Why not? Because the state law prohibits it.

Sec. 26-86a. Game management. Deer hunting; permitted weapons, locations, bag limits. Consent forms; permits, selection process. (a) The commissioner shall establish by regulation adopted in accordance with the provisions of chapter 54 standards for deer management, and methods, regulated areas, bag limits, seasons and permit eligibility for hunting deer with bow and arrow, muzzleloader and shotgun, except that no such hunting shall be permitted on Sunday. No person shall hunt, pursue, wound or kill deer with a firearm without first obtaining a deer permit from the commissioner in addition to the license required by section 26-27. Application for such permit shall be made on forms furnished by the commissioner and containing such information as he may require. Such permit shall be of a design prescribed by the commissioner, shall contain such information and conditions as the commissioner may require, and may be revoked for violation of any provision of this chapter or regulations adopted pursuant thereto. As used in this section, "muzzleloader" means a rifle or shotgun of at least forty-five caliber, incapable of firing a self-contained cartridge, which uses powder, a projectile, including, but not limited to, a standard round ball, mini-balls, maxi-balls and Sabot bullets, and wadding loaded separately at the muzzle end and "rifle" means a long gun the projectile of which is six millimeters or larger in diameter. The fee for a firearms permit shall be nineteen dollars for residents of the state and sixty-eight dollars for nonresidents, except that any nonresident who is an active full-time member of the armed forces, as defined in section 27-103, may purchase a firearms permit for the same fee as is charged a resident of the state. The commissioner shall issue, without fee, a private land deer permit to the owner of ten or more acres of private land and the husband or wife, parent, grandparent, sibling and any lineal descendant of such owner, provided no such owner, husband or wife, parent, grandparent, sibling or lineal descendant shall be issued more than one such permit per season. Such permit shall allow the use of a rifle, shotgun, muzzleloader or bow and arrow on such land from November first to December thirty-first, inclusive. Deer may be so hunted at such times and in such areas of such state-owned land as are designated by the Commissioner of Environmental Protection and on privately owned land with the signed consent of the landowner, on forms furnished by the department, and such signed consent shall be carried by any person when so hunting on private land. The owner of ten acres or more of private land may allow the use of a rifle to hunt deer on such land during the shotgun season. The commissioner shall determine, by regulation, the number of consent forms issued for any regulated area established by said commissioner. The commissioner shall provide for a fair and equitable random method for the selection of successful applicants who may obtain shotgun and muzzleloader permits for hunting deer on state lands. Any person whose name appears on more than one application for a shotgun permit or more than one application for a muzzleloader permit shall be disqualified from the selection process for such permit. No person shall hunt, pursue, wound or kill deer with a bow and arrow without first obtaining a bow and arrow permit pursuant to section 26-86c. "Bow and arrow" as used in this section and in section 26-86c means a bow with a draw weight of not less than forty pounds. The arrowhead shall have two or more blades and may not be less than seven-eighths of an inch at the widest point. No person shall carry firearms of any kind while hunting with a bow and arrow under this section and section 26-86c.

(b) Any person who takes a deer without a permit shall be fined not less than two hundred dollars or more than five hundred dollars or imprisoned not less than thirty days or more than six months or shall be both fined and imprisoned, for the first offense, and for each subsequent offense shall be fined not less than two hundred dollars or more than one thousand dollars or imprisoned not more than one year or shall be both fined and imprisoned.

Sec. 26-86c. Permits to hunt deer and small game with bow and arrow. Fees. Applications. Education course requirement. No person may hunt deer or small game with a bow and arrow under the provisions of this chapter without a valid permit issued by the Commissioner of Environmental Protection pursuant to this section or section 26-86a for persons hunting deer with bow and arrow under private land deer permits issued free to qualifying landowners, or their husbands or wives, parents, grandparents, lineal descendants or siblings under that section. The fee for such bow and arrow permit to hunt deer and small game shall be forty-one dollars for residents and one hundred thirty-five dollars for nonresidents, or nineteen dollars for any person twelve years of age or older but under sixteen years of age, except that any nonresident who is an active full-time member of the armed forces, as defined in section 27-103, may purchase a bow and arrow permit to hunt deer and small game for the same fee as is charged a resident of the state. Permits to hunt with a bow and arrow under the provisions of this chapter shall be issued only to qualified applicants therefor by the Commissioner of Environmental Protection, in such form as said commissioner prescribes. Applications shall be made on forms furnished by the commissioner containing such information as he may require and all such application forms shall have printed thereon: "I declare under the penalties of false statement that the statements herein made by me are true and correct." Any person who makes any material false statement on such application form shall be guilty of false statement and shall be subject to the penalties provided for false statement and said offense shall be deemed to have been committed in the town in which the applicant resides. No such application shall contain any material false statement. On and after January 1, 2002, permits to hunt with a bow and arrow under the provisions of this chapter shall be issued only to qualified applicants who have successfully completed the conservation education bow hunting course as specified in section 26-31 or an equivalent course in another state.