* Zwintz et al. (2017b) Zwintz, K., Van Reeth, T., Tkachenko, A., et al. 2017b, A&A, 608, A103

##  Appendix A TESS observations

###  A.1 Spectral window of TESS and CoRot observation

In the Fourier analysis of time series photometry, the time stamps of
observations often introduce alias frequencies due to the convolution of the
amplitude spectrum with the spectral window. The spectral window of different
light curves analysed in this work are shown in Fig. 18 \- 20. While there is
no signal whatsoever in the spectral window of the TESS observations, there
are some small peaks in the CoRot observations. However, their amplitude is
small enough that this does not lead to serious aliasing problems in our
analysis.

Figure 18: Same as Fig. 4 but including the spectral window. (red line) of the
light curves in the right panels. The spectral window is normalised to the
highest peak in the amplitude spectrum and magnified by a factor of $100$.
Figure 19: Same as Fig. 6 but including the spectral window. (red line) of the
light curves in the right panels. The spectral window is normalised to the
highest peak in the amplitude spectrum and magnified by a factor of $10$.
Figure 20: Same as Fig. 7 but including the spectral window. (red line) of the
light curves in the right panels. The spectral window is normalised to the
highest peak in the amplitude spectrum and magnified by a factor of $10$.

###  A.2 Frequency tables and FFI images of the newly discovered pre-main
sequence pulsators

In the following we provide the tables of extracted frequencies for the newly
discovered pre-main sequence pulsators from TESS data (i.e., Tables 5 to 9 and
the respective Full Frame Images (FFIs) that are described in the main part of
the article (i.e., Figures 21 to 31).

It is important to state that the reported uncertainties for frequency,
amplitude, and phase correspond only to the uncertainties obtain from the
modelling of the adopted light curve. Given that different apertures and and
corrections lead to slightly different light curves, we assume that our
uncertainties are strongly underestimated.

Table 5: Extracted frequencies for the pre-main sequence SPB star BD+65 1637.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
F1 | $3.076462(4)$ | $1.094(12)$ | $0.854(2)$ | $11.53$
F2 | $0.507339(4)$ | $1.085(12)$ | $0.692(2)$ | $13.70$
F3 | $2.151676(4)$ | $1.054(12)$ | $0.284(2)$ | $13.60$
F4 | $3.018586(6)$ | $0.772(12)$ | $0.555(3)$ | $11.60$
F5 | $0.199488(12)$ | $0.374(12)$ | $0.522(5)$ | $5.36$
F6 | $2.905707(14)$ | $0.335(12)$ | $0.592(6)$ | $5.31$
F7 | $3.190572(14)$ | $0.333(12)$ | $0.731(6)$ | $5.74$
F8 | $3.261994(14)$ | $0.334(12)$ | $0.744(6)$ | $6.14$
F8 | $5.228138(15)$ | $0.311(12)$ | $0.806(6)$ | $6.79$
F9 | $3.09616(2)$ | $0.292(12)$ | $0.500(7)$ | $5.40$
F10 | $2.98411(2)$ | $0.273(12)$ | $0.798(7)$ | $5.11$
F11 | $5.74287(2)$ | $0.230(12)$ | $0.433(8)$ | $5.59$
F12 | $2.54778(2)$ | $0.224(12)$ | $0.425(9)$ | $4.34$
F13 | $6.09464(2)$ | $0.210(12)$ | $0.975(9)$ | $5.78$
F14 | $6.16729(2)$ | $0.194(12)$ | $0.791(10)$ | $5.92$
F15 | $3.08344(2)$ | $0.191(12)$ | $0.846(10)$ | $4.00$
F16 | $8.59152(3)$ | $0.161(12)$ | $0.292(11)$ | $8.53$
F17 | $10.51292(3)$ | $0.150(12)$ | $0.139(13)$ | $10.06$
F18 | $5.41367(3)$ | $0.148(12)$ | $0.870(13)$ | $4.12$
F19 | $7.32194(3)$ | $0.134(12)$ | $0.647(14)$ | $6.02$
(F20 | $3.00381(2)$ | $0.191(12)$ | $0.078(10)$ |  $3.85$)
(F21 | $5.17067(3)$ | $0.139(12)$ | $0.843(14)$ |  $3.77$)

999 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Table 6: Extracted frequencies for the pre-main sequence SPB candidates found
in this work.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
HD 152822 – candidate
F1 | $2.25479(2)$ | $1.16(5)$ | $0.135(7)$ | $10.56$
F2 | $0.84580(4)$ | $0.73(5)$ | $0.310(11)$ | $6.48$
F3 | $2.31573(5)$ | $0.49(5)$ | $0.333(17)$ | $6.52$
F4 | $1.97877(7)$ | $0.34(5)$ | $0.40(3)$ | $4.93$
F5 | $2.47346(8)$ | $0.31(5)$ | $0.13(3)$ | $5.38$
F6 | $4.84655(12)$ | $0.21(5)$ | $0.31(4)$ | $4.56$
F7 | $5.15125(13)$ | $0.19(5)$ | $0.09(4)$ | $4.68$
F8 | $4.51676(14)$ | $0.18(5)$ | $0.55(5)$ | $4.90$
HD 152799 – candidate
F1 | $1.40530(2)$ | $1.33(5)$ | $0.257(6)$ | $10.04$
F2 | $0.70984(4)$ | $0.52(5)$ | $0.433(15)$ | $4.75$
F3 | $1.16862(7)$ | $0.35(5)$ | $0.70(2)$ | $4.41$
F4 | $2.25479(14)$ | $0.17(5)$ | $0.67(5)$ | $4.55$
F5 | $4.4988(2)$ | $0.11(5)$ | $0.68(7)$ | $4.64$
(F6 | $1.49842(9)$ | $0.25(5)$ | $0.81(3)$ |  $3.88$)
(F7 | $1.0898(1)$ | $0.24(5)$ | $0.92(3)$ |  $3.70$)
HD 329271 – candidate
F1 | $2.473508(2)$ | $1.54(6)$ | $0.301(6)$ | $13.84$
F2 | $2.51648(8)$ | $0.36(5)$ | $0.92(3)$ | $5.49$
CD-32 12908 – candidate
F1 | $2.73037(2)$ | $3.22(15)$ | $0.268(8)$ | $9.80$
F2 | $2.68658(3)$ | $2.59(15)$ | $0.91(1)$ | $11.63$
F3 | $0.08751(7)$ | $1.11(15)$ | $0.21(2)$ | $4.73$
F4 | $5.42128(7)$ | $0.95(15)$ | $0.66(3)$ | $6.06$
F5 | $1.21202(9)$ | $0.88(15)$ | $0.43(3)$ | $5.79$
F6 | $5.67506(9)$ | $0.84(15)$ | $0.14(3)$ | $6.60$
F7 | $5.37315(11)$ | $0.68(15)$ | $0.36(4)$ | $6.41$
F8 | $3.06725(11)$ | $0.66(15)$ | $0.37(4)$ | $6.20$
F9 | $1.26015(13)$ | $0.59(15)$ | $0.41(4)$ | $4.52$
F10 | $1.17264(15)$ | $0.51(15)$ | $0.04(5)$ | $4.31$
F11 | $2.7785(2)$ | $0.28(15)$ | $0.28(6)$ | $4.54$
F12 | $7.2940(3)$ | $0.24(15)$ | $0.16(10)$ | $5.51$
(F13 | $11.4945(3)$ | $0.37(15)$ | $0.137(15)$ |  $3.73$)
HD 76534 – candidate
F1 | $5.289865(6)$ | $1.131(13)$ | $0.108(2)$ | $13.33$
F2 | $2.611675(9)$ | $0.761(13)$ | $0.901(3)$ | $8.30$
F3 | $2.574505(10)$ | $0.645(13)$ | $0.149(3)$ | $8.38$
F4 | $0.491034(11)$ | $0.583(13)$ | $0.782(4)$ | $6.38$
F5 | $2.862083(12)$ | $0.573(13)$ | $0.178(4)$ | $10.37$
F6 | $1.67265(2)$ | $0.409(13)$ | $0.135(5)$ | $6.31$
F7 | $0.43235(2)$ | $0.387(13)$ | $0.195(5)$ | $4.93$
F8 | $5.14901(2)$ | $0.312(13)$ | $0.147(7)$ | $5.43$
F9 | $4.85361(2)$ | $0.324(13)$ | $0.903(6)$ | $5.95$
F10 | $5.090322(2)$ | $0.245(13)$ | $0.114(8)$ | $4.94$
F11 | $2.36518(3)$ | $0.220(13)$ | $0.909(9)$ | $4.44$
F12 | $5.22139(3)$ | $0.213(13)$ | $0.834(10)$ | $4.64$
F13 | $2.42974(3)$ | $0.209(13)$ | $0.166(10)$ | $4.56$
F14 | $9.360949(3)$ | $0.204(13)$ | $0.595(11)$ | $8.02$
F15 | $9.45681(4)$ | $0.164(13)$ | $0.457(13)$ | $7.78$
F16 | $2.59798(4)$ | $0.156(13)$ | $0.014(13)$ | $4.00$
(F17 | $4.57386(4)$ | $0.165(13)$ | $0.215(13)$ |  $3.77$)
(F18 | $5.35051(5)$ | $0.145(13)$ | $0.043(13)$ |  $3.66$)

101010 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Table 7: Extracted frequencies for the pre-main sequence $\gamma$ Doradus
candidate HD 317859.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
F1 | $1.21202(3)$ | $2.51(13)$ | $0.912(8)$ | $7.44$
F2 | $1.260153(3)$ | $2.01(13)$ | $0.915(10)$ | $7.16$
F3 | $1.17264(4)$ | $1.48(13)$ | $0.568(14)$ | $6.46$
F4 | $2.73471(8)$ | $0.78(13)$ | $0.66(3)$ | $6.44$
F5 | $2.68658(10)$ | $0.62(13)$ | $0.43(5)$ | $6.28$
F6 | $5.4213(2)$ | $0.28(13)$ | $0.16(7)$ | $4.95$
F7 | $5.3775(3)$ | $0.22(13)$ | $0.68(10)$ | $4.23$
F8 | $5.6794(4)$ | $0.18(13)$ | $0.47(11)$ | $4.33$
(F9 | $2.4022(2)$ | $0.28(13)$ | $0.99(7)$ |  $3.64$)

111111 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Table 8: Extracted frequencies for the pre-main sequence $\delta$ Scuti stars
found in this work.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
HD 287823
F1 | $20.32224(5)$ | $0.189(17)$ | $0.713(14)$ | $6.41$
F2 | $23.74746(5)$ | $0.186(17)$ | $0.428(14)$ | $5.22$
F3 | $19.69683(7)$ | $0.137(17)$ | $0.803(19)$ | $6.56$
F4 | $16.41862(8)$ | $0.107(17)$ | $0.39(2)$ | $6.07$
F5 | $20.99800(9)$ | $0.103(17)$ | $0.17(3)$ | $5.87$
F6 | $19.06234(11)$ | $0.083(17)$ | $0.69(3)$ | $5.01$
F7 | $23.22791(12)$ | $0.078(17)$ | $0.89(3)$ | $4.31$
F8 | $16.14735(13)$ | $0.072(17)$ | $0.09(4)$ | $4.87$
F9 | $21.40260(13)$ | $0.070(17)$ | $0.63(4)$ | $4.47$
F10 | $18.65773(14)$ | $0.063(17)$ | $0.08(4)$ | $4.46$
F11 | $19.44395(18)$ | $0.051(17)$ | $0.52(5)$ | $4.08$
(F12 | $23.97275(9)$ | $0.100(17)$ | $0.73(3)$ |  $3.58$)
HD 290500
F1 | $22.72214(3)$ | $0.85(5)$ | $0.995(9)$ | $8.24$
F2 | $20.92903(7)$ | $0.37(5)$ | $0.70(2)$ | $5.34$
F3 | $23.46700(8)$ | $0.34(5)$ | $0.01(2)$ | $5.05$
F4 | $23.75206(8)$ | $0.33(5)$ | $0.83(2)$ | $6.14$
F5 | $19.38418(8)$ | $0.32(5)$ | $0.23(2)$ | $5.85$
F6 | $21.76122(9)$ | $0.29(5)$ | $0.17(3)$ | $5.02$
F7 | $19.22786(10)$ | $0.26(5)$ | $0.70(3)$ | $5.67$
F8 | $21.03018(11)$ | $0.23(5)$ | $0.21(3)$ | $4.97$
F9 | $21.68766(13)$ | $0.20(5)$ | $0.72(4)$ | $5.06$
F10 | $13.14041(13)$ | $0.20(5)$ | $0.43(4)$ | $6.00$
F11 | $20.22557(13)$ | $0.20(5)$ | $0.50(4)$ | $6.38$
F12 | $2.84142(15)$ | $0.17(5)$ | $0.04(5)$ | $4.00$
F13 | $23.21872(17)$ | $0.16(5)$ | $0.68(5)$ | $4.22$
F14 | $17.61864(18)$ | $0.15(5)$ | $0.50(5)$ | $4.32$
F15 | $21.3842(2)$ | $0.12(5)$ | $1.00(6)$ | $4.21$
F16 | $19.8761(3)$ | $0.10(5)$ | $0.40(8)$ | $4.07$
(F17 | $1.42991(10)$ | $0.25(5)$ | $0.49(3)$ |  $3.85$)
(F18 | $21.2831(3)$ | $0.10(5)$ | $0.85(8)$ |  $3.67$)
HD 144432
F1 | $0.89620(2)$ | $2.08(10)$ | $0.558(8)$ | $8.36$
F2 | $14.50410(3)$ | $1.77(10)$ | $0.350(9)$ | $14.30$
F3 | $13.18489(7)$ | $0.67(10)$ | $0.25(2)$ | $11.35$
F4 | $23.16855(8)$ | $0.63(10)$ | $0.31(2)$ | $11.19$
F5 | $15.51142(8)$ | $0.58(10)$ | $0.38(3)$ | $9.17$
F6 | $21.46936(9)$ | $0.57(10)$ | $0.16(3)$ | $12.08$
F7 | $20.61618(10)$ | $0.47(10)$ | $0.46(3)$ | $11.25$
F8 | $18.8560(2)$ | $0.25(10)$ | $0.90(6)$ | $6.20$
F9 | $15.3609(2)$ | $0.24(10)$ | $0.38(7)$ | $6.79$
F10 | $13.2709(2)$ | $0.19(10)$ | $0.20(8)$ | $6.35$
F11 | $18.4617(3)$ | $0.18(10)$ | $0.56(9)$ | $5.64$
F12 | $16.3072(3)$ | $0.16(10)$ | $0.49(10)$ | $4.94$
F13 | $15.8233(4)$ | $0.11(10)$ | $0.69(14)$ | $4.47$
(F14 | $23.3693(4)$ | $0.11(10)$ | $0.89(14)$ |  $3.84$)
(F15 | $18.5406(5)$ | $0.09(10)$ | $0.31(17)$ |  $3.54$)
(F16 | $14.5686(5)$ | $0.09(10)$ | $0.32(17)$ |  $3.56$)

121212 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Table 9: Extracted frequencies for the pre-main sequence $\delta$ Scuti
candidates found in this work.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
V599 Ori – candidate
F1 | $23.22791(3)$ | $14.7(7)$ | $0.683(8)$ | $11.85$
F2 | $17.59105(4)$ | $9.4(7)$ | $0.234(12)$ | $8.42$
F3 | $21.53593(4)$ | $9.3(7)$ | $0.864(13)$ | $12.74$
F4 | $18.49681(7)$ | $5.6(7)$ | $0.39(2)$ | $8.18$
F5 | $17.83933(8)$ | $5.0(7)$ | $0.00(2)$ | $11.24$
F6 | $5.6415(3)$ | $1.3(7)$ | $0.30(9)$ | $8.24$
F7 | $23.1957(4)$ | $1.1(7)$ | $0.71(11)$ | $6.51$
F8 | $3.9449(4)$ | $0.9(7)$ | $0.38(12)$ | $4.50$
F9 | $4.7311(6)$ | $0.7(7)$ | $0.99(16)$ | $5.16$
F10 | $7.1449(6)$ | $0.6(7)$ | $0.77(18)$ | $4.99$
F11 | $23.2555(7)$ | $0.6(7)$ | $0.9(2)$ | $4.27$
F12 | $7.5541(9)$ | $0.5(7)$ | $0.1(2)$ | $4.06$
F13 | $12.5105(9)$ | $0.4(7)$ | $0.6(3)$ | $4.82$
F14 | $8.3817(10)$ | $0.4(7)$ | $0.5(3)$ | $4.12$
(F15 | $17.6324(6)$ | $0.7(7)$ | $0.54(17)$ |  $3.61$)
(F16 | $17.0485(7)$ | $0.6(7)$ | $0.4(2)$ |  $3.55$)
(F17 | $23.1635(10)$ | $0.4(7)$ | $0.9(3)$ |  $3.67$)
TYC 8581-2002-1 – candidate
F1 | $16.69326(5)$ | $0.155(14)$ | $0.915(15)$ | $10.66$
F2 | $17.50498(6)$ | $0.121(14)$ | $0.400(19)$ | $10.29$
F3 | $16.27781(6)$ | $0.120(14)$ | $0.993(19)$ | $9.85$
F4 | $15.60543(7)$ | $0.102(14)$ | $0.20(2)$ | $9.59$
F5 | $18.56213(7)$ | $0.100(14)$ | $0.81(2)$ | $8.14$
F6 | $1.75382(7)$ | $0.097(14)$ | $0.73(2)$ | $4.25$
F7 | $11.15057(7)$ | $0.095(14)$ | $0.33(2)$ | $8.81$
F8 | $18.74492(8)$ | $0.088(14)$ | $0.81(3)$ | $7.25$
F9 | $15.17848(11)$ | $0.066(14)$ | $0.07(3)$ | $6.94$
F10 | $22.23083(11)$ | $0.063(14)$ | $0.64(4)$ | $6.35$
F11 | $19.65890(11)$ | $0.062(14)$ | $0.30(4)$ | $5.50$
F12 | $20.30956(14)$ | $0.052(14)$ | $0.16(4)$ | $5.24$
F13 | $16.62806(17)$ | $0.042(14)$ | $0.58(5)$ | $4.62$
F14 | $16.98599(17)$ | $0.040(14)$ | $0.14(6)$ | $4.50$
(F15 | $14.3182(2)$ | $0.033(14)$ | $0.50(7)$ |  $3.55$)
HD 135344A – candidate
F1 | $74.26705(3)$ | $0.289(16)$ | $0.299(9)$ | $8.33$
F2 | $73.94123(6)$ | $0.134(16)$ | $0.927(19)$ | $6.48$
F3 | $60.53231(7)$ | $0.115(16)$ | $0.42(2)$ | $5.60$
F4 | $79.93974(7)$ | $0.114(16)$ | $0.62(2)$ | $5.32$
F5 | $63.05118(9)$ | $0.087(16)$ | $0.98(3)$ | $4.79$
F6 | $80.81278(10)$ | $0.075(16)$ | $0.15(3)$ | $4.19$
F7 | $77.27049(11)$ | $0.073(16)$ | $0.42(3)$ | $4.17$
(F8 | $259.40209(10)$ | $0.078(16)$ | $0.83(3)$ |  $3.54$)

131313 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Table 10: Extracted frequencies for HD 329379.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
F1 = 2$f_{\rm orb}$ | $0.889013(15)$ | $32.3(2)$ | $0.283(5)$ | $13.13$
F2 = $f_{\rm orb}$ | $0.44451(3)$ | $14.2(2)$ | $0.283(11)$ | $11.25$
F3 | $6.30912(14)$ | $3.5(2)$ | $0.91(5)$ | $13.41$
F4 = 4$f_{\rm orb}$ | $1.7816(2)$ | $2.2(2)$ | $0.45(7)$ | $5.71$
F5 = 6$f_{\rm orb}$ | $2.6670(4)$ | $1.1(2)$ | $0.35(15)$ | $5.99$
F6 = F3 - 2$f_{\rm orb}$ | $5.4165(5)$ | $1.0(2)$ | $0.72(16)$ | $7.15$
F7 = 8$f_{\rm orb}$ | $3.5632(5)$ | $1.0(2)$ | $0.62(16)$ | $6.91$
F8 = 14$f_{\rm orb}$ | $6.2123(6)$ | $0.9(2)$ | $0.79(19)$ | $8.35$
F9 = 10$f_{\rm orb}$ | $4.4558(8)$ | $0.6(2)$ | $0.8(3)$ | $6.38$
F10 = 13$f_{\rm orb}$ | $5.7642(10)$ | $0.5(2)$ | $0.7(3)$ | $6.71$
F11 | $7.4885(13)$ | $0.4(2)$ | $0.8(4)$ | $5.58$
F12 = 12$f_{\rm orb}$ | $5.3305(15)$ | $0.3(2)$ | $0.4(5)$ | $4.92$
F13 = F3 - 4$f_{\rm orb}$ | $4.5275(16)$ | $0.3(2)$ | $0.7(5)$ | $4.29$
F14 = 16$f_{\rm orb}$ | $7.1013(17)$ | $0.3(2)$ | $0.9(6)$ | $4.77$
F15 = 17$f_{\rm orb}$ | $7.5459(19)$ | $0.3(2)$ | $0.4(6)$ | $4.82$
F16 = F3 + 2 $f_{\rm orb}$ | $7.198(2)$ | $0.2(2)$ | $0.5(7)$ | $4.72$
F17 = 15$f_{\rm orb}$ | $6.653(3)$ | $0.2(2)$ | $0.7(9)$ | $4.36$

141414 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Table 11: Extracted frequencies for $\beta$ Cephei star HD 96042.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
F1 | $2.036134(15)$ | $3.10(9)$ | $0.079(5)$ | $12.55$
F2 | $1.94242(3)$ | $1.38(9)$ | $0.456(11)$ | $7.01$
F3 | $6.88504(7)$ | $0.64(9)$ | $1.06(2)$ | $13.93$
F4 | $4.41531(8)$ | $0.59(9)$ | $0.17(2)$ | $7.96$
F5 | $10.13048(8)$ | $0.55(9)$ | $0.53(3)$ | $15.11$
F6 | $8.85513(9)$ | $0.52(9)$ | $0.17(3)$ | $13.11$
F7 | $11.46647(13)$ | $0.35(9)$ | $0.33(4)$ | $13.09$
F8 | $11.2018(3)$ | $0.15(9)$ | $0.31(10)$ | $7.24$
F9 | $8.4343(4)$ | $0.13(9)$ | $0.98(12)$ | $4.58$
F10 | $9.6968(4)$ | $0.13(9)$ | $0.24(12)$ | $5.18$
F11 | $9.6214(4)$ | $0.12(9)$ | $0.46(12)$ | $5.32$
(F12 | $2.14640(7)$ | $0.67(9)$ | $0.60(2)$ |  $3.80$)
(F13 | $1.90015(7)$ | $0.65(9)$ | $0.24(2)$ |  $3.65$)
(F14 | $2.07473(8)$ | $0.56(9)$ | $0.19(3)$ |  $3.63$)
(F15 | $1.95896(8)$ | $0.55(9)$ | $0.84(3)$ |  $3.52$)
(F16 | $2.18867(9)$ | $0.53(9)$ | $0.49(3)$ |  $3.78$)
(F17 | $4.8490(2)$ | $0.22(9)$ | $0.57(7)$ |  $3.76$)
(F18 | $7.5375(3)$ | $0.14(9)$ | $0.15(11)$ |  $3.95$)

151515 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Table 12: Extracted frequencies for HD 53367.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
F1 | $2.245051(8)$ | $0.755(12)$ | $0.422(2)$ | $7.19$
F2 | $2.204158(12)$ | $0.542(12)$ | $0.323(4)$ | $6.41$
F3 | $2.281855(19)$ | $0.336(12)$ | $0.544(6)$ | $5.04$
F4 | $12.11828(14)$ | $0.047(12)$ | $0.15(4)$ | $4.65$

161616 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Table 13: Extracted frequencies for HD 152743.

Designation | $f$ | $A$ | $\phi$ | SNR
---|---|---|---|---
| ($\mathrm{d}^{-1}$) | (mmag) | ($\frac{\rm rad}{2\pi}$) |
F1 | $0.39791(3)$ | $1.05(7)$ | $0.722(10)$ | $4.40$
F2 | $2.19027(3)$ | $0.96(7)$ | $0.073(11)$ | $7.08$
F3 | $0.99297(4)$ | $0.84(7)$ | $0.215(13)$ | $4.78$
F4 | $2.44837(5)$ | $0.65(7)$ | $0.629(17)$ | $7.12$
F5 | $3.20475(13)$ | $0.26(7)$ | $0.96(4)$ | $4.36$
F6 | $4.45223(14)$ | $0.23(7)$ | $0.66(5)$ | $4.75$
F7 | $2.92155(16)$ | $0.20(7)$ | $0.41(5)$ | $4.27$
(F8 | $1.24032(6)$ | $0.56(7)$ | $0.801(19)$ |  $3.95$)

171717 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999).

Figure 21: TESS FFI image in a $20\times 20$ pixel cutout around the position
of HD 152822 in sector 12. Red filled area shows the chosen aperture. The red
dots show stars within a GAIA magnitude limit $\propto 6$ with varying size
according to their GAIA magnitude. The GAIA query failed for HD 152822 and
hence the magnitudes are relative to TYC 8327-955-1, which is marked by the
white cross.  Figure 22: TESS FFI image in a $20\times 20$ pixel cutout around
the position of HD 152799 in sector 12. Red filled area shows the chosen
aperture. The red dots show stars within a GAIA magnitude limit $\propto 6$
with varying size according to their GAIA magnitude and the white cross marks
the position of HD 152799.  Figure 23: Same as fig. 22 but for HD 329271 in
sector 12.  Figure 24: Same as fig. 22 but for CD-32 12908 in sector 12.
Figure 25: Same as fig. 22 but for HD 287823 in sector 6.  Figure 26: Same as
fig. 22 but for HD 290500 in sector 6.  Figure 27: TESS FFI image in a
$20\times 20$ pixel cutout around the position of V599 Ori in sector 6. Red
filled area shows the chosen aperture and the magenta area shows the chosen
pixel for the simple background flux. The red dots show stars within a GAIA
magnitude limit $\propto 6$ with varying size according to their GAIA
magnitude and the white cross marks the position of V599 Ori. The influence of
the naked eye star 49 Ori is clearly visible.  Figure 28: Same as fig. 22 but
for TYC 8581-2002-1 in sector 10. Sectors 8 and 9 have also been observed.
Figure 29: Same as fig. 22 but for HD 144432 in sector 6.  Figure 30: Same as
fig. 22 but for HD 317859 in sector 12.  Figure 31: Same as fig. 22 but for
2MASS 11120327-7637034 in sector 12.

###  A.3 Other variable stars

CD-49 11096 is a B9V (Aidelman et al. 2018) star that has been observed in
sector 12. No short cadence data is available for this star, hence we use the
light curve obtained using the RegressionCorrector. CD-4911096 has a TESS
magnitude of $10.584$. Multiple other stars lie within the chosen aperture as
the sky region is very crowded. Most notably are GAIA DR2 5938324042340525696
with a TESS magnitude of $12.277$ and GAIA DR2 5938324038039512192 with a TESS
magnitude of $11.533$ which do not have an entry in the SIMBAD database. We
extract two significant frequencies at $5.45237$ and $5.71764$
$\mathrm{d}^{-1}$ and the amplitude spectrum shows additional signal around
$15$ and $21$$\mathrm{d}^{-1}$ although no frequency is significant in this
range. The TIC reports $T_{\rm eff}=6501$ K for one of the two close-by
objects, GAIA DR2 5938324038039512192; an effective temperature as would be
expected for $\delta$ Scuti stars. Given the frequency ranges, we might see
the signal of a pre-main sequence $\delta$ Scuti star. However, more data will
be needed to verify this preliminary identification and we restrain from
adding CD-4911096 with $T_{\rm eff}=11450\pm 308$ K and $\log(g)=4.26\pm 0.12$
to our sample. The FFI and chosen aperture are shown in Fig. 32.

PDS 344 is most likely a B5 star (Vieira et al. 2003) that has been observed
in sector 10 and 11. No short cadence data is available for this star, hence
we use the light curve obtained using the RegressionCorrector. PDS 344 has a
TESS magnitude of $13.071$. The apertures for sector 10 and sector 11 are
slightly shifted with respect to each other, resulting in different stars
lying inside the aperture of each sector. Most notably GAIA DR2
5332670397352388608 with a TESS magnitude of $14.374$ lies within the aperture
in sector 11. We extract one frequency at $6.96842(3)$ $\mathrm{d}^{-1}$with
an amplitude of $0.69(4)$ mmag and a SNR of $16.18$. It is a clear detection
of the frequency, as it is evident in both sectors. However, given the crowded
nature of the FFIs we restrain from adding PDS 344 to our sample of pulsating
stars. More observations are needed for an identification. The FFI and chosen
aperture for sector 10 are shown in Fig. 33.

HD 36982 is a B1.5Vp star according to SIMBAD that has been observed in sector
6. No short cadence data is available for this star, hence we use the light
curve obtained using the RegressionCorrector. There are no other stars in the
aperture, but strong background light from nearby O-type stars can be
detected. We extract one frequency at $6.08744(3)$ $\mathrm{d}^{-1}$ with an
amplitude of $0.41(3)$ mmag and a SNR of $11.11$. This is similar to the case
of PDS 344. More observations are needed for an identification. The FFI and
chosen aperture for sector 10 are shown in Fig. 34.

HD 329379 is a B0II star (Aidelman et al. 2018) that has been observed in
sector 12. No short cadence data is available for this star, hence we use the
light curve obtained using the RegressionCorrector. There are no other stars
in the aperture, but two darker stars in the vicinity from which we expect no
contamination. We extract $17$ significant frequencies and interpret the light
curve and the frequency spectrum as ellipsoidal variability of a close binary
system with an orbital period of about $2.25$ d. All but five of the extracted
frequencies are multiples of the orbital frequency F2. The others constitute a
quartet (F3, F6, F13, F16) and a singlet F11 (see Table 10). The frequencies
F3 and F16 and F6 and F13, respectively are each split by exactly the orbital
frequency while there is a small additional offset of 0.008 $f_{\rm orb}$
between F3 and F6. Thus, this could also be two doublets that coincide to a
quartet. The light curve (see Fig. 35) clearly shows additional variability
that resembles pulsations. We constructed a simple model for the binary light
curve using a Savgol filter with a window length of 31. The residuals show an
envelope that rises whenever the light curve obtains a maximum. It is
difficult to speak on the origin of the observed light curve and frequencies.
Multiple explanations are possible that consist of, e.g. tidally trapped
pulsations (Handler et al. 2020; Kurtz et al. 2020; Fuller et al. 2020),
tidally perturbed pulsations (Reyniers & Smeyers 2003b, a; Steindl et al.
2021), and tidally induced pulsation including heartbeat effects (Fuller
2017). HD 36982 should therefore be studied in more detail. The FFI and chosen
aperture for sector 12 are shown in Fig. 36.

HD 96042 is a B1Vne star (Houk & Cowley 1975) that has been observed in
sectors 10 and 11. No short cadence data is available for this star, hence we
use the light curve obtained using the RegressionCorrector. Two other stars
lie within the chosen aperture, but HD 96042 outshines them by four TESS
magnitudes. We extract three significant frequencies in the g-mode ($<5$
$\mathrm{d}^{-1}$) and eight in the p-mode regime ($>6.5$ $\mathrm{d}^{-1}$).
Six more frequencies in the g-mode regime are just below our significance
criterion. This indicates that multiple g-modes are excited in HD 96042 and a
period spacing might be inferred with more observational data. Given the
stellar parameters of $T_{\rm eff}=25000\pm 1500$ K and $\log(g)=3.8\pm 0.2$
(Fairlamb et al. 2015), we classify HD 96042 as $\beta$ Cephei star on the
main sequence. The FFI and chosen aperture for sector 10 are shown in Fig. 37
and the frequencies are listed in Table 11.

HD 53367 is a B0IV/Ve star (Tjin A Djie et al. 2001) that has been observed in
sector 7. Short cadence data is available, hence we downloaded the PDCSAP flux
from the MAST portal. We extract three frequencies in the g-mode and one
additional frequency in the p-mode regime. The residuals show more variability
in the low frequency range, hence further observations of HD 53367 might
enable the identification of further modes, similar to HD 96042. Given the
stellar parameters of $T_{\rm eff}=29000\pm 2000$ K and $\log(g)=4$ (Alecian
et al. 2013), we classify HD 53367 as $\beta$ Cephei star on the main
sequence. The frequencies are listed in Table 12.

HD 152743 is a B2IV star (Houk 1978) that has been observed in sector 12. No
short cadence data is available for this star, hence we use the light curve
obtained using the RegressionCorrector. Five other stars lie within the chosen
aperture with up to only one magnitude less than HD 152743. We extract seven
significant frequencies clearly indicating g-mode pulsations. With $T_{\rm
eff}=24600\pm 2371$ K and $\log(g)=3.76\pm 0.34$ (Aidelman et al. 2018), HD
152743 could very well be the source of these g-mode pulsations. However, some
of the other stars also have effective temperatures expected for g-mode
pulsators according to the TIC. Hence, additional observations are needed for
a clear identification. The FFI and chosen aperture for sector 10 are shown in
Fig. 38 and the frequencies are listed in Table 13.

Figure 32: Same as fig. 22 but for CD-4911096 in sector 12.  Figure 33: Same
as fig. 22 but for PDS 344 in sector 10. The aperture was shifted in sector 11
which leads to having stars 4 and 5 within the aperture.  Figure 34: Same as
fig. 22 but for HD 36982 in sector 6. The aperture was shifted in sector 11
which leads to having stars 4 and 5 within the aperture.  Figure 35: TESS
observation of HD 329379. The top panel shows the the first orbit of the TESS
observation (black dots) and a simple model for the binary light curve (red
line). The second panel shows the residual of the top panel. The third panel
shows the folded TESS light curve. The bottom panel shows the amplitude
spectrum (black line). Significant frequencies are marked as open triangles
for multiples of the orbital frequency and filled triangles otherwise.  Figure
36: Same as fig. 22 but for HD 329379 in sector 12.  Figure 37: Same as fig.
22 but for HD 96042 in sector 10. The star has also been observed in sector
11.  Figure 38: Same as fig. 22 but for HD 152743 in sector 12.

##  Appendix B Pre-main sequence $\gamma$ Doradus and hybrid stars discovered
from CoRoT data: Frequency tables and spectroscopic data

###  B.1 Frequency tables

Tables 14, 15 and 16 list the frequencies extracted in the two pre-main
sequence $\gamma$ Doradus stars and the pre-main sequence $\delta$ Scuti –
$\gamma$ Doradus hybrid discovered from CoRoT data.

Table 14: Extracted frequencies for Cl⋆ NGC 2264 VAS 196.

Designation | $f$ | $A_{\rm SRa05}$ | $\phi$ | SNR | cross-ref
---|---|---|---|---|---
|  ($\mathrm{d}^{-1}$) | (mmag) |  ($\frac{\rm rad}{2\pi}$) |  |
F1 | 0.88674(8) | 1.344(7) | 0.4615(9) | 7.83 | SRa01, TESS
F2 | 1.3564(1) | 1.029(7) | 0.434(1) | 9.75 | SRa01, TESS
F3 | 1.3217(1) | 0.762(7) | 0.353(2) | 10.99 | SRa01, TESS
F4 | 1.4819(2) | 0.519(7) | 0.621(2) | 10.99 | SRa01
F5 | 0.9083(2) | 0.451(7) | 0.062(2) | 5.84 | SRa01
F6 | 0.1569(4) | 0.289(7) | 0.750(4) | 4.06 |
F7 | 1.2874(4) | 0.248(7) | 0.657(5) | 5.07 |
F8 | 2.7511(5) | 0.212(7) | 0.660(6) | 7.14 | SRa01
F9 | 1.4420(5) | 0.233(7) | 0.558(5) | 6.22 |

181818 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999). The
amplitudes ($A_{\rm SRa05}$) are determined from the CoRoT SRa05 data. The
last column (cross-ref) marks the frequencies that can also be detected in the
CoRoT SRa01 and TESS data.

Table 15: Extracted frequencies for Cl⋆ NGC 2264 VAS 219.

Designation | $f$ | $A_{\rm SRa05}$ | $\phi$ | SNR | cross-ref
---|---|---|---|---|---
|  ($\mathrm{d}^{-1}$) | (mmag) |  ($\frac{\rm rad}{2\pi}$) |  |
F1 | 2.09926(5) | 4.07(1) | 0.2272(5) | 7.37 | SRa01, TESS
F2 | 1.92554(6) | 3.34(1) | 0.9149(6) | 7.05 | SRa01, TESS
F3 | 1.36574(7) | 2.77(1) | 0.3421(7) | 6.44 | SRa01, TESS
F4 | 0.04759(7) | 2.51(1) | 0.8480(8) | 5.93 |
F5 | 0.1993(1) | 1.88(1) | 0.126(1) | 4.57 |
F6 | 2.0562(1) | 1.96(1) | 0.103(1) | 5.22 | SRa01, TESS
F7 | 1.3392(1) | 1.88(1) | 0.290(1) | 4.94 | SRa01
F8 | 2.1481(1) | 1.88(1) | 0.521(1) | 6.10 | SRa01
F9 | 3.4281(1) | 1.62(1) | 0.416(1) | 9.91 | SRa01
F10 | 1.8049(1) | 1.58(1) | 0.302(1) | 5.32 | SRa01, TESS
F11 | 3.7810(1) | 1.50(1) | 0.301(1) | 13.95 | SRa01
F12 | 0.1087(1) | 1.57(1) | 0.929(1) | 5.00 |
F13 | 1.6359(1) | 1.45(1) | 0.303(1) | 5.01 | SRa01
F14 | 0.5601(1) | 1.49(1) | 0.277(1) | 4.19 | SRa01
F15 | 2.4539(1) | 1.34(1) | 0.327(1) | 5.88 | SRa01
F16 | 1.5451(2) | 1.24(1) | 0.416(2) | 5.10 | SRa01
F17 | 0.1662(2) | 1.00(1) | 0.329(2) | 4.26 |
F18 | 1.0608(2) | 1.11(1) | 0.554(2) | 3.74 | SRa01
F19 | 0.3389(2) | 1.03(1) | 0.218(2) | 4.10 | SRa01
F20 | 2.3782(2) | 1.01(1) | 0.961(2) | 5.07 |
F21 | 0.0773(2) | 1.77(1) | 0.278(2) | 8.98 |
F22 | 1.8715(2) | 1.00(1) | 0.119(2) | 4.93 |
F23 | 1.9070(2) | 0.89(1) | 0.316(2) | 5.51 |
F24 | 1.6848(3) | 0.72(1) | 0.751(3) | 4.26 |

191919 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999). The
amplitudes ($A_{\rm SRa05}$) are determined from the CoRoT SRa05 data. The
last column (cross-ref) marks the frequencies that can also be detected in the
CoRoT SRa01 and TESS data.

Table 16: Extracted frequencies for Cl⋆ NGC 2264 VAS 230.

Designation | $f$ | $A_{\rm SRa01}$ | $\phi$ | SNR | cross ref
---|---|---|---|---|---
|  ($\mathrm{d}^{-1}$) | (mmag) |  ($\frac{\rm rad}{2\pi}$) |  |
F1 | 8.1866(4) | 0.77(1) | 0.336(3) | 6.71 | SRa05, TESS
F2 | 8.4660(6) | 0.62(1) | 0.730(3) | 9.10 | SRa05, TESS
F3 | 8.5591(8) | 0.42(1) | 0.891(4) | 7.98 | SRa05, TESS
F4 | 11.8419(9) | 0.35(1) | 0.175(6) | 10.87 | SRa05, TESS
F5 | 8.385(1) | 0.29(1) | 0.464(6) | 6.33 | TESS
F6 | 2.3203(7) | 0.58(1) | 0.282(4) | 4.18 | SRa05
F7 | 2.2407(6) | 0.54(1) | 0.650(4) | 5.07 | SRa05
F8 | 2.1643(6) | 0.49(1) | 0.718(3) | 4.15 | SRa05
F9 | 3.3006(8) | 0.46(1) | 0.275(5) | 5.48 | SRa05

202020 The values in parentheses give the $1\sigma$ uncertainty as reported by
the standard error estimates formulated by Montgomery & O’Donoghue (1999). The
amplitudes ($A_{\rm SRa01}$) are determined from the CoRoT SRa01 data. The
last column (cross ref) marks the frequencies that can also be detected in the
CoRoT SRa05 and TESS data. The first five frequencies are the p-modes and the
second four frequencies are the g-modes; both are sorted by decreasing
amplitude.

###  B.2 High-resolution spectroscopy: observations, data reduction, and
analysis

We obtained high-resolution spectra with the Robert G. Tull Coudé spectrograph
(TS) on the 2.7-m telescope of Mc Donald Observatory for Cl⋆ NGC 2264 VAS 196
(CoID0223991870) and Cl⋆ NGC 2264 VAS 219 (CoID0223997565). The cross-
dispersed échelle spectrograph has a resolving power of 60 000 in the
configuration adopted by us. Each spectrum covers the wavelength range from
3633 to 10 849 Å with gaps between the échelle orders at wavelengths longer
than 5880 Å. The spectra for the two stars have signal-to-noise (S/N) ratios
of 84 and 80, respectively, calculated at $\sim$5240 Å.

Bias and Flat Field frames were obtained at the beginning of each night, while
several Th-Ar comparison lamp spectra were obtained each night for wavelength
calibration purposes. The reduction was performed using the Image Reduction
and Analysis Facility212121IRAF (http://iraf.noao.edu) is distributed by the
National Optical Astronomy Observatory, which is operated by the Association
of Universities for Research in Astronomy (AURA) under cooperative agreement
with the National Science Foundation. (IRAF). The spectra were normalized by
fitting a low order polynomial to carefully selected continuum points.

For Cl⋆ NGC 2264 VAS 230 (CoID0223999101) and Cl⋆ NGC 2264 VAS 196 (CoID
0223991870), a spectrum obtained with the cross-dispersed échelle spectrograph
UVES at the ESO Very Large Telescope (VLT) was available in the ESO
archive222222http://archive.eso.org. UVES has a resolving power of 40 000 in
the standard mode. All reduction steps were performed within the UVES pipeline
(version 5.2.0) and the Reflex
software232323http://www.eso.org/sci/software/reflex. The UVES spectra for Cl⋆
NGC 2264 VAS 230 (CoID0223999101) and Cl⋆ NGC 2264 VAS 196 have S/N values of
139 and 157 calculated at $\sim$5845 Å, respectively.

The spectroscopic analysis was performed using the LLmodels model atmosphere
code (Shulyak et al. 2004), the VALD database for atomic line
parameters242424http://vald.astro.uu.se (Kupka et al. 1999), SYNTH3 (Kochukhov
2007) for the computation of synthetic spectra and an updated version of the
Spectroscopy Made Easy (SME, version 474) software package (Valenti & Piskunov
1996; Piskunov & Valenti 2017). Using these methods for each of the three pre-
main sequence pulsators, the effective temperature, surface gravity, projected
rotational velocity, and metallicity were determined. A detailed description
of the adopted procedure can be found for example in Zwintz et al. (2017b).

Table 17: Fundamental parameters for the six pre-main sequence $\gamma$ Doradus stars discovered from CoRoT data. Identifier | $T_{\rm eff}$ | log(g) |  $v$ sin $i$ | [Fe/H]
---|---|---|---|---
| (K) | (dex) |  (km s-1) | (dex)
Cl⋆ NGC 2264 VAS 196 | 6784(150) | 4.0(2) | 77(5) | -0.072
Cl⋆ NGC 2264 VAS 219 | 7170(150) | 4.3(3) | 74(5) | -0.286
Cl⋆ NGC 2264 VAS 230 | 7520(150) | 4.3(4) | 205(25) | -0.200

Cl⋆ NGC 2264 VAS 196 (CoID 0223991870): For this star, two spectra are
available: one obtained at Mc Donald Observatory and one observed with the
UVES spectrograph at the VLT that we retrieved from the ESO archive. As the
S/N value of the UVES spectrum (i.e., 157) is significantly higher than that
of the spectrum from Mc Donald observatory (i.e., 50), we base our analysis on
the UVES spectrum. We determine a $T_{\rm eff}$ of 6784 $\pm$ 150 K and a
log(g) of 4.0 $\pm$ 0.2. The star rotates moderately fast with a $v$ sin $i$,
of 77 km s-1 and it shows basically solar chemical abundances. The observed
and synthetic spectra are shown in Fig. 39.

Cl⋆ NGC 2264 VAS 219 (CoID 0223997565): The spectrum obtained at Mc Donald
observatory for Cl⋆ NGC 2264 VAS 219 shows many emission lines (as can be seen
in Fig. 40). Our analysis yielded a $T_{\rm eff}$ of 7170 $\pm$ 150 K, a
log(g) of 4.3 $\pm$ 0.3, a $v$ sin $i$, of 74 $\pm$ 5 km s-1, and slightly
less than solar abundance. The observed and synthetic spectra are shown in
Fig. 40.

Cl⋆ NGC 2264 VAS 230 (CoID 0223999101): The UVES spectrum retrieved from the
ESO archive shows that Cl⋆ NGC 2264 VAS 230 has an extremely high $v$ sin $i$
value. This makes it challenging to determine reliable values for the star’s
fundamental parameters. Our best fitting spectrum synthesis model yields a
$T_{\rm eff}$ of 7520 $\pm$ 150 K, a log(g) of 4.3 $\pm$ 0.4, and a $v$ sin
$i$ of 205 $\pm$ 25 km s-1 with slightly less than solar abundance. The
observed and synthetic spectra are shown in Fig. 41.

Figure 39: Observed (black line) and synthetic (red line) spectrum of Cl⋆ NGC
2264 VAS 196 in the region around the Mg ib triplet. The synthetic spectrum
was calculated with $T_{\rm eff}$ = 6784 K, log(g) of 4.0 and a $v$ sin $i$ of
77 km s-1.  Figure 40: Observed (black line) and synthetic (red line) spectrum
of Cl⋆ NGC 2264 VAS 219: the top panel shows the region around the Mg ib
triplet, the bottom panel the H$\gamma$ line. The synthetic spectrum was
calculated with $T_{\rm eff}$ = 7170 K, log(g) of 4.3 and a $v$ sin $i$ of 74
km s-1.  Figure 41: Observed (black line) and synthetic (red line) spectrum of
Cl⋆ NGC 2264 VAS 230 in the region around the Mg ib triplet. The synthetic
spectrum was calculated with $T_{\rm eff}$ = 7520 K, log(g) of 4.3 and a $v$
sin $i$ of 205 km s-1.

##  Appendix C MESA microphysics

The MESA EOS is a blend of the OPAL Rogers & Nayfonov (2002), SCVH Saumon et
al. (1995), PTEH Pols et al. (1995), HELM Timmes & Swesty (2000), and PC
Potekhin & Chabrier (2010) EOSes.

Radiative opacities are primarily from OPAL (Iglesias & Rogers 1993, 1996),
with low-temperature data from Ferguson et al. (2005) and the high-
temperature, Compton-scattering dominated regime by Buchler & Yueh (1976).
Electron conduction opacities are from Cassisi et al. (2007).

Nuclear reaction rates are a combination of rates from NACRE (Angulo et al.
1999), JINA REACLIB (Cyburt et al. 2010), plus additional tabulated weak
reaction rates Fuller et al. (1985); Oda et al. (1994); Langanke & Martínez-
Pinedo (2000). Screening is included via the prescription of Chugunov et al.
(2007). Thermal neutrino loss rates are from Itoh et al. (1996).

##  Appendix D Instability regions

In the following we show instability regions resulting from our calculations.
Figures 42 and 43 show the influence of the mixing length on the instability
regions of p-modes and g-modes, respectively. Figure 44 shows the comparison
between classical models and the models from accreting protostars.

Figure 42: P-mode instability regions for different mixing lengths. The green,
red, and blue lines show the boundary for $\alpha_{\rm MLT}=1.8,2.0$, and
$2.2$, respectively.  Figure 43: Same as Fig. 42 but for g-modes and the
$\gamma$ Doradus instability region.  Figure 44: Comparision of the
instability regions from accreting models versus classical models. The dashed
line is the evolutionary track from the accreting model. Black lines show the
evolutionary tracks of classical models and the colour code is the same as in
Fig. 11. The red lines show the border of instability regions of the classical
model while the blue lines show the same borders as in Fig. 11.


11institutetext: Fakultät für Physik und Astronomie, Institut für Theoretische
Physik II, Ruhr-Universität Bochum, 44780 Bochum, Germany

# Hadron Spectroscopy with lattice QCD: challenges and opportunities

John Bulava 11<EMAIL_ADDRESS>
###### Abstract

Ongoing challenges in computing the spectrum of hadronic resonances and
shallow bound-states from lattice QCD are reviewed. Since such states are
identified as poles in the scattering matrix, nearby non-analyticities must be
treated to analytically continue to complex center-of-mass energies.
Significant lattice spacing effects have also been observed in some channels,
necessitating a continuum limit. Recent achievements are also highlighted,
including lattice investigations of states in the charm region, baryon-baryon
scattering, and the first coupled channel meson-baryon amplitude in the
$\Lambda(1405)$ channel.

The spectrum of hadrons is a rich manifestation of the complex strong dynamics
of QCD. In addition to conventional mesons and baryons, more exotic color-
singlets are possible, such as tetra- and penta-quarks, or configurations with
constituent gluons like hybrids and glueballs. The nature of the atomic
nucleus suggests additional possibilities. Like the van der Waals forces
between atoms, residual interactions between hadrons could generate
‘molecular’ bound states akin to conventional nuclei but with meson or hyperon
constituents.

Much experimental and theoretical effort is oriented in this direction,
resulting in increasing evidence for many exotic states Chen:2022asf ;
Richard:2016eis ; Briceno:2015rlt . The exploration of these scenarios
requires the study of resonances and near-threshold bound states, which are
identified as poles in the scattering matrix analytically continued to complex
center-of-mass energies. Lattice QCD can contribute (in principle) not only by
computing scattering amplitudes at the physical point to match experiment, but
also by tracing the behaviour of states toward more tractable limits of QCD.
These limits include the isospin limit where $m_{\rm u}=m_{\rm d}$ and
electroweak interactions are absent, the chiral limit $m_{\rm u,d}\rightarrow
0$, the heavy-quark limit $m_{\rm b}\rightarrow\infty$, the $SU(3)$-flavor
limit with $m_{\rm u}=m_{\rm d}=m_{\rm s}$, and the large-$N_{\rm color}$ or
large-$N_{\rm flavor}$ limits.

However, the first-principles computation of scattering amplitudes from
lattice QCD is considerably more difficult than for stable single-hadron
states. Higher $n$-point Euclidean correlation functions are required, which
suffer from a computational cost which increases with $n$ and an exponentially
decreasing signal-to-noise ratio Beane:2010em . Furthermore, a matrix of
correlation functions is required between interpolating operators for all
relevant finite-volume states. In addition to improvements in available
computing power, these difficulties have been ameliorated by efficient
algorithms for quark propagation Peardon:2009gh ; Morningstar:2011ka , such
that precise determination of the finite-volume spectrum in many channels is
possible. These algorithms enable a factorization of the quark propagator and
reduce the construction of correlation functions to tensor contraction
operations, facilitating the computation of correlation matrices between many
interpolating operators.

A recent example demonstrating the efficacy of such methods is given by the
HadSpec collaboration in Refs. Wilson:2023hzu ; Wilson:2023anv , which
computes the finite volume charmonium spectrum (neglecting charm-quark
annihilation) on a single ensemble of $N_{\rm f}=2+1$ gauge configurations
with $m_{\pi}=391\,{\rm MeV}$. This comprehensive computation includes
interpolators for all two-hadron channels containing a charm and anti-charm
quark below $4100{\rm MeV}$, in addition to some three-meson operators. The
finite volume spectra from Refs. Wilson:2023hzu ; Wilson:2023anv  for some
irreps at total zero momentum are shown in Fig. 1. A list of the kinematically
relevant channels, all of which require corresponding interpolators, is also
given there.

Figure 1:  Taken from Refs. Wilson:2023hzu ; Wilson:2023anv  by the HadSpec
collaboration, the finite-volume spectrum of charmonium states (neglecting
charm annihilation) in some irreps with zero total momentum computed on a
single ensemble of $N_{\rm f}=2+1$ gauge configurations with
$m_{\pi}=391\,{\rm MeV}$. Three different spatial volumes are employed with
$L/a_{\rm s}=16,\,20,\,24$ as denoted on the horizontal axis. The horizontal
dashed lines indicate various thresholds, as denoted in the legend, while
solid curved lines correspond to non-interacting finite-volume energies. The
band of states in the $E^{-}$ and $T_{2}^{-}$ irreps indicates the presence of
a tensor resonance.

We turn now to the determination of scattering amplitudes from finite-volume
spectra like the one in Fig. 1. Due to the Euclidean metric signature in
lattice QCD, the Haag-Ruelle asymptotic formalism familiar to Minkowski space
Haag ; Ruelle  cannot be naively applied: the large time-separation limit of
Euclidean correlators does not generally yield on-shell scattering amplitudes
Maiani:1990ca . Fortunately, a work-around was developed by Lüscher
Luscher:1990ux  which relates the finite-volume spectrum of two-hadron states
below $n\geq 3$ hadron thresholds to the two-to-two scattering amplitude. As
demonstrated above, the finite-volume spectrum is directly computable from the
large-time limit of Euclidean correlators in lattice QCD using a variational
approach Michael:1982gb . In addition to two-to-two scattering, considerable
progress has been made on three-hadron amplitudes Romero-Lopez:2021zdo ;
Dawid:2023lcq . A recent review of the finite-volume formalism is given in
Ref. Hansen:2022csj .

The relation between two-to-two scattering amplitudes and the finite-volume
spectrum below $n\geq 3$ hadron thresholds is given by Morningstar:2017spu

${\rm det}[K^{-1}(E_{\rm cm}^{\rm FV})-B_{L}(\boldsymbol{P})]$ (1)

where $K=(2T^{-1}+i)^{-1}$ is related to the scattering $T$-matrix in the
usual way, and $B_{L}$ encodes the effect of the finite volume. The
determinant is taken over all open scattering channels and partial waves, and
is typically evaluated in a basis which is block-diagonal in the irreducible
representations of the little group for total momentum $\boldsymbol{P}$. This
infinite-dimensional relation must be truncated at some maximum angular
momentum $\ell_{\rm max}$, and holds up to corrections which are suppressed
exponentially with the spatial extent $L$.

It should be emphasized that Eq. 1 holds also for finite-volume energies
_below_ the two-particle scattering threshold, which occur for an sufficiently
attractive interaction. This uniquely provides direct constraints on the
scattering amplitude at complex center-of-mass momenta and aides the
determination of near-threshold bound-state poles. However, as first pointed
out in Ref. Green:2021qol  for $\Lambda\Lambda$ scattering, levels
significantly below threshold may overlap left-hand (cross-channel) cuts,
which are not accommodated by Eq. 1. This is not purely an academic
observation: Ref. Green:2021qol  determined a number of finite-volume levels
below the left hand cut but could not use them to constrain the amplitude
since Eq. 1 is inapplicable.

Theoretical effort is underway to augment the finite-volume formalism to
handle the closest left-hand cut which, for two heavy scattering particles
like nucleons, is due to the $t$\- and $u$-channel exchanges of a single pion.
Ref. Raposo:2023oru  replaces Eq. 1 by one constraining an alternative
infinite-volume quantity $\overline{\mathcal{K}}^{\rm os}$ and a coupling $g$.
These intermediate objects are then related to the desired amplitude by an
integral equation. By contrast, Ref. Meng:2023bmz  uses an effective field
theory approach with explicit one-pion exchange and solves an eigenvalue
problem to obtain levels both above and below the left hand cut. It is based
on earlier work which treats Eq. 1 in the plane-wave basis Meng:2021uhz , and
a previous EFT treatment of the same system Du:2023hlu . As a first
application, Ref. Meng:2023bmz  incorporates the leading left-hand cut due to
one-pion exchange in an analysis of the lattice QCD energy levels for $D^{*}D$
scattering from Ref. Padmanath:2022cvl , which are at quark masses such that
$m_{\pi}=280\,{\rm MeV}$ and the $D^{*}$ is stable. This application is
summarized in Fig. 2, which compares fits to the next-to-leading order (NLO)
effective range parametrization

$p_{\rm
cm}^{2\ell+1}\,{\cot}\,\delta_{\ell}=\frac{1}{a_{\ell}}+\frac{r}{2}p_{\rm
cm}^{2}$ (2)

to points obtained from Eq. 1 treating only the lowest contributing partial
wave against another fit including one-pion exchange. Both fits indicate the
presence of a (virtual) bound state pole.

Figure 2:  The ${}^{3}S_{1}$ and ${}^{3}P_{0}$ partial waves in $DD^{*}$
scattering from Ref. Meng:2023bmz . Shown are fits to the finite-volume
$DD^{*}$ energy levels of Ref. Padmanath:2022cvl  computed on a single $N_{\rm
f}=2+1$ ensemble with $m_{\pi}=280\,{\rm MeV}$. The points are obtained from
Eq. 1 in the leading partial wave approximation, and labelled ‘LQCs’, while a
fit to them using Eq. 2 is shown in blue and labelled ‘LQCs+ERE’. The real and
imaginary parts of fits including one-pion exchange are shown in red (labelled
‘Fit 2-Re’) and blue (‘Fit 2-Im’), respectively. The dashed lines are $ip$ and
intersect the fit at a (virtual) bound state pole. The position of the left-
hand cut is given by the vertical lines labelled ‘lhc’.

In addition to left-hand cuts, another source of systematic error which has
been studied recently is the finite lattice spacing. In practice it is
difficult to engineer gauge field ensembles at different lattice spacings with
the same physical volume, hampering a direct continuum limit of the finite-
volume spectrum. An alternative is therefore to extract a pole position at
each lattice spacing separately and perform a continuum extrapolation of the
results. This strategy was employed in Ref. Green:2021qol  to examine the
continuum limit of the $H$ dibaryon binding energy at the $SU(3)$ flavor-
symmetric point $m_{\pi}=m_{\rm K}=m_{\eta}=420{\rm MeV}$. The results are
shown in Fig. 3 and reveal significant cutoff effects in the extracted pole
position.

Figure 3:  From Ref. Green:2021qol , the binding energy of the $H$ dibaryon at
$m_{\pi}=m_{\rm K}=m_{\eta}=420\,{\rm MeV}$ determined from ensembles at six
different lattice spacings. The different points at each lattice spacing
correspond to various data selection procedures performed in the analysis.

To recap, the workflow for determining the pole positions of resonances and
bound states from lattice QCD simulations is as follows. Given an ensemble of
gauge configurations, determine the correlation functions between all relevant
two-hadron and single-hadron interpolating operators. From these correlation
functions, the finite-volume spectrum is determined as in Fig. 1. Then, Eq. 1
is employed to constrain parametrization of the scattering amplitude using the
finite-volume spectrum. Finally, these parametrizations are used to
analytically continue to the complex plane and search for poles. This last
step is illustrated in Refs. BaryonScatteringBaSc:2023ori ;
BaryonScatteringBaSc:2023zvt  which compute the coupled-channel $I=0$,
strangeness $S=1$, $\pi\Sigma-\bar{K}N$ scattering amplitude relevant for the
$\Lambda(1405)$ resonance. This analysis performed on single ensemble of
$N_{\rm f}=2+1$ gauge configurations with $m_{\pi}=200\,{\rm MeV}$ and lattice
spacing $a=0.065\,{\rm fm}$. The determination of the finite volume spectrum
requires the algorithms for all-to-all propoagation discussed above to
determine correlation matrices between $\pi\Sigma$, $\bar{K}{N}$, and single-
baryon interpolators at various total momenta. The finite volume spectrum is
used to constrain various parametrizations of the amplitude similar to Eq. 2.
The nearest left-hand cut for this system is far from the region of interest,
and is therefore ignored.

Figure 4:  From Refs. BaryonScatteringBaSc:2023ori ;
BaryonScatteringBaSc:2023zvt , the coupled-channel $\pi\Sigma-\bar{K}N$
scattering amplitude computed on a single $N_{\rm f}=2+1$ flavor ensemble with
$m_{\pi}=200\,{\rm MeV}$ and lattice spacing $a=0.065\,{\rm fm}$. The finite
volume spectrum is shown in the bottom panel, and the amplitude with
statistical errors in the top panel. The center panel shows the pole positions
found in the analytic continuation of the amplitude, indicating a virtual
bound state below $\pi\Sigma$ threshold and a resonance just below
$\bar{K}{N}$ threshold.

Parametrization constrained by the spectrum are then analytically continued to
the complex plane, resulting in the two poles shown in Fig. 4. This first
lattice QCD analysis of the coupled-channel $\pi\Sigma$-$\bar{K}N$ amplitude
clearly yields two poles in the $\Lambda(1405)$ system, albeit at the
unphysically large pion mass of $m_{\pi}=200{\rm MeV}$. The pole positions
determined here are consistent with the recent $SU(3)$ chiral effective theory
analysis of Ref. Guo:2023wes . Such approaches typically predict two poles,
generated by the flavor-$SU(3)$ octet and singlet channels. Future lattice
computations at the $SU(3)$ point employed by Ref. Green:2021qol  could test
this hypothesis. Furthermore, the methods employed here are directly
applicable at the physical point to compare with the recent (preliminary)
experimental analysis of the Glue-X collaboration Wickramaarachchi:2022mhi .
Cutoff effects in the pole positions must also be investigated for
quantitative accuracy.

In summary, the study of near-threshold bound states in lattice QCD has
reached a point where several new sources of systematic error must be
considered, including the influence of left-hand cuts and lattice spacing
effects. The finite-volume formalism is well-equipped to study these systems
since finite-volume energy levels below threshold provide direct constraints
on the amplitude in the complex plane. The ongoing theoretical developments
described here to treat the leading left-hand cuts due to single-particle
exchange will be incorporated in the analysis of numerical data, and systems
relevant for experiment are increasingly studied near or at the physical
point. Another interesting development not reviewed here concerns the
determination of spectral densities from Euclidean correlators. As reviewed in
Ref. Bulava:2023mjc , this approach has enabled the determination of inclusive
rates from lattice QCD for the first time. A formalism for extending this to
exclusive amplitudes has also been proposed Bulava:2019kbi , but has not yet
been applied.

We thank the organizers for an enjoyable MENU 2023 conference, as well as the
authors of Refs. Raposo:2023oru  and Meng:2023bmz  for discussions about their
work.

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