Jul 21, 2016 - By adopting the nonlinear realization of chiral symmetry, a holographic model for the baryon octet is proposed. The mass spectra of the...

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arXiv:1607.06197v1 [hep-ph] 21 Jul 2016

By adopting the nonlinear realization of chiral symmetry, a holographic model for the baryon octet is proposed. The mass spectra of the baryon octet and their low-lying excited states are calculated, which show good consistency with experiments. The couplings of pion to nucleons are derived in two gauges and are shown to be equivalent with each other. It also shows that only derivative couplings of pion to nucleons appear in this holographic model. The coupling constant is then calculated. I.

INTRODUCTION

In the recent decades, there have been many studies on AdS/QCD since the discovery of AdS/CFT [1–3]. Various holographic QCD models have been constructed, either in the top-down approach [4–6] or in the bottomup approach [7–9]. Many researches have been focused on the phenomenology of low-energy QCD, such as the hadron spectrum [10–13], the deconfining or chiral phase transition [14–18], and the hadron form factors [9] etc. The AdS/QCD method has shown powerful ability in the description of low-energy QCD phenomenon. In this paper, we focus on the baryon properties and try to construct a sensible holographic model for the baryon octet, based on the AdS/QCD model for mesons [7, 8], which have shown superiority in respect of phenomenology. Indeed, considerable efforts have been made in the construction of the holographic baryon models, such as in [19, 20]. In [21], the authors built a holographic model for spin- 21 baryons, which naturally incorporates the important phenomenon of parity-doublet [22]. However, one large limitation is that they only address the parity-doublet of nucleons and their excited states. To generalize the model, one cannot bypass the baryons with heavy quarks, which makes the issue about chiral symmetry troublesome. The meson physics in three-flavor case has been considered in the holographic framework in [23]. For baryons with three flavors, chiral symmetry is believed to be broken from the beginning and the picture of chiral symmetry breaking in [21] seems to be invalid. Note that the holographic construction in [21] shares similarities with the linear sigma model and the massless modes of baryons before chiral symmetry breaking form the representation of the chiral group. However, for heavy baryons, it is more suitable to accept the nonlinear realization of chiral symmetry in the construction of baryon models [24]. Following this way, a holographic model for the baryon octet will be given. The whole mass spectra of the baryon octet and their low-lying excited states will be calculated, which show good agreement with experiments. we also show that this holographic model only contains gradient couplings of pion

∗ Electronic

address: [email protected]

with baryon fields, at least in the chiral limit, which is in parallel with the nonlinear construction of the effective baryon model in the 4D field-theoretic method [25].

II.

OUTLINE OF THE MODEL

As in [7, 8], we take a slice of AdS5 space-time with the following metric ansatz: ds2 = e2A(z) ηµν dxµ dxν − dz 2 ,

ǫ 6 z 6 zm ,

(1)

where A(z) = −ln z, ηµν = diag(+, −, −, −) and the IR brane zm = 1/ΛQCD . To produce the possible parity-doubling pattern of excited baryon states, we follow [21] and introduce two bulk baryon fields B1,2 which correspond to the chiral baryon operators OL,R in the boundary respectively, but with the isodoublet in two-flavor case replaced by the baryon octet in three-flavor case: 8 a a X λ B √ = B= 2 a=1

Σ0 √ 2

+ Σ− Ξ−

Λ √ 6

Σ+ 0

Σ + −√ 2 Ξ0

p Λ √ 6

n . (2) 2Λ −√ 6

The holographic model is formulated on the basis of the chiral gauge group G = SU (3)L × SU (3)R , which is broken spontaneously into a vectorial subgroup H = SU (3)V . The bulk action comprises two sectors, one of which containing the covariant kinetic terms and the bulk mass terms can be written as Sb1 =

Z

2

√ X ¯ j Bj , ¯j i eM ΓA ∇M Bj − m5 B d5 x g Tr B A j=1

(3) B where the metric gMN = eA e η with the vielbein M N AB 1 A A µ 5 eA M , the Dirac matrices Γ = (γ , −iγ ) satisfy M = zη A B AB Γ ,Γ = 2η , and the mass of the bulk baryon fields m5 can be determined by the AdS/CFT correspondence as m5 = ± 25 with the sign dictated by the analysis of the chirality of baryon fields (conventionally, we set m5 = 52 for B1 and m5 = − 25 for B2 ) [21]. The Lorentz and gauge covariant derivatives of the bulk baryon fields have the

2 form as follows i AB ∇M B1 = ∂M B1 + ωM ΓAB B1 − i[LM , B1 ], (4) 4 i AB ΓAB B2 − i[RM , B2 ], (5) ∇M B2 = ∂M B2 + ωM 4 AB A B B where the spin ωM = ∂z A(z)(δM δz −δzA δM ), A connection 1 ΓAB = 2i Γ , ΓB is the Lorentz generator for spinor fields, and LM (RM ) can be formulated by the gauged Maurer-Cartan 1-forms as

LM = i ξ † DM ξ,

RM = i ξDM ξ †

(6)

with the covariant derivatives defined by DM ξ = ∂M ξ − i(AL )M ξ, DM ξ † = ∂M ξ † − i(AR )M ξ † ,

s

with vu = mu z + σu z 3 and vs = ms z + σs z 3 , where mu , σu denote the mass and chiral condensate of u, d quarks, and ms , σs denote the ones of s quark [7, 8]. In terms of Eq.(15), χ+ can be reduced to χ+ = X 0 + T 0 S0 + T a Sa ,

(7) (8)

M iTa πa where AM L , AR are the chiral gauge fields and ξ = e is the element of the coset space G/H (Ta = λa /2 is the generator of SU (3) group with λa the Gell-Mann matrices and πa denotes the bulk pseudoscalar field). We can see that LM , RM contain both the gauge fields and the pseudoscalar fields, and are indeed the only terms containing these fields in the model, which is different from that in [21]. Note that in the nonlinear representation the bulk baryon fields Bj only transforms under subgroup H as Bj′ = HBj H † . The element ξ(π) of the coset space G/H transforms under the chiral gauge group G as ξ ′ = Lξ(π)H † = Hξ(π)R† with L ∈ SU (3)L , R ∈ SU (3)R , M and the chiral gauge fields AM L and AR transform in the following way:

(AL )′M = L(AL )M L† + iL∂M L† , (AR )′M = R(AR )M R† + iR∂M R† .

where S0 , Sa represent the scalar singlet and octet, X0 is the vacuum expectation value (VEV) of the scalar field X and is presumed to be of the following form: 1 vu vu X0 = (16) 2 v

(9) (10)

(17)

which is only associated with the scalar mesons. As in [21], the bulk baryon fields Bj can be written in the chiral form: Bj = BjL +BjR with BjL,R decomposed by the Kaluza-Klein (KK) and Fourier decomposition as X Z d4 p (n) (n) e−ipx fjL,R (z)ψL,R (p), (18) BjL,R (x, z) = 4 (2π) n where fjL,R are the Kaluza-Klein profiles of the bulk baryon fields, ψL,R (p) are the 4D spinors satisfying γ 5 ψL (p) = ψL (p), γ 5 ψR (p) = −ψR (p) and /p ψL,R (p) = |p|ψR,L (p). Now in terms of the profiles fjL,R (note that the superscript n has been neglected), the equation of motion (EOM) for the baryon octet can be derived from the variation of the bulk action as follows ! 9 − v(z) ∂z − 2z f1L f1R 2z = −|p| , 1 f2L f2R − v(z) ∂z + 2z 2z ! v(z) 1 f1R f1L ∂z + 2z 2z = |p| , (19) v(z) 9 f f2L 2R ∂z − 2z 2z

Then the group transformations of LM and RM can be obtained as

where v(z) has the following form for different particles in the baryon octet: ′ † † LM = HLM H + iH∂M H , (11) (3 c2 − c1 ) vu (z) for (p, n), ′ † † RM = HRM H + iH∂M H . (12) (2 c − 4 c1 ) v (z) + ( c1 + c ) v (z) for Λ, 2 u 2 s 3 3 v(z) = The other sector of the bulk action which generates 2 c v (z) + (c − c ) v (z) for Σs, 2 1 s 2 u chiral symmetry breaking can be formulated as (c2 − c1 ) vu (z) + 2 c2 vs (z) for Ξs. Z (20) √ ¯1 {χ+ , B2 } + c2 Tr B ¯1 [χ+ , B2 ] Sb2 = − d5 x g c1 Tr B From the parity transformations of the baryon fields, we get f1L = f2R and f1R = −f2L for even-parity states, ¯1 B2 Trχ+ + h.c. , (13) while f = −f and f = f for odd-parity states +(c2 − c1 )Tr B 1L 2R 1R 2L [21, 26]. As in [21], by solving Eqs.(19) with the boundwhere h.c. denotes the Hermitian conjugate, χ+ is reary conditions f1L (ǫ → 0) = 0 and f1R (zm ) = 0, we lated to the three-flavor generalization of the bulk scalar obtain the mass spectra of the baryon octet and their field X (see also Sec. IV) by excited states. 1 † † † χ+ = (ξ Xξ + ξX ξ), (14) 2 and χ+ transforms in the nonlinear representation as χ′+ = Hχ+ H † . The bulk scalar field X can be defined as follows X ≡ ξ (X0 + T0 S0 + Ta Sa ) ξ,

(15)

III. MASS SPECTRA OF THE BARYON OCTET AND THEIR EXCITED STATES

To get the baryon spectrum, we first fix the same parameters as the ones in [21] by fitting the masses of

3 the nucleons (p, n) and their first excited state N(1440): mu = 0, σu = (198 MeV)3 , zm = (205 MeV)−1 and 3 c2 − c1 = 14.8. Then we take ms = 100 MeV and σs = σu = (198 MeV)3 , and the last parameter c2 fixed by the mass of Λ to be c2 = 4.52. The experimental and numerical results for the baryon-octet spectrum are shown in Tabel I, where we can see that the model predictions for the masses of Σs and Ξs match with experimental data very well. It can be easily shown that the mass hierarchy of different particles in the baryon octet is exclusively due to the large mass of the strange quark. Baryon octet p, n

Λ

Σs

Ξs

Exp.(MeV) 939 1115 1190 1320 Model (MeV) 939 1115 1184 1321 ∗

∗

TABLE I: Experimental value and model predictions for the mass spectrum of the baryon octet. * indicates input value, and the experimental data are taken from [27].

1st even

p, n

Λ

Σs Ξs

Exp.(MeV) 1440∗ 1600 1660 — Model (MeV) 1440 1548 1598 — 1st odd

p, n

Λ

Σs Ξs

Exp.(MeV) 1535 1405 1620 — Model (MeV) 1505 1588 1627 — TABLE II: Experimental value and model predictions for the mass spectra of the first excited even-parity and odd-parity states.

Table II lists the masses of the first excited baryonoctet states with both even and odd parities, which are also consistent with experiments, except for the first oddparity state of Λ. It should be noted that the masses of the excited states of Ξs are not given due to the lack of experimental data. For the masses of higher excited states, this model cannot give consistent results with experiments, the reason of which might be attributed to the sharp IR cutoff of the fifth dimension of AdS5 , which can be remedied through a soft dilaton term [26, 28]. It should be remarked that the baryon masses in this model are acquired by a different way from that in [21], where the VEV of the scalar field X in a Yukawa coupling term breaks the chiral symmetry and gives the nucleon mass. In our model with a nonlinear realization of chiral symmetry, the chiral symmetry of the baryon action per se has been broken from the beginning, which can be seen from the group transformations of the baryon fields Bj that keep the action invariant. However, it is still the VEV of the scalar field χ+ that contributes to the mass spectrum of the baryon octet. A unique feature of this holographic framework is that one cannot separate the explicit chiral symmetry break-

ing terms from the ones generating spontaneous symmetry breaking in the bulk action as the quark mass and the chiral condensate are sewed into a single term, i.e., the VEV of the bulk scalar field X0 . In the three-flavor case, X0 does not vanish due to the large mass of s quark, even if there is no spontaneous chiral symmetry breaking. It leads us to accept the nonlinear realization of chiral symmetry. However, as the bulk baryon action does not realize the full chiral symmetry in the nonlinear representation, the baryon states cannot form the Wigner-Weyl modes of the chiral group, which might cause some vagueness for the explanation of the parity-doublet pattern of excited baryons. We will not go into details about the reason of the parity-doubling phenomenon, which is still inconclusive [22].

IV.

THE PION-NUCLEON COUPLING

In the above, we construct a holographic model for the baryon octet using a nonlinear realization of chiral symmetry, which has been proved more suitable for describing baryons with larger mass [24]. The explanation of the parity-doubling pattern in terms of two baryon fields B1,2 with inverse chirality (parity) has been incorporated in this framework, though the bulk baryon action does not realize the full chiral symmetry. Another characteristic of this holographic model is that there are only derivative interactions of pions with baryons, which is also a desired feature in the 4D effective baryon models [24, 25]. Next, we will derive the expressions for the pion-nucleon couplings and show directly the nature of derivative interactions. As the pion wave function is necessary for the derivation of the pion-nucleon coupling, we first present the bulk action of the meson sector which has been studied in [7, 8]: √ 1 d5 x g Tr |DX|2 + 3|X|2 − 2 (FL2 + FR2 ) , 4g5 (21) where the covariant derivative of the bulk scalar field X is M DM X = ∂ M X−iAM L X+iXAR and the chiral gauge field N M MN M N strength is FL,R = ∂ AL,R − ∂ N AM L,R − i[AL,R , AL,R ]. The vector and axial-vector fields are defined as V M = 1 M M M M = 21 (AM L − AR ) respectively. 2 (AL + AR ) and A Note that the scalar part of the meson action (21) can also be recast into a nonlinear form using the Eq.(15), and the results do not depend on which forms we used. Below we will work in two gauges, as has been used in [7] and [8] respectively. The results in both gauges will be shown to be equivalent with each other, from which the gauge-independence manifests itself obviously. Let us first work in the A5 = 0 gauge accepted in [8]. The EOM of the pseudoscalar meson in this gauge can Sm =

Z

4 be derived as ∂z fϕ g2 v2 ∂z + 5 3u (fπ − fϕ ) = 0, (22) z z g2 v2 −q 2 ∂z fϕ + 5 2u ∂z fπ = 0, (23) z where fπ (z) comes from the KK decomposition of the bulk pion field, and fϕ (z) comes from that of the radial component of axial-vector field (Aaµ = Aaµ⊥ + ∂µ ϕa ). As in the calculation of baryon spectrum, we only consider the mu = 0 case in which the pion as Nambu-Goldstone boson has zero mass (q 2 = 0), so we have fπ′ (z) = 0, which also indicates with the normalization condition (fπ (z → 0) = 0) that fπ (z) = 0. Then Eq.(22) can be reduced to the following form: g2 v2 ∂z fϕ − 5 2u fϕ = 0. z ∂z (24) z z The pion-nucleon couplings are supplied by the covariant kinetic term of the bulk baryon action (3), which is the unique term containing the pseudoscalar and gauge fields, as noted above. In the A5 = 0 gauge, the interaction terms (Lagrange density) of pion and nucleons can be extracted from the action (3) as (1) ¯ 5 ∂π / a ta ψ LπN N = z (fπ − fϕ ) |f1L |2 − |f2L |2 ψγ ∗ ∗ ¯ 5 π a ta ψ +∂z fπ (f1L f1R − f2L f2R ) i ψγ ¯ 5 ∂/π a ta ψ, (25) = −zfϕ |f1L |2 − |f2L |2 ψγ

where the superscript * denotes complex conjugate, ta = σa a 2 with σ the Pauli matrices and ψ denotes the 4D isodoublet of nucleons. The Lagrange density (25) shows obviously the derivative coupling of pion to nucleons, which is a characteristic feature of the nonlinear representation. Note that it is different from the holographic baryon model proposed in [21], where a Yukawa coupling term was introduced for chiral symmetry breaking, which mimics the structure of the linear sigma model. For comparison, we take another gauge-fixing scheme used in [7] to work out the coupling of pion to nucleons, and show that the result is equal to the one in the A5 = 0 gauge. The pion meson in this gauge is associated with the axial-vector and pseudoscalar fields which can be extracted (up to quadratic terms) from the action (21) as follows: Z 1 5 SπA = d x − 2 (∂µ Aaν ∂ µ Aaν − ∂µ Aaν ∂ ν Aaµ 2g5 z v2 −(∂µ Aa5 − ∂z Aaµ )2 + u3 (∂µ π a − Aaµ )2 2z (26) −(∂z π a − Aa5 )2 .

To eliminate the mixing terms between Aa5 (π a ) and a Aµ , we add the following gauge-fixing term: a 2 Z −d5 x A5 ξA g52 vu2 a µ a Sg.f. = ∂ A − ξ z∂ + π . A z µ 2ξA g52 z z z2 (27)

In the unitary we take ξA → ∞ to decouple the a gauge, 2 A5 g52 vu term z∂z z − z2 π a from the Aaµ terms. Note that the orthogonal combination of Aa5 and π a remains massless if a A5 g2 v2 − 5 2u π a = 0. z ∂z (28) z z With the ansatz Aa5 (x, z) = f0 (z)π a (x), Eq.(28) becomes g2 v2 f0 (z) − 5 2u fπ (z) = 0. z ∂z (29) z z After fixing the gauge, the pseudoscalar part of the action (26) can be written as Z 1 vu2 a µ a Sπ = d5 x ∂ A ∂ A + ∂µ π a ∂ µ π a µ 5 5 2g52 z 2z 3 v2 (30) − u3 (∂z π a − Aa5 )2 , 2z from which one can see that the condition for pion to be massless is ∂z π a − Aa5 = 0, which implies f0 (z) = fπ′ (z). In this gauge, we can also derive the coupling of pion to nucleons as (2) ¯ 5 ∂/π a ta ψ LπN N = z fπ |f1L |2 − |f2L |2 ψγ ∗ ∗ ¯ 5 π a ta ψ + (∂z fπ − f0 ) (f1L f1R − f2L f2R ) i ψγ ¯ 5 ∂π / a ta ψ. (31) = zfπ |f1L |2 − |f2L |2 ψγ

To prove the equivalence of LπN N in Eqs.(25) and (31), we note that Eq.(29) is just the same as Eq.(24) in view of the relation f0 (z) = fπ′ (z) derived in the unitary gauge. It indicates that the couplings of pion with nucleons are equivalent with each other in the two gauges. The gradient coupling constant can be defined as Z zm gA dz (32) ≡ fπ (z) |f1L (z)|2 − |f2L (z)|2 , 4 Fπ z 0 where Fπ denotes the pion decay constant, f1L (z) and f2L (z) are normalized by the condition: Z zm dz (33) |f1L (z)|2 + |f2L (z)|2 = 1, 4 z 0

and fπ (z) is solved from Eq.(29) with the normalization of f0 (z): " 2 # Z zm f0 1 2 z3 dz = 1. (34) ∂z 2 z f0 + g 4 v 2 g z 0 5 5 The pion decay constant Fπ can be computed from the meson action (21) as Fπ2 = −(g52 z)−1 ∂z A(0, z)|z=ǫ with A(q, z) the bulk-to-boundary propagator of the axialvector field [8]. Using the parameters fixed by the baryon spectrum, we get Fπ ≃ 54.5MeV and gA ≃ 0.33, which are too small compared with the experimental value: Fπ ≃ 92.4MeV and gA ≃ 1.27 [27].

5 The discrepancy between the holographic model for spin- 12 baryons and the one for mesons has been revealed in [21], where the masses of N (1440) and N (1535) cannot be obtained from the fitting of the meson sectors [7], for which they argued by the possibility of the nonvanishing anomalous dimension of baryons. Another important reason for this inconsistency, as has been noted above, is that the sharp IR cutoff of the AdS5 metric causes no linear confinement which is manifest in the hadron spectrum [26, 28]. The coupling of pion to nucleons may also receive contributions from non-minimal gauge interactions, such as the following magnetic gauge coupling term [29]: ¯1 ΓMN (FL )MN B1 − B ¯2 ΓMN (FR )MN B2 , L = iκ B (35) where the transformed gauge field strengths are FLMN = MN ξ † FLMN ξ and FR = ξFRMN ξ † . Note that this term also only generates gradient coupling of pion to nucleons. V.

SUMMARY AND CONCLUSION

In this paper, we propose a holographic model for the baryon octet with a nonlinear realization of chiral symmetry on the basis of the AdS/QCD model for mesons [7, 8]. The mass spectra of the baryon octet and their low-lying excited states have been calculated, which show good agreement with experiments, especially for the ground states. However, the model cannot give consistent results with experiments for highly excited baryon states, which might be attributed to the sharp IR cutoff of the AdS5 metric [26, 28]. We have also shown that the model only contains gradient coupling of pion to nucleons, as in the 4D effective baryon models with nonlinear realization of chiral symmetry [24]. However, the derivative coupling constant is much smaller than that obtained from experimental measurements, which might be rooted

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VI.

ACKNOWLEDGEMENT

The author thanks Doctor Danning Li and Professor Yue-Liang Wu for valuable discussions and generous help, and also thanks Professor M. Traini for useful communications. In addition, the author is also grateful to Professor Cong-Feng Qiao for the application of postdoctoral position.

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