# Zweig rule violation in the scalar sector

and values of low-energy constants

###### Abstract

We discuss the role of the Zweig rule (ZR) violation in the scalar channel for the determination of low-energy constants and condensates arising in the effective chiral Lagrangian of QCD. The analysis of the Goldstone boson masses and decay constants shows that the three-flavor condensate and some low-energy constants are very sensitive to the value of the ZR violating constant . A similar study is performed in the case of the decay constants. A chiral sum rule based on experimental data in the channel is used to constrain , indicating a significant decrease between the two- and the three-flavor condensates. The analysis of the scalar form factors of the pion at zero momentum suggests that the pseudoscalar decay constant could also be suppressed from to 3.

INPO-DR/00-33

SHEP 00/16

hep-ph/0012221

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^{†}thanks: e-mail:

## 1 Introduction

The low-energy constants (LEC’s) of the effective chiral Lagrangian of QCD [1] are quantities of great theoretical interest, since they reflect the way chiral symmetry is spontaneously broken. However, their determination remains a particularly awkward problem. In most cases [1, 3, 4, 5, 6], their values have been inferred from observables for the pseudoscalar mesons, with the help of two assumptions: (1) the quark condensate is the dominant order parameter to describe the Spontaneous Breakdown of Chiral Symmetry (SBS) [1], and (2) the pattern of SBS agrees correctly with a large- description of QCD [7], in which quantum fluctuations are treated as small perturbations.

If we admit both assumptions, the SU(2)SU(2) quark condensate should not depend much on the mass of the strange quark. We could then set the latter to zero with no major effect on the quark condensate: . We end up with only one large condensate for SU(2)SU(2) and SU(3)SU(3) chiral limits, which is not very sensitive to fluctuations. The LEC’s suppressed by the Zweig rule, and , are consistently supposed to be very small when considered at a typical hadronic scale.

However, several arguments may be raised against this ”mean-field approximation” of SBS, in which the Zweig rule applies and the chiral structure of QCD vacuum is more or less independent of the number of massless quarks. On the one hand, the scalar sector does not comply with large -predictions [8], and some lattice simulations with dynamical fermions suggest a strong -sensitivity of SBS signals [9]. On the other hand, the behaviour of the perturbative QCD -function indicates that chiral symmetry should be restored for large enough values of . In the vicinity ot the corresponding critical point, chiral order parameters should strongly vary with . Various approaches, based on the investigation of the QCD conformal window [10], gap equations [11], or the instanton liquid model [12], have been proposed to investigate the variations of chiral order parameters with the number of massless flavors and to determine the critical value of for the restoration of chiral symmetry.

In Ref. [13], the -sensitivity of chiral order parameters has been investigated without relying on perturbative methods, but rather by exploiting particular properties of vector-like gauge theories. The mechanism of SBS is indeed related to the dynamics of the lowest eigenvalues of the Dirac operator: , considered on an Euclidean torus [14, 15, 16]. Two main chiral order parameters can be expressed in this framework. The quark condensate is related to the average density of eigenvalues around zero [14] and the pion decay constant in the chiral limit can be interpreted as a conductivity [16]. The paramagnetic behaviour of Dirac eigenvalues leads to a suppression of both order parameters when the number of flavors increases:

(1) |

This sensitivity of chiral order parameters to light-quark loops is suppressed in the large- limit and is considered as weak for QCD according to the second hypothesis of the Standard framework. However, the -dependence of chiral order parameters can be measured by correlators that violate the Zweig rule in the scalar (vacuum) channel. For instance, the difference (and the LEC ) is related to the correlator [17, 18] (this correlator can be interpreted as fluctuations of the density of Dirac eigenvalues [13]). The large ZR violations observed in the channel could therefore support a swift evolution in the chiral structure of the vacuum from to . The quantum fluctuations of pairs would then play an essential role in the low-energy dynamics of QCD.

Hence, it is worth reconsidering the determination of LEC’s without supposing (1) the dominance of the quark condensate and (2) the suppression of quantum fluctuations. This determination starts with quark mass expansion of measured observables such as or , using Chiral Perturbation Theory (PT) [1]:

(2) |

where denotes formally light quark masses (, , ) and the remainder is of order . The coefficients , , are combinations of LEC’s. The chiral logarithms stem from meson loops. The coefficient of each power of does not depend on the renormalization scale of the effective theory ( and are independent of this scale).

Series like Eq. (2) are assumed to converge on the basis of a genuine dimensional estimate [19]. The LEC’s involved in the coefficients are related to Green functions of axial and vector currents, and scalar and pseudoscalar densities. The dimensional estimate consists in saturating the correlators by the exchange of resonances with masses of order [4]. We obtain coefficients of order for the power . The quark mass expansion would therefore lead to (convergent) series in powers of .

Notice that this genuine estimate cannot be applied to the linear term corresponding to the quark condensate (there is no colored physical state to saturate ). Moreover, the convergence of the whole series does not imply that the linear term is dominant with respect to the quadratic term. In this article, we will precisely address (1) the possibility of such a competition between the first two orders in the quark mass expansions, and (2) the implications of large values for the ZR-suppressed constants and , in particular for the determination of LEC’s.

Unfortunately, the masses and decay constants of the Goldstone bosons do not provide enough information to estimate the actual size of quantum fluctuations in QCD. To reach this goal, Refs. [17, 18] have proposed a sum rule to estimate [or ] from experimental data in the scalar channel. Starting with Standard assumptions (two- and three-flavor condensates of large and similar sizes), Ref. [17] ended up with a ratio at the Standard order, whereas Ref. [18] confirmed a large decrease of the quark condensate when Standard contributions were taken into account. Even though these results suggest a significant variation in the pattern of SBS from to , it is seems necessary to reevaluate this sum rule without any supposition about the size of the condensates. This analysis will be performed in the second part of this article.

We will follow mainly the line of Ref.[20], which can be considered as an orientation guide to this article. The first part is devoted to the determination of the LEC’s from the pseudoscalar spectrum. Sec. 2 considers the role played by for the Goldstone boson masses and the quark condensates, whereas the decay constants and are treated in Sec. 3. Sec. 4 deals mainly with the implication of ZR violation in the channel for the determination of LEC’s. The second part of this article focuses on the estimate of from data in the scalar sector. Sec. 5 introduces the sum rule for , and sketches the Operator Product Expansion of the involved Green function. In Sec. 6, we estimate this sum rule, with a special emphasis on the the scalar form factors of the pion and the kaon. In Sec. 7, we present the results obtained for the quark condensates and LEC’s from the sum rule, and we discuss two other quantities related to the pion scalar form factors: the slope of the strange form factor and the scalar radius of the pion. Sec. 8 sums up the main results of the article. App. A collects the expansions of pseudoscalar masses and decay constants in powers of quark masses. App. B deals with the Operator Product Expansion of the correlator . App. C provides logarithmic derivatives of the pseudoscalar masses with respect to the quark masses.

## 2 Constraints from the pseudoscalar meson masses

### 2.1 Role of

Let us first study the pseudoscalar masses , , , starting from their expansion at the Standard order, Eqs. (10.7) in Ref. [1]. We reexpress them as:

(3) | |||||

where and and are scale-independent constants, containing respectively the LEC’s and ,

(5) | |||||

(6) |

with and . The remaining
chiral logarithms are contained in :^{1}^{1}1In this article, we use the following
values of masses and decay constants:
MeV, MeV, MeV, MeV, .

(7) |

There is a similar equation for :

with the scale-independent constant . A factor is included in the expression of , and in terms of , so that they do not diverge in the limit . The corresponding equations for the pseudoscalar decay constants (= , , ) will be treated in Sec. 3.1.

We take and as independent observables, in order to separate in a straightforward way the “mass” constants , , from , that appear only in the expansion of decay constants . There is a second argument supporting the choice of and as independent observables of the pseudoscalar spectrum. We expand observables in powers of quark masses, supposing a good convergence of the series. We have sketched in the introduction how a naive dimensional estimate justifies this assumption : LEC’s are related to QCD correlation functions, which can be saturated by massive resonances, leading to series in . We should therefore expect good convergence properties for ”primary” observables obtained directly from the low-energy behavior of QCD correlation functions, like and . For such quantities, the higher-order remainders should thus remain small. On the other hand, we have to be careful when we deal with ”secondary” quantities combining ”primary” observables. The higher-order remainders may then have a larger influence. In particular, ratios of ”primary” observables (like ) might be dangerous if higher-order terms turned out to be sizable [leading to untrustworthy approximations like with a large ].

In Eqs. (3), (2.1) and (2.1), all terms linear and quadratic in quark masses are shown. The remaining contributions, of order and higher, are collected in the remainders . We can consider that the latter are given to us, so that Eqs. (3), (2.1) and (2.1) can be seen as algebraic identities relating the 3-flavor condensate the quark mass ratio , and the LEC’s , , and . The three-flavor quark condensate is measured in physical units, using the Gell-Mann–Oakes ratio: [21].

We are going to assume that the remainders are small (), and investigate then the consequences of Eqs. (3) and (2.1) for the values of LEC’s. Before starting, we should comment the status of Eqs. (3), (2.1) and (2.1) with respect to Chiral Perturbation Theory (PT). Even if we imposed , we would not work in the frame of one-loop Standard PT [1]: we do not suppose that the condensate is dominant in these equations, we do not treat as a small expansion parameter, and accordingly, we do not replace (for instance) by in higher-order terms. However, we are not following Generalized PT either [22], since is not treated as an expansion parameter: even with , Eqs. (3), (2.1) and (2.1) exceed the Generalized tree level, since these equations include chiral logarithms.

It is useful to rewrite Eqs. (3) and (2.1) as:

(9) | |||||

(10) |

with

(11) |

and are linear combinations of the remainders and :

(12) | |||||

(13) |

For large , we expect . Similarly to Ref. [6], we consider as parameters [i.e. ], the ZR violating constant and the quark mass ratio . Eq. (9) ends up with a non-perturbative formula (no expansion) for the three-flavor Gell-Mann–Oakes–Renner ratio :

(14) |

where contains :

Eq. (14) is an exact identity, useful if the remainder in Eq. (12) is small, i.e. if the expansion of QCD correlators in powers of the quark masses , , is globally convergent. It means that in Eqs. (3) and (2.1), but the linear term in these equations (related to the condensate) does not need to dominate.

describes quantum fluctuations of the condensate, and actually . has to be fixed carefully to keep small. is equal to zero for at the scale , which is close to the value usually claimed in Standard PT analysis [3]. In this case, Eq. (14) yields near 1, unless the quark mass ratio decreases significantly, leading to . This effect is well-known in GPT [22]: the minimal value of (corresponding to and ) is . Notice that for these very small values of , the combination of higher-order remainders cannot be neglected any more in Eq. (14).

But quantum fluctuations can modify this picture: the number before the curly brackets in Eq. (2.1) is very large ( for and = 85 MeV). Hence, even a small positive value of can lead to a strong suppression of , whatever the value of . This effect can be seen on Fig. 1, where is plotted as a function of for , , and = 85 and 75 MeV. The decrease of is slightly steeper for lower values of .

Once is known, Eq. (10) leads to :

(16) |

This constant depends on only through . Notice that this dependence is smaller when decreases ( depends actually on through ).

The ZR violating constant can be obtained from Eq. (2.1):

(17) | |||||

where will be discussed in Sec. 3.1. The pseudoscalar spectrum satisfies with a good accuracy the relation , which reduces at the leading order to the Gell-Mann–Okubo formula [23]. This relation leads to a strong correlation between and : . This correlation can also be seen in Standard PT between and , and remains to be explained in both frameworks. No obvious reason forces this particular combination of two low-energy constants to be much smaller than the typical size of the effective constants.

### 2.2 Paramagnetic inequality for

In Ref. [13], fluctuations were shown to increase the two-flavor condensate with respect to , so that . The two-flavor quark condensate can be obtained through the limit:

(18) |

keeping fixed. We have the quantities: and , with:

(19) |

and .
The effect of is very small^{2}^{2}2It can be evaluated
following the procedure of Sec. A.3..
should be compared to the logarithmic terms included in
, Eq. (5), at a typical scale
. reaches hardly 10% of this logarithmic piece.

Once is eliminated from Eqs. (9) and (18), we obtain the two-flavor Gell-Mann–Oakes–Renner ratio :

with:

(21) | |||||

In the expression of , the remainders are suppressed by a factor : this suppression is obvious for [], whereas the operator applied to cancels the terms of order . For , we expect thus . The dependence of on is completely hidden in , and therefore marginal, as shown in Fig. 1.

The paramagnetic inequality constrains the maximal value reached by . If we neglect in Eq. (18), the inequality can be translated into a lower bound for :

(23) |

Fig. 1 shows clearly the lower bound: .

is loosely related to , but it is very strongly correlated with , specially for small values of . Eq. (2.2) yields the estimate , up to small correcting terms due to . We are going to study the effect of these correcting terms.

If we neglect , is a quadratic function of , which is not monotonous when varies from 0 to : it first increases, and then decreases (see Fig. 1). The decrease of for close to its upper bound is caused by the negative term, quadratic in , in Eq. (2.2). This decrease of is more significant for small , because the factor in front of in Eq. (2.2) becomes larger.

Therefore, does not reach its maximum for the paramagnetic bound
, whereas its minimum is the smallest of the two
values obtained for and
^{3}^{3}3This updates
Ref. [13], where the minimum and maximum of were claimed to
be obtained for and .. The dependence on
of the minimal value of can be guessed rather easily. For large , the
term linear in in Eq. (2.2) can be neglected: the minimum of
occurs for . For small
, and tend to 0, and the term quadratic in should be
small with respect to the linear term. Therefore, reaches its minimum for
when is small.

The numerical analysis of Eq. (2.2), including , supports this intuitive description. In Fig. 2, the variation ranges of are plotted for several values of . The curve for the minimum of exhibits a cusp when the minimum of corresponds no more to , but to . appears to be strongly correlated to , even though a large value of can be associated to a broad range of .

## 3 Constraints from the pseudoscalar decay constants

### 3.1 Role of

The decomposition used for the Goldstone boson masses can be adapted to the decay constants:

(25) |

with the scale-independent constants related to and :

(26) | |||||

(27) |

Eqs. (3.1) and (25) contain all the terms constant or linear in quark masses in the expansion of and , whereas denote remainders of order . There is also a formula for , which can be written as:

The two scale-independent constants can be extracted from Eqs. (3.1) and (25):

with:

(31) |

where the latter estimate is obtained for .

(i.e. ) turns out to depend essentially on and , whereas (i.e. ) is related to the difference between and . Eq. (3.1) leads to a quadratic equation for , involving :

(32) |

with:

(34) |

Eq. (32) has the solution:

(35) |

Notice that this formula is very close to Eq. (14), that relates to through the parameter . For , the factor in front of the curly brackets in the definition of is of order , and vanishes for .

The variations of with are plotted for three values of and and 0.5 in Fig. 3. When decreases, the allowed range for broadens. This is due to the definition of , which relates to the low-energy behavior of a QCD Green function. starts at 0.9 (for the lowest possible value of ) and decreases until 0.5. can thus vary from 87 to 65 MeV, depending on the value of .

### 3.2 Paramagnetic inequality for

We obtain by taking the limit:

(37) |

keeping fixed. We have and , with

(38) |

has a tiny effect on the results, similarly to for . We get the equation:

(39) |

with:

(40) | |||||

(41) | |||||

(42) |

It is interesting to compare the expression of as a function of with Eq. (2.2) that relates to . Even though the equations look rather similar, we can notice that Eq. (2.2) is a quadratic function of whereas Eq. (39) involves and its inverse . Moreover, Eq. (39) is an increasing function of , whereas is not monotonous with . On the other hand, suppresses the remainders by a factor , in a comparable way to .

The paramagnetic inequality is translated into an upper bound for :

(43) |

A quick estimate shows that the condition is satisfied for any between