Transformada de Borel

Embora a compara¸c˜ao dos lados da QCD e Fenomenol´ogico seja garantida pelo princ´ıpio da Dualidade Quark-H´adron, o uso das aproxima¸c˜oes que fizemos em ambos os lados introduzem al- gumas complica¸c˜oes. No lado da QCD, ao lidarmos com contribui¸c˜oes n˜ao-perturbativas, `a rigor, dever´ıamos calcular os infinitos termos da OPE. Contudo, a impraticabilidade deste procedimento

terça-feira, 21 de julho de 15

a

Lado Fenomenológico

a

CAP´ITULO 2. REGRAS DE SOMA DA QCD (QCDSR) 21 que ´e v´alido desde que exista um valor m´ınimo, tal que, q² > s_min. Fazendo a substitui¸c˜ao em (2.60), obtemos a rela¸c˜ao de dispers˜ao:

Π(q²) = 1 π

!∞

s_min

ds Im [Π(s)]

s − q² − i" ≡

!∞

s_min

ds ρ(s)

s − q² − i" (2.61) onde

ρ(s) = 1

π Im [Π(s)] (2.62)

´e a densidade espectral fenomenol´ogica.

Usualmente, a densidade espectral ρ(s) pode ser descrita por um p´olo bem definido, represen- tado pela part´ıcula no estado fundamental, e mais um cont´ınuo, representado pelo seus estados ressonantes, de tal forma que:

ρ(s) = λ² δ(s − m²) + ρ^cont(s) (2.63)

Fenomenologicamente, s´o h´a contribui¸c˜ao das ressonˆancias quando estamos acima de um valor limiar (ou threshold) do cont´ınuo, indicado por s₀. Acima deste threshold, ´e razo´avel afirmar que [40]:

ρ^cont(s) = ρ^ope(s) Θ(s − s₀), (2.64) com ρ^ope(s) dado por (2.45). Portanto,

ρ(s) = λ² δ(s − m²) + ρ^ope(s) Θ(s − s₀) (2.65) Inserindo (2.65) em (2.61), obtemos:

Π(q²) = λ²

m² − q² +

!∞

s₀

ds ρ^ope(s)

s − q² (2.66)

Note que o primeiro termo desta equa¸c˜ao ´e exatamente igual ao primeiro termo de (2.59). Com isto, obtemos uma express˜ao mais elegante para as contribui¸c˜oes dos estados ressonantes:

"

H_i

1

3q² (q² − m²)#0| j_µ | H_i(q) $# H_i(q) | j^{† µ} |0$ ≡

!∞

s₀

ds ρ^ope(s)

s − q² (2.67)

De fato, (2.66) ´e a que ser´a utilizada na compara¸c˜ao de ambos os lados da teoria.

2.5 Transformada de Borel

Embora a compara¸c˜ao dos lados da QCD e Fenomenol´ogico seja garantida pelo princ´ıpio da Dualidade Quark-H´adron, o uso das aproxima¸c˜oes que fizemos em ambos os lados introduzem al- gumas complica¸c˜oes. No lado da QCD, ao lidarmos com contribui¸c˜oes n˜ao-perturbativas, `a rigor, dever´ıamos calcular os infinitos termos da OPE. Contudo, a impraticabilidade deste procedimento

polo

terça-feira, 21 de julho de 15

a

Lado Fenomenológico

a

CAP´ITULO 2. REGRAS DE SOMA DA QCD (QCDSR) 21 que ´e v´alido desde que exista um valor m´ınimo, tal que, q² > s_min. Fazendo a substitui¸c˜ao em (2.60), obtemos a rela¸c˜ao de dispers˜ao:

Π(q²) = 1 π

!∞

s_min

ds Im [Π(s)]

s − q² − i" ≡

!∞

s_min

ds ρ(s)

s − q² − i" (2.61) onde

ρ(s) = 1

π Im [Π(s)] (2.62)

´e a densidade espectral fenomenol´ogica.

Usualmente, a densidade espectral ρ(s) pode ser descrita por um p´olo bem definido, represen- tado pela part´ıcula no estado fundamental, e mais um cont´ınuo, representado pelo seus estados ressonantes, de tal forma que:

ρ(s) = λ² δ(s − m²) + ρ^cont(s) (2.63)

Fenomenologicamente, s´o h´a contribui¸c˜ao das ressonˆancias quando estamos acima de um valor limiar (ou threshold) do cont´ınuo, indicado por s₀. Acima deste threshold, ´e razo´avel afirmar que [40]:

ρ^cont(s) = ρ^ope(s) Θ(s − s₀), (2.64) com ρ^ope(s) dado por (2.45). Portanto,

ρ(s) = λ² δ(s − m²) + ρ^ope(s) Θ(s − s₀) (2.65) Inserindo (2.65) em (2.61), obtemos:

Π(q²) = λ²

m² − q² +

!∞

s₀

ds ρ^ope(s)

s − q² (2.66)

Note que o primeiro termo desta equa¸c˜ao ´e exatamente igual ao primeiro termo de (2.59). Com isto, obtemos uma express˜ao mais elegante para as contribui¸c˜oes dos estados ressonantes:

"

H_i

1

3q² (q² − m²)#0| j_µ | H_i(q) $# H_i(q) | j^{† µ} |0$ ≡

!∞

s₀

ds ρ^ope(s)

s − q² (2.67)

De fato, (2.66) ´e a que ser´a utilizada na compara¸c˜ao de ambos os lados da teoria.

2.5 Transformada de Borel

Embora a compara¸c˜ao dos lados da QCD e Fenomenol´ogico seja garantida pelo princ´ıpio da Dualidade Quark-H´adron, o uso das aproxima¸c˜oes que fizemos em ambos os lados introduzem al- gumas complica¸c˜oes. No lado da QCD, ao lidarmos com contribui¸c˜oes n˜ao-perturbativas, `a rigor, dever´ıamos calcular os infinitos termos da OPE. Contudo, a impraticabilidade deste procedimento

polo

contínuo

terça-feira, 21 de julho de 15

a

Lado Fenomenológico

a

CAP´ITULO 2. REGRAS DE SOMA DA QCD (QCDSR) 21 que ´e v´alido desde que exista um valor m´ınimo, tal que, q² > s_min. Fazendo a substitui¸c˜ao em (2.60), obtemos a rela¸c˜ao de dispers˜ao:

Π(q²) = 1 π

!∞

s_min

ds Im [Π(s)]

s − q² − i" ≡

!∞

s_min

ds ρ(s)

s − q² − i" (2.61) onde

ρ(s) = 1

π Im [Π(s)] (2.62)

´e a densidade espectral fenomenol´ogica.

Usualmente, a densidade espectral ρ(s) pode ser descrita por um p´olo bem definido, represen- tado pela part´ıcula no estado fundamental, e mais um cont´ınuo, representado pelo seus estados ressonantes, de tal forma que:

ρ(s) = λ² δ(s − m²) + ρ^cont(s) (2.63)

Fenomenologicamente, s´o h´a contribui¸c˜ao das ressonˆancias quando estamos acima de um valor limiar (ou threshold) do cont´ınuo, indicado por s₀. Acima deste threshold, ´e razo´avel afirmar que [40]:

ρ^cont(s) = ρ^ope(s) Θ(s − s₀), (2.64) com ρ^ope(s) dado por (2.45). Portanto,

ρ(s) = λ² δ(s − m²) + ρ^ope(s) Θ(s − s₀) (2.65) Inserindo (2.65) em (2.61), obtemos:

Π(q²) = λ²

m² − q² +

!∞

s₀

ds ρ^ope(s)

s − q² (2.66)

Note que o primeiro termo desta equa¸c˜ao ´e exatamente igual ao primeiro termo de (2.59). Com isto, obtemos uma express˜ao mais elegante para as contribui¸c˜oes dos estados ressonantes:

"

H_i

1

3q² (q² − m²)#0| j_µ | H_i(q) $# H_i(q) | j^{† µ} |0$ ≡

!∞

s₀

ds ρ^ope(s)

s − q² (2.67)

De fato, (2.66) ´e a que ser´a utilizada na compara¸c˜ao de ambos os lados da teoria.

2.5 Transformada de Borel

Embora a compara¸c˜ao dos lados da QCD e Fenomenol´ogico seja garantida pelo princ´ıpio da Dualidade Quark-H´adron, o uso das aproxima¸c˜oes que fizemos em ambos os lados introduzem al- gumas complica¸c˜oes. No lado da QCD, ao lidarmos com contribui¸c˜oes n˜ao-perturbativas, `a rigor, dever´ıamos calcular os infinitos termos da OPE. Contudo, a impraticabilidade deste procedimento

polo

contínuo

terça-feira, 21 de julho de 15

dominância do polo ←

a

Lado Fenomenológico

a

CAP´ITULO 2. REGRAS DE SOMA DA QCD (QCDSR) 21 que ´e v´alido desde que exista um valor m´ınimo, tal que, q² > s_min. Fazendo a substitui¸c˜ao em (2.60), obtemos a rela¸c˜ao de dispers˜ao:

Π(q²) = 1 π

!∞

s_min

ds Im [Π(s)]

s − q² − i" ≡

!∞

s_min

ds ρ(s)

s − q² − i" (2.61) onde

ρ(s) = 1

π Im [Π(s)] (2.62)

´e a densidade espectral fenomenol´ogica.

Usualmente, a densidade espectral ρ(s) pode ser descrita por um p´olo bem definido, represen- tado pela part´ıcula no estado fundamental, e mais um cont´ınuo, representado pelo seus estados ressonantes, de tal forma que:

ρ(s) = λ² δ(s − m²) + ρ^cont(s) (2.63)

Fenomenologicamente, s´o h´a contribui¸c˜ao das ressonˆancias quando estamos acima de um valor limiar (ou threshold) do cont´ınuo, indicado por s₀. Acima deste threshold, ´e razo´avel afirmar que [40]:

ρ^cont(s) = ρ^ope(s) Θ(s − s₀), (2.64) com ρ^ope(s) dado por (2.45). Portanto,

ρ(s) = λ² δ(s − m²) + ρ^ope(s) Θ(s − s₀) (2.65) Inserindo (2.65) em (2.61), obtemos:

Π(q²) = λ²

m² − q² +

!∞

s₀

ds ρ^ope(s)

s − q² (2.66)

Note que o primeiro termo desta equa¸c˜ao ´e exatamente igual ao primeiro termo de (2.59). Com isto, obtemos uma express˜ao mais elegante para as contribui¸c˜oes dos estados ressonantes:

"

H_i

1

3q² (q² − m²)#0| j_µ | H_i(q) $# H_i(q) | j^{† µ} |0$ ≡

!∞

s₀

ds ρ^ope(s)

s − q² (2.67)

De fato, (2.66) ´e a que ser´a utilizada na compara¸c˜ao de ambos os lados da teoria.

2.5 Transformada de Borel

Embora a compara¸c˜ao dos lados da QCD e Fenomenol´ogico seja garantida pelo princ´ıpio da Dualidade Quark-H´adron, o uso das aproxima¸c˜oes que fizemos em ambos os lados introduzem al- gumas complica¸c˜oes. No lado da QCD, ao lidarmos com contribui¸c˜oes n˜ao-perturbativas, `a rigor, dever´ıamos calcular os infinitos termos da OPE. Contudo, a impraticabilidade deste procedimento

polo

contínuo

dominância do polo ←

terça-feira, 21 de julho de 15

5 This quantity has the advantage to be less sensitive to the perturbative radiative corrections than the individual moments. Therefore, we expect that our results obtained to leading order in α_s will be quite accurate.

In the numerical analysis of the sum rules, the values used for the quark masses and condensates are (see e.g. [19, 39, 40, 41]) ⁴ :

m_c(m_c) = (1.23 ± 0.05) GeV, m_b(m_b) = (4.24 ± 0.06) GeV,

m_u = 2.3 MeV, m_d = 6.4 MeV,

m_q ≡ (m_u + m_d)/2 = 4.3 MeV, "qq¯ # = −(0.23 ± 0.03)³ GeV³, m_s = 100 MeV, "ss¯ #/"qq¯ # = 0.8 ± 0.2,

"qg¯ σ.Gq# = m²₀"qq¯ # with m²₀ = 0.8 GeV²,

"g²G²# = 0.88 GeV⁴. (16)

We evaluate the sum rules in the range 2.0 ≤ M ² ≤ 2.8 for two values of s₀: s¹₀^/2 = 4.1 GeV, s¹₀^/2 = 4.2 GeV.

Comparing the relative contribution of each term in Eqs. (11) to (14), to the right hand side of Eq. (9) we obtain a quite good OPE convergence for M ² > 1.9 GeV², as can be seen in Fig. 1. This analysis allows us to determine the lower limit constraint for M² in the sum rules window. This figure also shows that, although there is a change of sign between dimension-six and dimension-eight condensates contributions, the contribution of the latter being smaller, where, we have assumed, in Fig. 1 to Fig. 4, the validity of the vacuum saturation for these condensates. The relatively small contribution of the dimension-eight condensates may justify the validity of our approximation, unlike in the case of the 5-quark current correlator, as noticed in [42]. However, the partial compensation of these two terms indicate the sensitivity of the central value of the mass prediction on the way the OPE is truncated.

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

M²(GeV ²) 3.4

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

M X (GeV)

FIG. 2: The OPE convergence for MX in the region 1.9 ≤ M² ≤ 2.8 GeV² for s¹₀^/2 = 4.1 GeV. We start with the perturbative contribution plus a very small mq contribution (long-dashed line) and each subsequent line represents the addition of one extra condensate dimension in the expansion: +"qq¯ # (solid line), +"g²G²# + mq"qq¯ # (dotted-line in top of the solid line), +"qq¯ #² (dashed line), +m²0"qq¯ # (solid line with triangles).

In Fig. 2 are shown the contributions of the individual condensates to M_X obtained from Eq. (15), From Fig. 2, it appears that the results oscillates around the perturbative result, and that the results

4 To leading order approximation in α_s, at which we are working, we do not consistently consider the running scale dependence of these parameters. We shall use here the values of the quark masses obtained within the same QCD spectral sum rules methods compiled in [19]. They are defined in the M S-scheme, and have obtained within the same truncation of the QCD series from different channels and by different authors.

√s0=4.1 GeV

4 where the integration limits are given by α_min = (1 − !

1 − 4m²_c/s)/2, α_max = (1 + !

1 − 4m²_c/s)/2 and (β_min = αm²_c)/(sα − m²_c). We have also included the dominant contributions from the dimension-five condensates:

ρ^mix(s) = m_c"qgσ.Gq¯ # 2⁶π⁴

αmax

"

αmin

dα

#

− 2

α (m²_c − α(1 − α)s) +

1−α

"

βmin

dβ $

(α + β)m²_c − αβs%

& 1

α + α + β β²

' ( ,

(12)

where the contribution of dimension-six condensates "g³G³# is neglected, since assumed to be suppressed by the loop factor 1/16π². The usual estimate "g³G³# $ 1GeV²"α_sG²# [19] would deserve to be checked in more detail. We have included the contribution of the dimension-six four-quark condensate:

ρ^{"qq¯ #2}(s) = m²_c"qq¯ #² 12π²

)s − 4m²_c

s , (13)

and (for completeness) a part of the dimension-8 condensate contributions ³: Π^mix"₁ ^{qq¯ #}(M²) = − m²_c"qg¯ σ.Gq#"qq¯ #

24π²

" 1

0

dα

#

1 + m²_c

α(1 − α)M ² − 1

2(1 − α) (

exp

#

− m²_c

α(1 − α)M² (

. (14)

III. LSR PREDICTIONS OF M^X

1.6 1.8 2.0 2.2 2.4 2.6 2.8

1 2 3 4

5 s₀^1/2 = 4.17 GeV

Pert + m_q + <qq>

+ <g²G²> + m_q<qq>

+ m₀²<qq>

+ <qq>² + m₀²<qq>²

Condensate/RHS

M² (GeV ²)

FIG. 1: The OPE convergence in the region 1.6 ≤ M² ≤ 2.8 GeV² for s¹₀^/2 = 4.17 GeV. We start with the perturbative contribution (plus a very small m_q contribution) and each subsequent line represents the addition of one extra condensate dimension in the expansion.

In order to extract the mass M_X without worrying about the value of the decay constant f_X, we take the derivative of Eq. (9) with respect to 1/M², divide the result by Eq. (9) and obtain:

M_X² =

* s⁰

4m²_c ds e^−s/M2 s ρ(s)

* s⁰

4m²_c ds e^−s/M2 ρ(s) . (15)

3 We should note that a complete evaluation of these contributions require more involved analysis including a non-trivial choice of the factorization assumption basis [38]. We wish that we can perform this analysis in the future.

terça-feira, 21 de julho de 15

5 This quantity has the advantage to be less sensitive to the perturbative radiative corrections than the individual moments. Therefore, we expect that our results obtained to leading order in α_s will be quite accurate.

In the numerical analysis of the sum rules, the values used for the quark masses and condensates are (see e.g. [19, 39, 40, 41]) ⁴ :

m_c(m_c) = (1.23 ± 0.05) GeV, m_b(m_b) = (4.24 ± 0.06) GeV,

m_u = 2.3 MeV, m_d = 6.4 MeV,

m_q ≡ (m_u + m_d)/2 = 4.3 MeV, "qq¯ # = −(0.23 ± 0.03)³ GeV³, m_s = 100 MeV, "ss¯ #/"qq¯ # = 0.8 ± 0.2,

"qg¯ σ.Gq# = m²₀"qq¯ # with m²₀ = 0.8 GeV²,

"g²G²# = 0.88 GeV⁴. (16)

We evaluate the sum rules in the range 2.0 ≤ M ² ≤ 2.8 for two values of s₀: s¹₀^/2 = 4.1 GeV, s¹₀^/2 = 4.2 GeV.

Comparing the relative contribution of each term in Eqs. (11) to (14), to the right hand side of Eq. (9) we obtain a quite good OPE convergence for M ² > 1.9 GeV², as can be seen in Fig. 1. This analysis allows us to determine the lower limit constraint for M² in the sum rules window. This figure also shows that, although there is a change of sign between dimension-six and dimension-eight condensates contributions, the contribution of the latter being smaller, where, we have assumed, in Fig. 1 to Fig. 4, the validity of the vacuum saturation for these condensates. The relatively small contribution of the dimension-eight condensates may justify the validity of our approximation, unlike in the case of the 5-quark current correlator, as noticed in [42]. However, the partial compensation of these two terms indicate the sensitivity of the central value of the mass prediction on the way the OPE is truncated.

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

M²(GeV ²) 3.4

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

M X (GeV)

FIG. 2: The OPE convergence for MX in the region 1.9 ≤ M² ≤ 2.8 GeV² for s¹₀^/2 = 4.1 GeV. We start with the perturbative contribution plus a very small mq contribution (long-dashed line) and each subsequent line represents the addition of one extra condensate dimension in the expansion: +"qq¯ # (solid line), +"g²G²# + mq"qq¯ # (dotted-line in top of the solid line), +"qq¯ #² (dashed line), +m²0"qq¯ # (solid line with triangles).

In Fig. 2 are shown the contributions of the individual condensates to M_X obtained from Eq. (15), From Fig. 2, it appears that the results oscillates around the perturbative result, and that the results

4 To leading order approximation in α_s, at which we are working, we do not consistently consider the running scale dependence of these parameters. We shall use here the values of the quark masses obtained within the same QCD spectral sum rules methods compiled in [19]. They are defined in the M S-scheme, and have obtained within the same truncation of the QCD series from different channels and by different authors.

√s0=4.1 GeV

pert.

4 where the integration limits are given by α_min = (1 − !

1 − 4m²_c/s)/2, α_max = (1 + !

1 − 4m²_c/s)/2 and (β_min = αm²_c)/(sα − m²_c). We have also included the dominant contributions from the dimension-five condensates:

ρ^mix(s) = m_c"qgσ.Gq¯ # 2⁶π⁴

αmax

"

αmin

dα

#

− 2

α (m²_c − α(1 − α)s) +

1−α

"

βmin

dβ $

(α + β)m²_c − αβs%

& 1

α + α + β β²

' ( ,

(12)

where the contribution of dimension-six condensates "g³G³# is neglected, since assumed to be suppressed by the loop factor 1/16π². The usual estimate "g³G³# $ 1GeV²"α_sG²# [19] would deserve to be checked in more detail. We have included the contribution of the dimension-six four-quark condensate:

ρ^{"qq¯ #2}(s) = m²_c"qq¯ #² 12π²

)s − 4m²_c

s , (13)

and (for completeness) a part of the dimension-8 condensate contributions ³: Π^mix"₁ ^{qq¯ #}(M²) = − m²_c"qg¯ σ.Gq#"qq¯ #

24π²

" 1

0

dα

#

1 + m²_c

α(1 − α)M ² − 1

2(1 − α) (

exp

#

− m²_c

α(1 − α)M² (

. (14)

III. LSR PREDICTIONS OF M^X

1.6 1.8 2.0 2.2 2.4 2.6 2.8

1 2 3 4

5 s₀^1/2 = 4.17 GeV

Pert + m_q + <qq>

+ <g²G²> + m_q<qq>

+ m₀²<qq>

+ <qq>² + m₀²<qq>²

Condensate/RHS

M² (GeV ²)

FIG. 1: The OPE convergence in the region 1.6 ≤ M² ≤ 2.8 GeV² for s¹₀^/2 = 4.17 GeV. We start with the perturbative contribution (plus a very small m_q contribution) and each subsequent line represents the addition of one extra condensate dimension in the expansion.

In order to extract the mass M_X without worrying about the value of the decay constant f_X, we take the derivative of Eq. (9) with respect to 1/M², divide the result by Eq. (9) and obtain:

M_X² =

* s⁰

4m²_c ds e^−s/M2 s ρ(s)

* s⁰

4m²_c ds e^−s/M2 ρ(s) . (15)

3 We should note that a complete evaluation of these contributions require more involved analysis including a non-trivial choice of the factorization assumption basis [38]. We wish that we can perform this analysis in the future.

terça-feira, 21 de julho de 15

5 This quantity has the advantage to be less sensitive to the perturbative radiative corrections than the individual moments. Therefore, we expect that our results obtained to leading order in α_s will be quite accurate.

In the numerical analysis of the sum rules, the values used for the quark masses and condensates are (see e.g. [19, 39, 40, 41]) ⁴ :

m_c(m_c) = (1.23 ± 0.05) GeV, m_b(m_b) = (4.24 ± 0.06) GeV,

m_u = 2.3 MeV, m_d = 6.4 MeV,

m_q ≡ (m_u + m_d)/2 = 4.3 MeV, "qq¯ # = −(0.23 ± 0.03)³ GeV³, m_s = 100 MeV, "ss¯ #/"qq¯ # = 0.8 ± 0.2,

"qg¯ σ.Gq# = m²₀"qq¯ # with m²₀ = 0.8 GeV²,

"g²G²# = 0.88 GeV⁴. (16)

We evaluate the sum rules in the range 2.0 ≤ M ² ≤ 2.8 for two values of s₀: s¹₀^/2 = 4.1 GeV, s¹₀^/2 = 4.2 GeV.

Comparing the relative contribution of each term in Eqs. (11) to (14), to the right hand side of Eq. (9) we obtain a quite good OPE convergence for M ² > 1.9 GeV², as can be seen in Fig. 1. This analysis allows us to determine the lower limit constraint for M² in the sum rules window. This figure also shows that, although there is a change of sign between dimension-six and dimension-eight condensates contributions, the contribution of the latter being smaller, where, we have assumed, in Fig. 1 to Fig. 4, the validity of the vacuum saturation for these condensates. The relatively small contribution of the dimension-eight condensates may justify the validity of our approximation, unlike in the case of the 5-quark current correlator, as noticed in [42]. However, the partial compensation of these two terms indicate the sensitivity of the central value of the mass prediction on the way the OPE is truncated.

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

M²(GeV ²) 3.4

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

M X (GeV)

FIG. 2: The OPE convergence for MX in the region 1.9 ≤ M² ≤ 2.8 GeV² for s¹₀^/2 = 4.1 GeV. We start with the perturbative contribution plus a very small mq contribution (long-dashed line) and each subsequent line represents the addition of one extra condensate dimension in the expansion: +"qq¯ # (solid line), +"g²G²# + mq"qq¯ # (dotted-line in top of the solid line), +"qq¯ #² (dashed line), +m²0"qq¯ # (solid line with triangles).

In Fig. 2 are shown the contributions of the individual condensates to M_X obtained from Eq. (15), From Fig. 2, it appears that the results oscillates around the perturbative result, and that the results

4 To leading order approximation in α_s, at which we are working, we do not consistently consider the running scale dependence of these parameters. We shall use here the values of the quark masses obtained within the same QCD spectral sum rules methods compiled in [19]. They are defined in the M S-scheme, and have obtained within the same truncation of the QCD series from different channels and by different authors.

√s0=4.1 GeV

pert.

+ ＜qq＞

4 where the integration limits are given by α_min = (1 − !

1 − 4m²_c/s)/2, α_max = (1 + !

1 − 4m²_c/s)/2 and (β_min = αm²_c)/(sα − m²_c). We have also included the dominant contributions from the dimension-five condensates:

ρ^mix(s) = m_c"qgσ.Gq¯ # 2⁶π⁴

αmax

"

αmin

dα

#

− 2

α (m²_c − α(1 − α)s) +

1−α

"

βmin

dβ $

(α + β)m²_c − αβs%

& 1

α + α + β β²

' ( ,

(12)

where the contribution of dimension-six condensates "g³G³# is neglected, since assumed to be suppressed by the loop factor 1/16π². The usual estimate "g³G³# $ 1GeV²"α_sG²# [19] would deserve to be checked in more detail. We have included the contribution of the dimension-six four-quark condensate:

ρ^{"qq¯ #2}(s) = m²_c"qq¯ #² 12π²

)s − 4m²_c

s , (13)

and (for completeness) a part of the dimension-8 condensate contributions ³: Π^mix"₁ ^{qq¯ #}(M²) = − m²_c"qg¯ σ.Gq#"qq¯ #

24π²

" 1

0

dα

#

1 + m²_c

α(1 − α)M ² − 1

2(1 − α) (

exp

#

− m²_c

α(1 − α)M² (

. (14)

III. LSR PREDICTIONS OF M^X

1.6 1.8 2.0 2.2 2.4 2.6 2.8

1 2 3 4

5 s₀^1/2 = 4.17 GeV

Pert + m_q + <qq>

+ <g²G²> + m_q<qq>

+ m₀²<qq>

+ <qq>² + m₀²<qq>²

Condensate/RHS

M² (GeV ²)

FIG. 1: The OPE convergence in the region 1.6 ≤ M² ≤ 2.8 GeV² for s¹₀^/2 = 4.17 GeV. We start with the perturbative contribution (plus a very small m_q contribution) and each subsequent line represents the addition of one extra condensate dimension in the expansion.

In order to extract the mass M_X without worrying about the value of the decay constant f_X, we take the derivative of Eq. (9) with respect to 1/M², divide the result by Eq. (9) and obtain:

M_X² =

* s⁰

4m²_c ds e^−s/M2 s ρ(s)

* s⁰

4m²_c ds e^−s/M2 ρ(s) . (15)

3 We should note that a complete evaluation of these contributions require more involved analysis including a non-trivial choice of the factorization assumption basis [38]. We wish that we can perform this analysis in the future.

terça-feira, 21 de julho de 15

5 This quantity has the advantage to be less sensitive to the perturbative radiative corrections than the individual moments. Therefore, we expect that our results obtained to leading order in α_s will be quite accurate.

In the numerical analysis of the sum rules, the values used for the quark masses and condensates are (see e.g. [19, 39, 40, 41]) ⁴ :

m_c(m_c) = (1.23 ± 0.05) GeV, m_b(m_b) = (4.24 ± 0.06) GeV,

m_u = 2.3 MeV, m_d = 6.4 MeV,

m_q ≡ (m_u + m_d)/2 = 4.3 MeV, "qq¯ # = −(0.23 ± 0.03)³ GeV³, m_s = 100 MeV, "ss¯ #/"qq¯ # = 0.8 ± 0.2,

"qg¯ σ.Gq# = m²₀"qq¯ # with m²₀ = 0.8 GeV²,

"g²G²# = 0.88 GeV⁴. (16)

We evaluate the sum rules in the range 2.0 ≤ M ² ≤ 2.8 for two values of s₀: s¹₀^/2 = 4.1 GeV, s¹₀^/2 = 4.2 GeV.

Comparing the relative contribution of each term in Eqs. (11) to (14), to the right hand side of Eq. (9) we obtain a quite good OPE convergence for M ² > 1.9 GeV², as can be seen in Fig. 1. This analysis allows us to determine the lower limit constraint for M² in the sum rules window. This figure also shows that, although there is a change of sign between dimension-six and dimension-eight condensates contributions, the contribution of the latter being smaller, where, we have assumed, in Fig. 1 to Fig. 4, the validity of the vacuum saturation for these condensates. The relatively small contribution of the dimension-eight condensates may justify the validity of our approximation, unlike in the case of the 5-quark current correlator, as noticed in [42]. However, the partial compensation of these two terms indicate the sensitivity of the central value of the mass prediction on the way the OPE is truncated.

1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

M²(GeV ²) 3.4

3.5 3.6 3.7 3.8 3.9 4 4.1 4.2

M X (GeV)

FIG. 2: The OPE convergence for MX in the region 1.9 ≤ M² ≤ 2.8 GeV² for s¹₀^/2 = 4.1 GeV. We start with the perturbative contribution plus a very small mq contribution (long-dashed line) and each subsequent line represents the addition of one extra condensate dimension in the expansion: +"qq¯ # (solid line), +"g²G²# + mq"qq¯ # (dotted-line in top of the solid line), +"qq¯ #² (dashed line), +m²0"qq¯ # (solid line with triangles).

In Fig. 2 are shown the contributions of the individual condensates to M_X obtained from Eq. (15), From Fig. 2, it appears that the results oscillates around the perturbative result, and that the results

4 To leading order approximation in α_s, at which we are working, we do not consistently consider the running scale dependence of these parameters. We shall use here the values of the quark masses obtained within the same QCD spectral sum rules methods compiled in [19]. They are defined in the M S-scheme, and have obtained within the same truncation of the QCD series from different channels and by different authors.

√s0=4.1 GeV

pert.

+ ＜qq＞ + ＜g²G²＞

4 where the integration limits are given by α_min = (1 − !

1 − 4m²_c/s)/2, α_max = (1 + !

1 − 4m²_c/s)/2 and (β_min = αm²_c)/(sα − m²_c). We have also included the dominant contributions from the dimension-five condensates:

ρ^mix(s) = m_c"qgσ.Gq¯ # 2⁶π⁴

αmax

"

αmin

dα

#

− 2

α (m²_c − α(1 − α)s) +

1−α

"

βmin

dβ $

(α + β)m²_c − αβs%

& 1

α + α + β β²

' ( ,

(12)

where the contribution of dimension-six condensates "g³G³# is neglected, since assumed to be suppressed by the loop factor 1/16π². The usual estimate "g³G³# $ 1GeV²"α_sG²# [19] would deserve to be checked in more detail. We have included the contribution of the dimension-six four-quark condensate:

ρ^{"qq¯ #2}(s) = m²_c"qq¯ #² 12π²

)s − 4m²_c

s , (13)

and (for completeness) a part of the dimension-8 condensate contributions ³: Π^mix"₁ ^{qq¯ #}(M²) = − m²_c"qg¯ σ.Gq#"qq¯ #

24π²

" 1

0

dα

#

1 + m²_c

α(1 − α)M ² − 1

2(1 − α) (

exp

#

− m²_c

α(1 − α)M² (

. (14)

III. LSR PREDICTIONS OF M^X

1.6 1.8 2.0 2.2 2.4 2.6 2.8

1 2 3 4

5 s₀^1/2 = 4.17 GeV

Pert + m_q + <qq>

+ <g²G²> + m_q<qq>

+ m₀²<qq>

+ <qq>² + m₀²<qq>²

Condensate/RHS

M² (GeV ²)

FIG. 1: The OPE convergence in the region 1.6 ≤ M² ≤ 2.8 GeV² for s¹₀^/2 = 4.17 GeV. We start with the perturbative contribution (plus a very small m_q contribution) and each subsequent line represents the addition of one extra condensate dimension in the expansion.

In order to extract the mass M_X without worrying about the value of the decay constant f_X, we take the derivative of Eq. (9) with respect to 1/M², divide the result by Eq. (9) and obtain:

M_X² =

* s⁰

4m²_c ds e^−s/M2 s ρ(s)

* s⁰

4m²_c ds e^−s/M2 ρ(s) . (15)

3 We should note that a complete evaluation of these contributions require more involved analysis including a non-trivial choice of the factorization assumption basis [38]. We wish that we can perform this analysis in the future.

terça-feira, 21 de julho de 15

No documento Marina Nielsen - MESONPI (páginas 37-54)