Bled Workshops in Physics Vol. 17, No. 1 p. 20
A
Proceedings of the Mini-Workshop
Quarks, Hadrons, Matter
Bled, Slovenia, July 3 - 10, 2016

Angular momentum content of the p(1450) from chiral lattice fermions*
C. Rohrhofer, M. Pak, L. Ya. Glozman
Institut für Physik, FB Theoretische Physik, Universität Graz, Universitatsplatz 5, 8010 Graz, Austria
Abstract. We identify the chiral and angular momentum content for the leading quarkantiquark Fock component for the p(770) and p(1450) mesons using a lattice simulation with chiral fermions. Our analysis shows that in the angular momentum basis the p(770) is a 3Si state, in accordance with the quark model. The p(1450) is a 3Di state, showing that the quark model wrongly assumes the p(1450) to be a radial excitation of the p(770).
1 Introduction
An interesting question in hadronic physics is the origin of spin and distribution of angular momentum. How the spin of a hadron is generated, and by which constituents it is carried, is a priori not clear. In the non-relativistic, constituent quark model [2], which has been quite successful in delivering a classification scheme for the low-lying hadron spectrum, the spin of a hadron is assigned solely to its valence quarks. Being an effective classification scheme, it does not care about foundations in terms of underlying QCD dynamics. Despite its successes the non-relativistic description clearly has limitations.
In this project we investigate the angular momentum content of the p(770) and p(1450) mesons. In the spectroscopic notation n 2S+1 lj the p(770) is assigned to the 1 3S1 state by the quark model. The p(1450) is assigned to the 2 3S1 state, hence being the first radial excitation of the p(770). However, this assumption is by far not clear from the underlying QCD dynamics, and is an output of the non-relativistic potential description of a meson as a two-body system.
The angular momentum content of the leading quark-antiquark Fock components of mesons can in principle be identified by lattice simulations. Studies like [3], which rely on heavy quarks for the non-relativistic reduction of hadrons, find good agreement with the quark model classification. However, there is an alternative approach to project non-perturbative lattice results onto the quark model assuming ultra-relativistic quarks. Latter method, which is explained and has been applied in previous studies [4-8], makes use of the chiral-parity group and an unitary transformation to the 2S+1J basis.
Main ingredients to such an investigation are the overlap factors of operators obtained in lattice calculations. In our study it is crucial that these operators
* Talk delivered by C. Rohrhofer
Angular momentum content of the p(1450)
21
form a complete set with respect to the chiral-parity group. From these overlap factors the chiral content of a state can be identified, and using the unitary transformation also the angular momentum content. Since the chiral properties are important for such a study, we need a proper lattice fermion discretization, which respects chiral symmetry. For this purpose we use overlap fermions, which distinguishes the present study from the previous ones.
2 Method and Simulation
The full details of this study, its methodology and simulation parameters, can be found in the main paper [1] and references therein. Here we present the idea and summarize the most important components.
To generate states with p quantum numbers (1,1 ) two different local interpolators can be used, which belong to two distinct chiral representations
JV(x)= Y(x)(Ta ® Yi)¥(x) G (0,1) 0 (1,0)	(1)
JP(x) = Y(x)(Ta ®yVMx) g (1/2,1/2)b.	(2)
We denote them according to their Dirac structure as vector (V) and pseudotensor (T) interpolators. In a next step we connect the chiral basis to the angular momentum basis with quantum numbers isospin I and 2S+1 lj. For spin-1 isovector mesons there are only two allowed states |1;3 S1) and |1;3 D1), which are connected to the chiral basis by a unitary transformation:
lp(0,i)®(i,0)> = |1;3 Si) + 11;3 °1> ,	(3)
IP(V2,i/2)b> = \/ïli;3Si)-^n;3Di> .	(4)
Note that the operators (1),(2) form a complete and orthogonal basis with respect to the chiral group. Through the unitary transformation (3),(4) they also form a complete and orthogonal basis with respect to the angular momentum content.
On the lattice we evaluate the correlators ( J(t) J (0)}. We apply the variational technique, where different interpolators are used to construct the correlation matrix (Ji(t) JÎa(0)} = C(t)lm. By solving the generalized eigenvalue problem
C(t)lmumn) = A(n)(t,tc)C(to)imumi)	(5)
the masses of states can be extracted in a standard way. Denoting aln) = (0| Jl |n} as the overlap of interpolator Jl with the physical state |n}, the relative weight of the chiral representations is now given by
C(t)ljujn) = o^ C(t)kjujn) okn) .
(6)
We can extract the ratio aV/aT for each state n. Then via the unitary transformation (3),(4) we arrive at the angular momentum content of the p mesons.
22
C. Rohrhofer, M. Pak, L. Ya. Glozman
1
0,5 0
-0,5 -1
Fig. 1. Partial wave content of p mesons in dependence of the relative chiral contribution av/aT, which are connected via transformation (3),(4).
avt
For any lattice simulation an intrinsic resolution scale is set by the lattice spacing a. This means that probing the hadron structure with point-like sources gives results at a scale fixed by the ultraviolet regularization a.
In order to measure the structure close to the infrared region we introduce a different resolution scale by smearing the sources of the quark propagators. We use four different smearing widths in this study. The radius ct of a given source S(x; x0) is calculated by
-xo)2|S(x;xo)|2 LJS(x; xo)|2
(7)
where we define the resolution scale as R = 2aa. The smeared profiles of the sources used in this study are pictured in Figure 2. The Ultra Wide source does not resolve details smaller than ~ 0.9 fm and marks our infrared end, where we ultimatively extract the resolution scale dependent quantities.
12345678 12345678 r / a	r / a
Fig. 2. Different source profiles. a is their radius in lattice units.
2
CT
Angular momentum content of the p(1450)
23
3 Results
To study the ratio aV/aT at different resolution scales R we solve the eigenvalue problem (5) with operators (1) and (2) and four different smearings. Then using (6) we extract the ratio aV/aT as a function of R. In Fig. 4 we show the ratio aV/aT at different resolution scales R. We find a clear R-effect for the ratio aV/aT: both p and p' states are linear dependent on the resolution scale.
Fig. 3. Normalized eigenvalues and effective masses.
2 1,5 1
I- 0,5 (0
""> 0 (0
-0,5 -1 -1,5 -2
I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I 1 I
L...........................I /T*I I
1	I Wl


I
I . I . I . I . I . I . I . I . I
o	OP
o	OP'
	P''
0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 1,1 1,2
R / fm
Fig. 4. av/aT ratio for different resolutions.
Using now transformations (3),(4) we find:
|p(770)) =+(0.998 ± 0.002) |3St >	(8)
(0.05 ± 0.025) |3Dt ) ,

|p(1450)> =-(0.106 ± 0.09 ) |3St )	(9)
-	(0.994 ± 0.005) |3Dt ) .
|p(1700)> =+(0.99 ± 0.01) |3St )	(10)
-	(0.01 ± 0.12) |3Dt).
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C. Rohrhofer, M. Pak, L. Ya. Glozman
The ground state p is therefore practically a pure 3Si state, in agreement with the potential quark model assumption.
The first excited p is, however, a 3Di state with a very small admixture of a 3Si wave. The second excited state is almost pure 3Si state. The latter results are in clear contradiction with the potential constituent quark model that attributes the first excited state of the p-meson to a radially excited 3Si state and the next excited state to a 3Di state.
Acknowledgments
The speaker wishes to thank the organizers for the warm welcome and hospitality during the workshop. We thank the JLQCD collaboration for supplying us with the Overlap gauge configurations, M. Denissenya and C.B. Lang for discussions and help. The calculations have been performed on local clusters at ZID at the University of Graz and the Graz University of Technology. Support from the Austrian Science Fund (FWF) through the grants DK W1203-N16 and P26627-N27 is acknowledged.
References
1.	C. Rohrhofer, M. Pak and L. Y. Glozman, arXiv:1603.04665 [hep-lat].
2.	K. A. Olive et al. [Particle Data Group Collaboration], Chin. Phys. C 38 (2014) 090001. doi:10.1088/1674-1137/38/9/090001
3.	J. J. Dudek, Phys. Rev. D 84 (2011) 074023 doi:10.1103/PhysRevD.84.074023 [arXiv:1106.5515 [hep-ph]].
4.	L. Y. Glozman, C. B. Lang and M. Limmer, Phys. Rev. Lett. 103, 121601 (2009) doi:10.1103/PhysRevLett.103.121601 [arXiv:0905.0811 [hep-lat]].
5.	L. Y. Glozman, C. B. Lang and M. Limmer, Few Body Syst. 47 (2010) 91 [arXiv:0909.2939 [hep-lat]].
6.	L. Y. Glozman, C. B. Lang and M. Limmer, Phys. Rev. D 82, 097501 (2010) doi:10.1103/PhysRevD.82.097501 [arXiv:1007.1346 [hep-lat]].
7.	L. Y. Glozman, C. B. Lang and M. Limmer, Phys. Lett. B 705 (2011) 129 [arXiv:1106.1010 [hep-ph]].
8.	L. Y. Glozman, C. B. Lang and M. Limmer, Prog. Part. Nucl. Phys. 67 (2012) 312 [arXiv:1111.2562 [hep-ph]].