The shape of quark flavors

Shinsuke Kawai; Nobuchika Okada

hep-phSep 19, 2026

The shape of quark flavors

Authors: Shinsuke Kawai, Nobuchika Okada

Organizations: Faculty of Science, Yamagata University, 1-4-12 Kojirakawa-machi, Yamagata, 990-8560 Japan · Department of Physics and Astronomy, University of Alabama, Tuscaloosa, Alabama, AL35487 USA

Abstract

We construct the Yukawa couplings of the quark sector as overlap integrals of Gaussian wave functions in extra spatial dimensions. Assuming that the wave function of each chiral fermion is localised at a point in the extra dimensions, while that of the Higgs doublet is flat, the Yukawa matrix elements are determined by the relative positions of the three left-handed quark doublets, $q_1$ , $q_2$ , $q_3$ , and the six right-handed quark singlets, $u_1$ , $u_2$ , $u_3$ , $d_1$ , $d_2$ , $d_3$ . We employ numerical optimisation techniques from machine learning to identify field configurations that satisfy all current experimental constraints on the quark Yukawa sector. In this framework, the large hierarchy of quark masses arises naturally, since the overlap integrals of Gaussian wave functions are exponentially sensitive to ${\mathcal O}(1)$ separations in the extra dimensions.

Figures & tables

Sample	Optimised parameter values
	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$	$x_{6}$	$x_{7}$	$x_{8}$	$x_{9}$	$x_{10}$	$x_{11}$	$x_{12}$
#1	-0.74222	1.3478	5.4744	-4.4582	6.8517	-0.022080	4.4409	6.3030	-3.5100	-4.5038	3.9302	-4.1567
#2	-1.6859	1.6635	-0.17823	0.20985	0.056992	6.8938	3.5297	6.3481	-4.6365	-4.2183	3.9720	-4.1796
#3	-0.83353	1.4553	-1.0119	-4.6180	6.8937	0.056854	6.3481	3.5295	-4.6370	-4.2187	3.9719	-4.1800
#4	4.9318	-0.38551	-1.2008	4.2447	-6.9569	0.55474	-3.4003	-6.4104	-4.2314	5.2905	-3.9216	-3.9156
#5	0.74028	0.87891	0.85937	0.31770	-0.019286	-7.0584	-6.5114	3.4890	-4.4305	-3.9903	4.6432	-4.0458

Table 1: Optimised parameter values

x_{1},x_{2},\ldots,x_{12}

for the 5 best fit samples for the one extra dimension model. The optimised loss function values of these 5 samples are 0.942, 1.34, 1.37, 2.20, and 2.54.

	Standard Model parameters
	$y_{u}/10^{-6}$	$y_{d}/10^{-5}$	$y_{s}/10^{-4}$	$y_{c}/10^{-3}$	$y_{b}/10^{-2}$	$y_{t}$	$\theta_{12}$	$\theta_{23}/10^{-2}$	$\theta_{13}/10^{-3}$	$\delta_{\rm CP}$	$J/10^{-5}$
Reference	$6.15$	$1.35$	$2.68$	$3.11$	$1.401$	$0.8616$	$0.22704$	$4.275$	$3.77$	$1.139$	$3.23$
	$\pm 0.14$	$\pm 0.02$	$\pm 0.03$	$\pm 0.05$	$\pm 0.009$	$\pm 0.0043$	$\pm 0.00082$	$\pm 0.042$	$\pm 0.08$	$\pm 0.023$	$\pm 0.08$
Sample
#1	6.1507	1.3491	2.6854	3.1097	1.4028	0.86196	0.22722	4.2634	3.7786	1.1328	3.1981
#2	6.1822	1.3507	2.6809	3.1105	1.4016	0.86165	0.22741	4.2737	3.7671	1.1389	3.2075

Table 2: Predictions for the Standard Model parameters obtained from the five best-fit samples listed in Table 1 . The reference experimental values are the

\overline{\rm MS}

quantities evaluated at 1 TeV [ 13 ] . The

1\sigma

uncertainties are used to define the weights in the loss function.

Figure 1: Field configurations of the 10 best-fit samples in the one-extra-dimension model. The left-handed quark fields

q_{i}

are shown in red, while the right-handed up- and down-type quark fields,

u_{i}

and

d_{i}

, are shown in green and blue, respectively. The field

q_{3}

is fixed at the origin.

Sample	Optimised parameter values
(type)	$x_{1}$	$x_{2}$	$x_{3}$	$x_{4}$	$x_{5}$	$x_{6}$	$x_{7}$	$x_{8}$	$x_{9}$	$x_{10}$	$x_{11}$	$x_{12}$
#1 (d)	-1.4574	1.2402	4.3009	-2.1939	-0.043863	0.97873	2.7282	0.0056122	-1.9345	-3.8797	-2.8570	-1.2032
#2 (7)	2.1362	1.7137	2.6881	-2.2426	0.034977	0.99208	2.7684	-0.0086459	-1.9004	-3.8944	-2.8538	-1.1421
#3 (d)	-0.43696	2.4282	0.83581	-2.5946	-0.10209	-0.87904	3.0463	0.21507	-3.2692	4.3672	-2.7787	-1.6416
#4 (5)	1.1424	1.2041	1.2432	-2.0543	-0.0062153	0.94375	2.5559	0.24881	-2.0393	-3.9090	-2.8006	-1.1090
#5 (7)	-0.66948	2.3715	0.70856	-3.0679	0.15047	-0.83097	2.9951	-0.55299	-4.0532	-4.3160	2.9002	-1.8187

Table 3: Optimised parameter values

x_{1},x_{2},\ldots,x_{12}

for the 5 best fit samples for the two extra dimension model. The optimised loss function values for these 5 samples are 0.457, 0.462, 0.482, 0.486, and 0.499.

Figure 2: Optimised field configuration of the best performing sample in the two extra dimension model. The left-handed quark fields

q_{i}

are shown in red, while the right-handed up- and down-type quark fields,

u_{i}

and

d_{i}

, are shown in green and blue, respectively. The field

q_{1}

is fixed at the origin and the field

q_{2}

is placed on the vertical axis.

	Standard Model parameters
	$y_{u}/10^{-6}$	$y_{d}/10^{-5}$	$y_{s}/10^{-4}$	$y_{c}/10^{-3}$	$y_{b}/10^{-2}$	$y_{t}$	$\theta_{12}$	$\theta_{23}/10^{-2}$	$\theta_{13}/10^{-3}$	$\delta_{\rm CP}$	$J/10^{-5}$
Reference	$6.15$	$1.35$	$2.68$	$3.11$	$1.401$	$0.8616$	$0.22704$	$4.275$	$3.77$	$1.139$	$3.23$
	$\pm 0.14$	$\pm 0.02$	$\pm 0.03$	$\pm 0.05$	$\pm 0.009$	$\pm 0.0043$	$\pm 0.00082$	$\pm 0.042$	$\pm 0.08$	$\pm 0.023$	$\pm 0.08$
Sample
#1	6.1634	1.3474	2.6793	3.1125	1.4014	0.86186	0.22679	4.2766	3.7698	1.1394	3.2045
#2	6.1785	1.3493	2.6805	3.1128	1.4008	0.86157	0.22701	4.2756	3.7717	1.1394	3.2083

Table 4: Predictions for the Standard Model parameters obtained from the five best-fit samples listed in Table 3 . The reference experimental values are the

\overline{\rm MS}

quantities evaluated at 1 TeV [ 13 ] . The

1\sigma

uncertainties are used to define the weights in the loss function.

Figure 3: Planar visualisation of 12 dimension parameters $\{x_{1},x_{2},\ldots,x_{12}\}$ of the two extra dimension model, using dimensionality reduction algorithm of t-SNE. The plot shows 502 best-fit samples with optimised loss values $<12$ . The colour represents the values of the optimised loss function: darker blue for smaller loss values, indicating better performance.

Figure 4: Optimised field configuration of the 48 best-fit samples in the two extra dimension model. The left-handed quark fields

q_{i}

are shown in red, while the right-handed up- and down-type quark fields,

u_{i}

and

d_{i}

, are shown in green and blue, respectively. The field

q_{1}

is fixed at the origin and the field

q_{2}

is constrained on the vertical axis.

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