs (we count the external field X which includes the symmetry breaking piece proportional to rh as an object of order p2). Graphs with n loops are suppressed by p2n. The loop graphs do therefore not modify the leading low energy behaviour which is given by the classical theory, the one-loop graphs do however contribute at first nonleading order. The lagrangian (8) does of course not specify a renormalizable theory in four dimensions. Accordingly, one needs counter terms of increasing complexity as one tries to calculate graphs with an increasing number of loops. This however is not a dis- ease of the effective lagrangian (8) that one ought to try to cure. It directly reflects the character of the low energy expansion and is related to the second category of contributions that have to be included to obtain the correct low energy expansion at nex-to-leading order: there is no reason for the lowest order effective lagrangian to determine the low energy expansion to all orders. Instead one has to expect a set of new constants to appear at every level of the expansion - this is precisely what happens in a nonrenormalizable theory. (In principle, all of these constants are determined by the parameters A, mu, rod, m s .... which occur

in the QCD lagrangian, but the available theoretical methods to not allow us at this time to make quanti- tative use of this information).

The general effective lagrangian describing the first nonleading contributions in the low energy expansion contains all terms of dimension 4 that are consistent with gauge invariance and with parity:

Z 2 = . f dx [/1 ( v u u T Vu U)2

+ 12(Via U T V v U) ( 7 i a u T 7 vU)

+ 13 (X TU) 2 + 14 ( V lax T Via U) + l 5 UrFIa~Fu~ U

+ l 6 ( Via U r F Ia" V,, U) + 17 (~ T U)2 + h 1X TX

+ h 2 tr FiaVFiav + h 3 ~ T~ ]. (10)

Note that U(x) obeys the classical equation of motion (9); we have used this constraint to reduce the number of independent terms. The field ~ stands for the second chiral four vector that can be constructed out of the external scalar and pseudoscalar fields, ~/1 = 2B(p 0 , - s i ) . The field strength Fia v is an antisym- metric 4 X 4 matrix defined by ( Via V~ - V v Via) U = Fia v U; it contains the field strengths associated with the right- and left-handed combinations of via and aia. The constants h 1, h2 and h 3 multiply terms which do not involve the pion field U - these terms depend on the renormalization scheme used in QCD [the lagrangian (2) requires a set of counter terms of dimension 4]. For this reason the constants h l , h2, h 3 are not directly measurable and accordingly do not occur in the low energy expansion of physical quantities.

To next-to leading order the generating functional Z is given by

exp( iZ)= e x p ( i Z 2 f d / a [ul exp(iZ1) ) (11)

where the Feynman path integral over the field U(x) is to be calculated in the one-loop approximation. To evaluate this integral [12] one expands the functional Z 1 in the vicinity of the classical solution defined by (9) and retains only the quadratic terms. In one-loop approximation the integral is given by (det D ) - 112 where D is the differential operator associated with the quadratic terms in Z 1 . Regularizing the determi- nant by going to d dimensions one finds that the di- vergences at d = 4 are removed by the following renormalization of the constants l i and hi which occur

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in Z 2 :

l i = lr+'yiX; h i =hri + 6 i ~ ,

1 2 1 "[1 = '3 ' "[2 "-"3' T3 = - - 5 , "~4 = 2 ,

1 1 ")'5 = -- -g, "[6 = - - 5 , "Y7 = O,

1 6 1 = 2 62=F2 , 8 3 =0 . (12)

We reformmate the result in the language of the perturbative expansion of eq. (11): The Green's functions are generated by the vacuum-to-vacuum amplitude in the presence of external fields. The first two terms in the low energy expansion are obtained by evaluating (i) tree and one-loop graphs in the nonlin- ear o-model coupled to external fields (Z 1), (ii) tree graphs which contain one vertex of Z 2 together with any number of o-model vertices. The limit d ~ 4 of the sum of these contributions is finite, once it is ex- pressed in terms of the renormalized constants l~ and h~ defined in (12). The occurrence of counter terms in Z 2 which are not linear in the external field X (or contain derivatives thereof) is related to the problems which one has to solve [13] ,4 if one calculates the Green's functions of the pion field in the standard manner. In contrast to the procedure the external field technique retains the full symmetry of the theory at every stage of the calculation and, furthermore, specifies the Green's functions associated with the currents. Note also, that an effective lagrangian which only allows one to deal with o n s h e l l matrix elements does not determine the manner in which the low parameters depend on the quark mass. In our framework all low energy constants refer to the massless theory; the quark mass appears as an explicit symmetry breaking parameter contained in the external field, ×0

= (rn u + m d ) B + 2 0 .

As an example we quote the quark mass expansion of (0 [au 10) which one obtains in this manner [ 14] :

(Ol~ulO> _ 1+ 2 2 r ( M ~ / F ~ ) ( 2 l 3 + 2hrl -(3/327r 2) ln(M~/g2)] (010ul0> 0

+ 0 (M~). (]3)

In contrast to (01flu 10> 0 the vacuum expectation value <0 ]gu [0) in the real world depends on the "high energy constant" h 1, i.e. is not directly measurable, but depends on the renormalization prescription used

,4 The literature on the subject may be traced from this reference.

in QCD (the same remark applies to the difference

<0 ;t~ui0) - <0[~sl0>) [1]. The left-hand side of (13) is independent of the scale g at which the o-model loops are renormalized - the g-dependence of the renormalized constants l~, h~ cancels the g-dependence of the logarithms. This observation allows one to extract the nonanalytic terms in the quark mass expansion of any physical constant without actually calculating the finite pieces of the one-loop graphs. It suffices to work out the tree-graph contributions generated by Z 1 and Z 2 and to observe that the loop contributions must be such as to convert the result into a g-independent expression. Since the loop contributions to a constant can depend on g only through the factor ln(g2/M 2) one can read off the nonanalytic dependence on the pion mass by simply looking at the g-dependence of the constants l~, hi. (This is a pedestrian form of a more elegant renormalization group argument.) The relation (1) follows if one applies this recipe to a0 0.

We report about the comparison of the low energy expansion for the 1rTr-scattering amplitude with data in a subsequent paper [15].

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R. Dashen and M. Weinstein, Phys. Rev. 183 (1969) 1291.

[4] S. Weinberg, Phys. Rev. Lett. 17 (1966) 616. [5] H. Lehmann, Phys. Lett. 41B (1972) 529; Acta Phys.

Austr. Suppl. II (1973) 139; H. Lehmann and I-t. Trute, Nucl. Phys. B52 (1973) 280; G. Ecker and J. Honerkamp, Nucl. Phys. B52 (1973) 211.

[6] S. Weinberg, Physica 96A (1979) 327. [7] B.R. Martin, D. Morgan and G. Shaw, Pion-pion interac-

tions in particle physics (Academic Press, London, 1976). [8] S.L. Adler, Phys. Rev. 177 (1969) 2426;

W.A. Bardeen, Phys. Rev. 184 (1969) 1848; J. Wess and B. Zumino, Phys. Lett. 37B (1971) 95.

[9] S. Weinberg, Phys. Rev. Lett. 18 (1967) 188; Phys. Rev. 166 (1968) 1568; S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1968) 2239; C. Callan, S. Coleman, J. Wess and B. Zumino, Phys. Rev. 177 (1969) 2247.

[10] D.G. Boulware and L.S. Brown, Ann. Phys. 138 (1982) 392.

[ 11] M. Gell-Mann, R.J. Oakes and B. Renner, Phys. Rev. 175 (1968) 2195.

[12] G. 't Hooft, Nucl. Phys. B62 (1973) 444. [13] T. Appelquist and C. Bernard, Phys. Rev. D23 (1981)

425. [ 14] P. Langacker and H. Pagels, Phys. Rev. D8 (1973) 4595. [15] J. Gasser and H. Leutwyler, Phys. Lett. 125B (1983) 325.

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On the low energy structure of QCD