First read thoroughly the Arch Wiki entry on bluetooth
Verify that your Broadcom bluetooth device is recognized with lsusb
. You should see something like:
...
Bus 002 Device 005: ID xxxx:xxxx Lite-On Technology Corp. Broadcom BCM43142A0 Bluetooth Device
...
Install the appropriate firmware package from the AUR https://aur.archlinux.org/packages/?O=0&K=broadcom. In my case (BCM43142A0), I installed bcm43142a0-firmware.
git clone https://aur.archlinux.org/packages/bcm43142a0-firmware/
makepkg -si
Then enable and start bluetooth.service
# systemctl enable bluetooth.service && systemctl start bluetooth.service