text:
Interceptor and reconnaissance aircraft
Mikoyan-Gurevich OKB
Limited active service
Primary users
Russian Air Force
Algerian Air Force
Syrian Air Force
Military of Turkmenistan
Number built
1,186,[1] or 1,190[2]
Developed into
The Mikoyan-Gurevich MiG-25 (Russian: Микоян и Гуревич МиГ-25) (NATO reporting name: Foxbat) is a supersonic interceptor and reconnaissance aircraft that was among the fastest military aircraft to enter service. Designed by the Soviet Union's Mikoyan-Gurevich bureau the first prototype flew in 1964 with entry into service in 1970. It has a top speed of Mach 2.83+ (as high as Mach 3.2, but at risk of significant damage to the engines), and features a powerful radar and four air-to-air missiles.
When first seen in reconnaissance photography, the large wing planform suggested an enormous and highly maneuverable fighter. This was during a period of time when U.S. design theories were also evolving towards higher maneuverability due to combat performance in the Vietnam War. The appearance of the MiG-25 sparked off serious concern in the west, and prompted dramatic increases in performance for the McDonnell Douglas F-15 Eagle in late 1960s. The capabilities of the MiG-25 were better understood in 1976 when Soviet pilot Viktor Belenko defected in a MiG-25 to the United States via Japan. The large wing turned out to be due to the aircraft's very heavy weight.
Production of the MiG-25 series ended in 1984 after completion of 1,190 aircraft. A symbol of the Cold War, the MiG-25 flew with a number of Soviet allies and former Soviet republics, remaining in limited service in Russia and several other nations. It remains the fastest combat aircraft ever produced.[3]
1 Design and development
1.2 Designing a new interceptor
1.2.1 Record breaker
1.2.2 Technical description
1.3 Into production
1.4 Western intelligence and the MiG-25
1.5 Later versions
2 Operational history
2.1 Over Israel
2.2 Iran-Iraq War
2.3 Persian Gulf War
4.1 Current operators
4.2 Former operators
5 Survivors
6 Specifications (MiG-25P 'Foxbat-A')
During the Cold War, Soviet Air Defence Forces, PVO (not to be confused with Soviet Air Force, VVS) was tasked with the strategic air defense of the USSR. In the decades after World War II, this meant not only to deal with accidental border violations, but more importantly to defend the vast airspace of the USSR against US reconnaissance aircraft and strategic bombers carrying free-fall nuclear bombs. The performance of these types of aircraft was steadily improved. Overflights by the very high altitude American Lockheed U-2 in the late 1950s revealed a need for higher altitude interceptor aircraft than what was then available.[4]
The subsonic Boeing B-47 Stratojet and Boeing B-52 Stratofortress strategic bombers were followed by the Mach 2-capable Convair B-58 Hustler, with the even faster North American B-70 Valkyrie on the drawing board. A major upgrade in the PVO defence system was required, and at the start of 1958 a requirement was issued for manned interceptors capable of going 3,000 km/h and fly at heights up to 27 km (88,583 ft). Mikoyan and Sukhoi responded.[5]
YE-152 and YE-152M experimental interceptor
The Mikoyan-Gurevich OKB had been working on a series of interceptors during the second half of the 1950s: the I-1, I-3U, I-7U, I-75, Ye-150, Ye-150A, Ye-152, Ye-152A, Ye-152P, and Ye-152M. The Ye-150 was noteworthy because it was built specifically to test the Tumansky R-15 engine, two of which would later be used for the MiG-25. This led to Ye-152, alternatively known as Ye-166, which set several world records.[6] The Ye-152M (converted from one of the two Ye-152 aircraft) was intended to be the definite heavy interceptor design. But before it was finished, the PVO had selected the Tupolev Tu-128. As the work on the MiG-25 was well under way, the single-engine Ye-152M was abandoned.
Designing a new interceptor
Work on the new Russian interceptor that would become the MiG-25 started in mid-1959,[7] a year before Soviet intelligence learned of the American Mach 3 A-12 reconnaissance aircraft.[8] It is not clear if the design was influenced by the American A-5 Vigilante.[7] Requirements could easily have led the design team to use a similar layout. The promise of the new design roused the military's interest in both VVS and PVO. In February 1961 the Central Committee of the Communist Party of the Soviet Union issued a joint directive with the Council of Ministers of the USSR, tasking the Mikoyan OKB with the development of an aircraft designated Ye-155, the interceptor and reconnaissance versions of which were designated Ye-155P (Perekhvatchik - radar-directed, all weather interceptor) and Ye-155R (Razvedchik-reconnaissance) respectively. On 10 March 1961, Mikoyan signed a formal order to start design work on the Ye-155.[citation needed]
The design bureau studied several possible layouts for the new aircraft. One had the engines located side-by-side, as on the MiG-19. The second had a stepped arrangement with one engine amidships, with exhaust under the fuselage, and another in the aft fuselage. The third project had an engine arrangement similar to that of the English Electric Lightning, with two engines stacked vertically. Option two and three were both rejected because the size of the engines meant any of them would result in a very tall aircraft which would complicate maintenance.[7]
The idea of placing the engines in underwing nacelles was also rejected because of the dangers of any thrust asymmetry during flight. Having decided on engine configuration there was thoughts on giving the machine variable-sweep wings and a second crew member, a navigator. Variable geometry would improve maneuverability at subsonic speed, but at the cost of decreased fuel tank capacity. Because the reconnaissance aircraft would operate at high speed and high altitude the idea was soon dropped. Another interesting but impractical idea was to improve the field performance using two RD36-35 lift-jets. Vertical takeoff and landing would allow for use of damaged runways during wartime and was studied on both sides of the Iron Curtain. The problem has always been that engines dedicated to vertical lift do not contribute with any power in horizontal flight, and occupy space in the airframe needed for fuel. The MiG interceptor would need all the fuel it could get so the idea went nowhere.[citation needed]
A variety of other roles were considered for the aircraft, including cruise missile carrier, and even a small five to seven-passenger supersonic transport, but none of these notions bore any fruit, and only the main directive of a high-altitude reconnaissance aircraft for the VVS' Frontal Aviation and heavy interceptor for PVO was ever realized.[citation needed]
Ye-155R3 Reconnaissance prototype with a 5,280 litre drop tank under the belly, 1964
The first prototype was a reconnaissance variant, designated "Ye-155-R1", and made its first flight on 6 March 1964. It had some characteristics that was unique to that prototype, and some of these were visually very evident: The wings had fixed wingtip tanks (600 litre capacity) to which small winglets were attached for stability purposes, but when it was found fuel sloshing around in the tanks caused vibrations they were eliminated. The aircraft also had attachments for movable foreplanes, canards, to help with pitch control at high speed (provisions for canards had previously been installed, but not used, on the Ye-152P.)[citation needed]
The first flight of the interceptor prototype, "Ye-155-P1", took place on 9 September 1964. Development of the MiG-25, which represented a major step forward in Soviet aerodynamics, engineering and metallurgy, took several more years to complete.[citation needed]
On July 9, 1967, the new aircraft was first shown to the public at Domodedovo air show.[citation needed]
The Mikoyan-Gurevich design bureau soon realized that the performance of the new aircraft gave it great potential to set new flight records. In addition to their normal duties, the prototypes Ye-155-P1, Ye-155-R1, Ye-155-R3 were made lighter by removing some unneeded equipment, and were used for these attempts. Under Federation Aeronautique Internationale (FAI) classification the Ye-155 type belonged to class C1 (III), which specifies jet-powered land planes with unlimited maximum take-off weight. A few records can be mentioned here:
The first claim was for world speed records with payloads of 1,000 and 2,000 kilogram. Test pilot Alexander Fedotov reached an average speed of 2,319.12 km/h over a 1,000 km circuit in 16 March 1965.
For pure speed, with no payload, test pilot Mikhail M. Komarov averaged 2,982.5 km/h over a 500 km closed circuit in 1967, three months after the new aircraft was first shown to the public at Domodedovo. On the same day A. Fedotov reached an altitude of 29,977 metres (98,350 ft) with a 1,000 kilogram payload. But the MiG could go higher and eventually it became the first aircraft to go higher than 35,000 metres (115,000 ft).[citation needed]
Time to height records were recorded on 4 June 1973 when Boris A. Orlov climbed to 20,000 m in 2 min 49.8 sec. The same day, Pyotr M. Ostapenko reached 25,000 in 3 min 12.6 and 30,000 m in 4 min 3.86 sec.
On 25 July 1973, A. Fedotov reached 35,230m with 1,000 kg payload, and 36,240 m with no load (an absolute world record).[citation needed] In the thin air, the engines flamed out and the aircraft coasted on in a ballistic trajectory by inertia alone. At the apex the speed had dropped to 75 km/h.
A few years later, on 31 August 1977, "Ye-266M" flown by MiG OKB Chief Test Pilot Alexander V. Fedotov, set the recognized absolute altitude record for a jet aircraft under its own power.[citation needed] He reached 37,650 metres (123,520 ft) at Podmoskovnoye, USSR in zoom climb (the absolute altitude is a different record than the sustained altitude in horizontal flight). The aircraft was actually a MiG-25RB re-engined with the powerful R15BF2-300. It had earlier been part of the program to improve the aircraft's top speed that resulted in the MiG-25M prototype.[7]
In all 29 records were claimed, of which seven were all-time world records for time to height, altitudes of 20,000 m and higher, and speed. Several records still stand.[7]
Because of the thermal stresses incurred in flight above Mach 2, the Mikoyan-Gurevich OKB had difficulties choosing what materials to use for the aircraft. They had to use E-2 heat-resistant plexiglass for the canopy, and high-strength stainless steel for the wings and fuselage. Using titanium rather than steel would have been ideal, but it was expensive and difficult to work. The problem of cracks in welded titanium structures with thin walls could not be solved, so the heavier nickel steel was used instead. It cost far less than titanium and allowed for welding, along with heat resistant seals.[7] The MiG-25 was constructed from 80% nickel steel alloy, 11% aluminium, and 9% titanium.[9] The steel components were formed by a combination of spot-welding, automatic machine welding and hand arc welding methods.
The MiG-25 was theoretically capable of a maximum speed of Mach 3+ and a ceiling of 90,000 ft (27,000 m). Its high speed was problematic: although sufficient thrust was available to reach Mach 3.2, a limit of Mach 2.8 had to be imposed as the turbines tended to overspeed and overheat at higher speeds, possibly damaging them beyond repair.[10][11][12][13]
Into production
Cameras of the MiG-25RB
Full scale production of the MiG-25R ('Foxbat-B') began in 1969 at the Gorkii aircraft factory (Plant No.21). The MiG-25P ("Foxbat-A") followed in 1971, and 460 of this variant was built until production ended in 1982. The improved PD variant that replaced it was built from 1978 till 1984 with 104 aircraft completed.[7] But from then on the Gorkii factory switched over production to the new MiG-31.
A non-combat trainer variant ("Foxbat-C") was also developed for each version, the MiG-25PU and MiG-25RU, respectively. The MiG-25R evolved several subsequent derivatives, including the MiG-25RB reconnaissance-bomber, the MiG-25RBS ("Foxbat-D") with side-looking airborne radar (SLAR), the MiG-25RBK ELINT aircraft, and the MiG-25BM ("Foxbat-F") SEAD variant, carrying four Raduga Kh-58 (NATO reporting name AS-11 'Kilter') anti-radiation missiles. The BM variant was introduced late into service, in 1988.[citation needed]
Western intelligence and the MiG-25
MiG-25RBSh with markings of 2nd Sqn/47th GvORAP (Guards independent recce Regiment)
Inaccurate intelligence analysis caused the West initially to believe the MiG-25 was an agile air-combat fighter rather than an interceptor. In response, the United States started a new program which resulted in the McDonnell Douglas F-15 Eagle.[14] NATO obtained a better understanding of the MiG-25's capabilities on 6 September 1976, when a Soviet Air Defence Forces pilot, Lt. Viktor Belenko, defected, landing his MiG-25P at Hakodate Airport in Japan.[15]
The pilot overshot the runway on landing, damaging the landing gear and making the MiG-25 un-airworthy. It was carefully dismantled and analyzed by the Foreign Technology Division (now the National Air and Space Intelligence Center) of the United States Air Force, at the Wright-Patterson Air Force Base. After 67 days, the aircraft was returned to the Soviets in pieces. The analysis, based on technical manuals and ground tests of engines and avionics, revealed unusual technical information:
Belenko's particular aircraft was brand new, representing the very latest Soviet technology.
The aircraft was assembled very quickly, and was essentially built around its massive Tumansky R-15(B) turbojets.
Welding was done by hand. Rivets with non-flush heads were used in areas that would not cause adverse aerodynamic drag.[16]
The aircraft was built of a nickel alloy and not titanium as was assumed (though some titanium was used in heat-critical areas). The steel construction contributed to the craft's 64,000 lb (29,000 kg) unarmed weight.
Maximum acceleration (g-load) rating was just 2.2 g (21.6 m/s²) with full fuel tanks, with an absolute limit of 4.5 g (44.1 m/s²). One MiG-25 withstood an inadvertent 11.5 g (112.8 m/s²) pull during low-altitude dogfight training, but the resulting deformation damaged the airframe beyond repair.[17]
Combat radius was 186 miles (299 km), and maximum range on internal fuel (at subsonic speeds) was only 744 miles (1,197 km) at low altitude (< 1000 meter).[7]
The airspeed indicator was redlined at Mach 2.8, with typical intercept speeds near Mach 2.5 in order to extend the service life of the engines.[15] A MiG-25 was tracked flying over Sinai at Mach 3.2 in the early 1970s, but the flight led to the destruction of its engines.[16]
The majority of the on-board avionics were based on vacuum-tube technology, not solid-state electronics. Although they represented aging technology, vacuum tubes were more tolerant of temperature extremes, thereby removing the need for providing complex environmental controls inside the avionics bays. In addition, the vacuum tubes were easy to replace in remote northern airfields where sophisticated transistor parts might not have been readily available. With the use of vacuum tubes, the MiG-25P's original Smerch-A (Tornado, NATO reporting name "Foxfire") radar had enormous power – about 600 kilowatts. As with most Soviet aircraft, the MiG-25 was designed to be as rugged as possible. The use of vacuum tubes also makes the aircraft's systems resistant to an electromagnetic pulse, for example after a nuclear blast.[18]
Later versions
As the result of Belenko's defection and the compromise of the MiG-25P's radar and missile systems, beginning in 1976, the Soviets started to develop an advanced version, the MiG-25PD ("Foxbat-E").[7] This upgrade consisted of new RP-25 Sapfir (Sapphire) look-down/shoot-down radar (the same radar as used in the MiG-23 fighter), infrared search and track (IRST) system, other electronic improvements and more powerful R15B-300 engines. About 370 earlier MiG-25Ps were converted to this standard and redesignated MiG-25PDS.[citation needed]
Plans for a new aircraft to develop the Foxbat's potential to go faster than the in-service limit of Mach 2.8 were designed as a flying prototype. Unofficially designated MiG-25M, it had new powerful engines R15BF2-300, improved radar, and missiles. This work never resulted in a machine for series production, as the coming MiG-31 showed more promise.[7]
Operational history
Over Israel
The unarmed 'B' version had greater impact than the interceptor when the USSR sent two MiG-25R, and two MiG-25RB to Egypt in March 1971 and stayed until July 1972. They were operated by the Soviet 63rd Independent Air Detachment (Det 63) set up specially for this mission. Det 63 flew over Israeli held territory in Sinai on reconnaissance missions roughly 20 times. The flights were in pairs at maximum speed and high altitude (between 17,000–23,000 m).[7]
On 6 November 1971, an Egyptian MiG-25 flying at Mach 2.5 was met by Israeli F-4Es and fired upon unsuccessfully.[11][19] A MiG-25 was tracked flying over Sinai at Mach 3.2 during this period. The MiG-25 overspeeded its engines, which led to their later destruction.[8] Unit Det 63 was sent back home in 1972, though reconnaissance Foxbats were sent back to Egypt in 19–20 October 1973 during the Yom Kippur War.[19] Unit Det 154 remained there until late 1974.
On 13 February 1981, Israeli Air Force sent two RF-4Es over Lebanon as a decoy for Syrian MiG-25 interceptors. As the MiGs scrambled, the RF-4Es turned back delivering chaffs and using ECM pods. Two IDF/AF F-15A were waiting for the MiGs and shot one of them down with AIM-7F missiles. The other MiG was able to escape. The trap was repeated on 29 July 1981 and 7 June 1982, when another two Syrian MiG-25s were downed by IDF/AF F-15As, one on each date. Finally, on 31 August 1983, a fourth Syrian MiG-25 was damaged by a Israeli Hawk SAM and then dispatched by an IDF/AF F-15.[20] No Israeli aircraft were lost to the MiG-25.
Iran-Iraq War
The MiG-25 was in service with the Iraqi Air Force during the Iran–Iraq War, but its success against Iranian fighters is largely unknown. Research by journalist Tom Cooper shows that as many as 14 MIG-25s may have been shot down by Iranian fighters during the period from 1978 to 1988,[21][22][23][page needed] although it is difficult to determine the validity of these claims. Iraqi MiG-25s made their first kill against Iran in February 1983, where an Iraqi MiG-25PD shot down an Iranian C-130. On 21 March 1984, an Iraqi MiG-25PD shot down an Iranian F-4E and on 5 June 1985 an Iraqi MiG-25PD shot down a second Iranian F-4E. On 23 February 1986 an Iraqi MiG-25PD shot down an Iranian EC-130E and on 10 June an RF-4E, later in October 1986, an Iraqi MiG-25PDS shot down a second RF-4E.[24] The most successful Iraqi MiG-25 pilot of the war was Colonel Mohommed "Sky Falcon" Rayyan, who was credited with 10 kills. Eight of these were while flying the MiG-25P from 1981 to 1986.[25]
During the Persian Gulf War, a US Navy F/A-18, piloted by Lt Cdr Scott Speicher, was shot down on the first night of the war by a missile[26] probably fired by a MiG-25.[27] The kill was reportedly made with a Bisnovat R-40TD missile fired from a MiG-25PDS flown by Lt. Zuhair Dawood of the 84th squadron of the IrAF.[28]
In another incident, an Iraqi MiG-25PD, after eluding eight U.S. Air Force F-15s, fired three missiles at General Dynamics EF-111A Raven electronic warfare aircraft, forcing them to abort their mission and leave attacking aircraft without electronic jamming support.[N 1][29]
In yet another incident, two MiG-25s approached a pair of F-15s, fired missiles (which were evaded by the F-15s), and then outran the American fighters. Two more F-15s joined the pursuit, and a total of 10 air-to-air missiles were fired at the MiG-25s, although none reached them.[30] According to the same sources, at least one F-111 was also forced to abort its mission by a MiG-25 on the first 24 hours of hostilities, during an air raid over Tikrit.[31]
Two MiG-25s were shot down by USAF F-15Cs during the Gulf War, both using AIM-7s.[32][33] After the war, on 27 December 1992, a U.S. F-16D downed a MiG-25 that violated the no-fly zone in southern Iraq with an AMRAAM missile. It was the first USAF F-16 air to air victory and the first AMRAAM kill.[34][35]
On 23 December 2002, an Iraqi MiG-25 shot down a U.S. Air Force unmanned MQ-1 Predator drone, which was performing armed reconnaissance over Iraq. This was the first time in history that an aircraft and an unmanned drone had engaged in combat. Predators had been armed with AIM-92 Stinger air-to-air missiles, and were being used to "bait" Iraqi fighter aircraft, then run. In this incident, the Predator did not run, but instead fired one of the Stingers, which missed, while the MiG's missile did not.[36][37]
No Iraqi aircraft were deployed in the U.S. invasion of Iraq in 2003, with most Iraqi aircraft being hidden or destroyed on the ground. In August 2003, several dozen Iraqi aircraft were discovered buried in the sand.
The MiG-25 was kept a guarded secret in India, and was nicknamed Garuda, after the large mythical bird-like creature from Hindu and Buddhist mythology.[38] It was used extensively in the Kargil War and Operation Parakram, conducting aerial reconnaissance sorties of Pakistan.[N 2][39]
In May 1997, an Indian Air Force Mikoyan MiG-25RB reconnaissance aircraft created a furor when the pilot flew faster than Mach 2 over Pakistani territory following a reconnaissance mission into Pakistan airspace. The MiG-25 broke the sound barrier while flying at an altitude of around 65,000 feet, otherwise the mission would have remained covert, at least to the general public. The Pakistan Government considered the breaking of the sound barrier was deliberate to make the point that the Pakistan Air Force (PAF) had no aircraft in its inventory which can come close to the MiG-25's cruising height (up to 74,000 feet). India denied the incident but Pakistan's Foreign Minister, Gohar Ayub Khan, believed that the Foxbat photographed strategic installations near the capital, Islamabad.[40]
Lack of spare parts and India's acquiring of unmanned aerial vehicles and satellite imagery eventually led to its retirement in 2006.[N 3][38][39]
Ye-133
This designation was given to one aircraft, it was used by Svetlana Savitskaya to establish a number of women's speed and height records. The Ye-133 set a women's speed record of 2,683.44-km (1.667.412 mph) on 22 June 1975.
Ye-155-R1
Reconnaissance prototype.
Ye-155-P1
Interceptor fighter prototype.
High-speed record breaking aircraft. The Ye-266 set a new speed record in a 1,000 km (620 mi) close circuit in April 1965.
Ye-266M
High-altitude record breaking aircraft. The Ye-266M set an absolute altitude record of 37,650 m (123,524 ft).
MiG-25R "Foxbat-B"
Single-seat reconnaissance aircraft, fitted with cameras, sensors and ECCM equipment.
MiG-25P "Foxbat-A"
Single-seat all-weather interceptor fighter aircraft, powered by two Tumansky R-15B-300 turbojet engines, armed with four R-40 air-to-air missiles[41][page needed]
MiG-25PU "Foxbat-C"
Two-seat trainer aircraft, with a new nose section fitted with two separate cockpits, lacks radar and has no combat capability.
MiG-25RB "Foxbat-B"
Single-seat reconnaissance-bomber[citation needed] aircraft, fitted with cameras and a Peleng automatic bombing system.[citation needed] The aircraft can carry a bombload of six 1,102 lb (500 kg) bombs or one nuclear weapon.
MiG-25RBS "Foxbat-D"
Single-seat reconnaissance-bomber aircraft, fitted with cameras and a side looking airborne radar (SLAR), equipped with a Sabla radio location system.
MiG-25RBSh
MiG-25RBS "Foxbats" fitted with new equipment.
MiG-25RBK
Single-seat Elint aircraft, fitted with a side looking airborne radar.
MiG-25RBT
Single-seat Elint aircraft.
MiG-25RBV
Single-seat reconnaissance-bomber[citation needed] aircraft, fitted improved equipment.
MiG-25RU
Two-seat reconnaissance, trainer aircraft, it has no operational equipment.
MiG-25BM "Foxbat-F"
Single-seat defence-suppression aircraft, armed with Kh-58 or Kh-31 air-to-surface missiles.[42]
MiG-25PD "Foxbat-E"
Single-seat all-weather interceptor fighter aircraft, fitted with a RP-25 look-down/shoot down radar, and an undernose IR search and track system, powered by two Tumansky R-15BD-300 turbojet engines, armed with two R-40 and four R-60 air-to-air missiles.
MiG-25PDS
This designation was applied to all surviving MiG-25P "Foxbat A"s. The aircraft were subsequently upgraded in the early 1980s.
MiG-25 Operators 2010 (former operators in red)
Current operators
Algerian Air Force – 48 MiG-25s purchased from Ukrainian stocks. These included MiG-25PDS, RBV, PU, and RU types.[7] 11 were in service in January 2010, including five MiG-25Ps, three MiG-25PDs, and three MiG-25RBSHs.[43]
Armenian Air Force – one in inventory as of November 2008.[44] One MiG-25PD based at Gyumri.[7]
Azerbaijan Air Force – received eight MiG-25PDs, 14 MiG-25RBs and six MiG-25 trainers.[7] Five of these aircraft were in use in November 2008.[44]
Military of Kazakhstan – 16 in inventory as of November 2008.[44]
MiG-25.
Russian Air Force – 42 MiG-25RBs are in service as of January 2010.[43]
Syrian Air Force – received 16 MiG-25PDs, eight MiG-25RBs and two trainers;[7] 40 were in use as of November 2008,[44] including MiG-25Rs and a MiG-25U.[45]
Military of Turkmenistan – 24 (MiG-25PD/MiG-25PUs[7]) in inventory as of November 2008.[44]
Former operators
Iraqi MiG-25 at the National Museum of the United States Air Force in Dayton, Ohio. October 2007
MiG-25 of the Libyan Air Force
MiG-25 at the Ukrainian Air Force Museum in Vinnitsa
Bulgarian Air Force – Three MiG-25RBTs (#731, #736 and #754) and one MiG-25RU (#51) aircraft were delivered in 1982. On 12 April 1984, #736 crashed near Balchik. The pilot ejected successfully. They were operated by 26th RAB at Dobrich until their withdrawal. In May 1991, the surviving MiG-25s were returned to the USSR in exchange for five MiG-23MLDs.
Belarus Air Force – Had up to 50 MiG-25s, including 13 MiG-25PDs; by 1995 the type had been withdrawn.[7]
Indian Air Force – Took delivery of six MiG-25RBKs and two MiG-25RUs in 1981. They were operated by No. 102 Squadron "Trisonics" based at Bakshi-ka-talab AB in Bareilly, Uttar Pradesh. One RBK crashed on 3 August 1994.[7] Retired from service in May 2006.[38] The Trishul air-base in Bareilly had Foxbats capable of flying up to 80,000 ft.
Iraqi Air Force – Had seven MiG-25PUs, nine MiG-25Rs, and 19 MiG-25PD/PDSs as of January 1991. During Operation Desert Storm most of them were destroyed on the ground,[46] Two were shot down during and seven were flown over to Iran.[47]
Georgian Air Force
Libyan Air Force – Operated a large number of MiG-25s, some sources say more than 60 were delivered. Types were of the MiG-25PD, MiG-25RBK, MiG-25PU and MiG-25RU variants. They were operated by No.1025 Squadron at Jufra-Hun, No.1055 Squadron at Ghardabiya and an unidentified squadron at Sabha Air Base. Current status of the force is unknown.[7]
Soviet Union – The largest operator historically, Soviet aircraft were passed on to its successor states in 1991.
Soviet Air Force
Soviet Anti-Air Defence
Ukrainian Air Force – Took over 79 aircraft after the breakup of the USSR.[7] They have been withdrawn from service. It is reported some have been sold to countries in the third-world.[citation needed]
MiG-25RB (s/n 25105) is in the restoration facility at the National Museum of the United States Air Force in Dayton, Ohio. This aircraft was found in 2003 during the opening months of Operation Iraqi Freedom by American forces, buried in the sand near Al Taqaddum Airbase, about 250 km west of Baghdad. The aircraft had been buried to prevent its destruction on the ground by coalition aircraft. When uncovered, the MiG-25RB was incomplete, as the wings could not be located. This aircraft was one of two MiG-25s transported by a Lockheed C-5A Galaxy from Iraq to Wright-Patterson Air Force Base for examination. It was donated to the National Museum of the United States Air Force in December 2006. The museum's restoration staff is currently attempting to locate a set of wings to complete the aircraft for display.[48]
MIG - 25 is in display at the Indian Air Force Museum at Palam, New Delhi.
Specifications (MiG-25P 'Foxbat-A')
Data from The Great Book of Fighters,[49][page needed] International Directory of Military Aircraft[50]Combat Aircraft since 1945 [1]
Crew: One
Length: 19.75 m (64 ft 10 in)
Wingspan: 14.01 m (45 ft 11.5 in)
Height: 6.10 m (20 ft 0.25 in)
Wing area: 61.40 m² (660.93 ft²)
Empty weight: 20,000 kg (44,080 lb)
Loaded weight: 36,720 kg (80,952 lb)
Powerplant: 2 × Tumansky R-15B-300 afterburning turbojets
Dry thrust: 73.5 kN (16,524 lbf) each
Thrust with afterburner: 100.1 kN (22,494 lbf) each
Maximum speed:
High altitude: Mach 3.2[8] (3,600 km/h, 2,170 mph); Mach 2.83 (3,200 km/h, 1,920 mph) continuous engine limit[8]
Low altitude: 1,200 km/h (648 knots, 746 mph)[50]
Range: 1,730 km (935 nmi, 1,075 mi) with internal fuel
Ferry range: 2,575 km (1,390 nmi)
Service ceiling: 20,700 m (with four missiles) (67,915 ft; over 80,000 ft (24.4 km) for RB models)
Rate of climb: 208 m/s (40,950 ft/min)
Wing loading: 598 kg/m² (122.5 lb/ft²)
Thrust/weight: 0.41
Time to altitude: 8.9 min to 20,000 m (65,615 ft)
2x radar-guided R-40R (AA-6 "Acrid") air-to-air missiles, and
2x infrared-guided R-40T missiles
RP-25 Smerch radar
A RV-UM or a RV-4 radar altimeter
Aviation portal
Related development
Aircraft of comparable role, configuration and era
Lockheed YF-12
North American XF-108 Rapier
List of military aircraft of the Soviet Union and the CIS
^ Quote: "But as the Ravens began their second orbit in a counterclockwise turn toward the Syrian border (over Al-Qaim), a MiG-25 suddenly darted toward them at high speed. The Iraqi fired one air-to-air missile at the lead Raven and two at his wingman. The missiles flew wide, but the Ravens dived to escape and then, uncertain where the MiG was lurking, turned back to Saudi Arabia."
^ Quote: "The MIG 25 was extensively used in the Kargil conflict in 1999 and also during Operation Parakram 2001."
^ Quote: "UAVs and Satellite Imagery have made these aircraft obsolete to an extent, however these are still useful for strategic reconnaissance. Spares are a major problem as per Air Marshal A K Singh, C in C Western Air Command."
^ a b Wilson 2000, p. 103.
^ "Aircraft Museum: MiG-25 'Foxbat'." Aerospaceweb.org. Retrieved: 5 September 2010.
^ "Global Aircraft – Top 50 Fastest Aircraft." The Global Aircraft Organization, 24 April 2007. Retrieved: 30 June 2011.
^ Rich and Janos 1994, p. 15.
^ Gordon and Gunston 2000, p. 166.
^ a b c d e f g h i j k l m n o p q r s t Gordon 2008
^ a b c d Spick 2000
^ Eden 2004, p. 308.
^ "Intelligence: Big-Mouth Belenko." Time, 11 October 1976. Retrieved: 12 May 2010.
^ a b Goebel, Greg. "MiG-25." vectorsite.net. Retrieved: 5 September 2010.
^ "Mikoyan MiG-25 Foxbat."[dead link] aeroweb.lucia.it. Retrieved: 5 September 2010.
^ Gunston and Spick pp. 132–133.
^ Jenkins 1998, pp. 6–7.
^ a b Barron 1980, p. 15.
^ a b Barron 1980, pp. 169–171.
^ "MiG-25." globalaircraft.org. Retrieved: 5 September 2010.
^ Broad, William J. "Nuclear Pulse (I): Awakening to the Chaos Factor." Science, Volume 212, 29 May 1981, pp. 1009–1012.
^ a b "Foxbats over Sinai." spyflight.co.uk. Retrieved: 5 September 2010.
^ "MiG-25 Foxbat." spyflight.co. Retrieved: 26 July 2011.
^ "Iranian Air-to-Air Victories 1982-today." acig.org, 19 May 2006.
^ "Iranian Air-to-Air Victories 1976-1981." acig.org, 19 May 2006.
^ Cooper and Bishop 2004
^ Mevlutoglu, Arda. "Airshow Turkiye 2011."[dead link] ACIG, 11 June 2011. Retrieved; 30 June 2011.
^ Nicolle and Cooper 2004, pp. 82, 86.
^ "Intelligence Community Assessment of the Lieutenant Commander Speicher Case". CIA, FOIA Electronic Reading Room, 27 March 2001. Retrieved: 10 September 2006.
^ Weiner, Tim. "With Iraq's O.K., a U.S. Team Seeks War Pilot's Body." The New York Times, 14 December 1995, p. A1.
^ Sadik, A. and D. Zampini. "Tretij Den' (i posledujuschie...)" ["The Third Day (and beyond...)."] Aviacija i vremja (Aviation and Time), No. 6, 2005.
^ Atkinson 1993, pp. 125–126.
^ Atkinson 1993, p. 75.
^ http://www.youtube.com/watch?v=OZ5N58z9UUM
^ http://www.rjlee.org/air/ds-aakill/
^ http://www.sci.fi/~fta/amraamsrc.htm
^ http://www.youtube.com/watch?v=UrYcxXIg5fI&NR=1
^ Krane, Jim. "Pilotless Warriors Soar To Success." CBS News, 25 April 2003.
^ "Video of Shoot-Down." CBS. Retrieved: 5 September 2010.
^ a b c "India retires Cold War spy MiGs." BBC News. 9 April 2006. Retrieved: 30 June 2011.
^ a b Bhonsle 2006, p. 256.
^ Steinemann, Peter. "VayuSena: Recce Incursion." Air Power International. Retrieved: 5 September 2010.
^ Gunston 1980
^ Gunston and Spick 1983, pp. 132–133.
^ a b "World Military Aircraft Inventory". 2010 Aerospace, Aviation Week and Space Technology, January 2010.
^ a b c d e "Directory: World Air Forces." Flight International, 11–17 November 2008.
^ "Syria - Air Force Equipment." globalsecurity.org. Retrieved: 5 September 2010.
^ "Iraqi Perspectives Project Phase II. Um Al-Ma'arik (The Mother of All Battles): Operational and Strategic Insights from an Iraqi Perspective, Volume 1 (Revised May 2008)." Institute for Defense Analysis, May 2008.
^ "Iraqi Air Force Equipment - Introduction." globalsecurity.org. Retrieved: 26 July 2011.
^ "MiG-25 fact sheet." nationalmuseum.af.mil. Retrieved: 27 October 2010.
^ Green and Swanborough 2001
^ a b Frawley 2002, p. 123.
Atkinson, Rick. Crusade: The Untold History of the Persian Gulf War. New York: Houghton Mifflin Company, 1993. ISBN 978-0395710838.
Barron, John. MiG Pilot: The Final Escape of Lt. Belenko. New York: Mcgraw-Hill, 1980. ISBN 0-380-53868-7.
Bhonsle, Brig. Rahul K. India: Security Scope 2006 The New Great Game". Delhi, India: Kalpaz Publications, 2006. ISBN 81-7835-512-4.
Cooper, Tom and Farzad Bishop. Iranian F-14 Units in Combat. London: Osprey Publishing, 2004. ISBN 978-1841767871.
Davies, Steve. Combat Legend, F-15 Eagle and Strike Eagle. London: Airlife Publishing, Ltd., 2002. ISBN 1-84037-377-6.
Eden, Paul, ed. "Mikoyan MiG-25 'Foxbat'". "Mikoyan MiG-31 'Foxhound'". Encyclopedia of Modern Military Aircraft. London: Amber Books, 2004. ISBN 1-90468-784-9.
Frawley, Gerald. "Mikoyan MiG-25." The International Directory of Military Aircraft, 2002/2003. Fyshwick, ACT, Australia: Aerospace Publications, 2002. ISBN 1-875671-55-2.
Gordon, Yefim. Mikoyan MiG-25 Foxbat: Guardian of the Soviet Borders (Red Star Vol. 34). Hinckley, UK: Midland Publishing Ltd., 2008. ISBN 978-1857802597.
Gordon, Yefim and Bill Gunston. Soviet X-Planes. Earl Shilton, Leicester, UK: Midland Publishing Ltd., 2000. ISBN 978-1857800999.
Green, William and Gordon Swanborough. The Great Book of Fighters. St. Paul, Minnesota: Motorbooks International Publishing, 2001. ISBN 0-7603-1194-3.
Gunston, Bill. An Illustrated Guide to Modern Fighters and Attack Aircraft. London: Salamander Books, 1980. ISBN 0-66804-964-2.
Gunston, Bill and Mike Spick. "Mikoyan/Gurevich MiG-25." Modern Air Combat: The Aircraft, Tactics and Weapons Employed in Aerial Combat Today. New York: Crescent Books, 1983. ISBN 978-0517412657.
Jenkins, Dennis R. McDonnell Douglas F-15 Eagle: Supreme Heavy-Weight Fighter. Hinckley, UK: Midland Publishing, 1998. ISBN 1-85780-081-8.
Nicolle, David and Tom Cooper. Arab MiG-19 and MiG-21 Units in Combat (Osprey Combat Aircraft 044). Oxford, UK: Osprey Publishing, 2004. ISBN 978-1841766553.
Rich, Ben and Leo Janos. Skunk Works: A Personal Memoir of My Years of Lockheed. New York: Little, Brown & Company, 1994. ISBN 0-316-74300-3.
Spick, Mike. The Great Book of Modern Warplanes. St. Paul, Minnesota: Motorbooks International Publishing, 2000. ISBN 0-7603-0893-4.
Wilson, Stewart. Combat Aircraft since 1945. Fyshwick, Australia: Aerospace Publications, 2000. ISBN 1-875671-50-1.
MiG-25 page on GlobalSecurity.org
MiG-25/31 at Greg Goebel's Air Vectors site
MiG-25 page on Milavia.net
MiG-25 vs. SR-71 Blackbird discussion
MiG-25 Foxbat at Russian Military Analysis
Foxbat and Foxhound - Australian Aviation
Recce Incursion - Famous incident of IAF MiG-25 intruding into Pakistan airspace
Photo Russian MiG-25 in flight - 2007
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text:
Zgniła Rega – struga na Wybrzeżu Trzebiatowskim, w woj. zachodniopomorskim, w gminie Trzebiatów; prawobrzeżny dopływ Starej Regi.

Struga bierze źródła w na północny zachód od kolonii wsi Trzebusz. Biegnie w kierunku północno-wschodnim do drogi wojewódzkiej nr 109, a po jej minięciu i ok. 2 km swojego biegu odbija na północ w kierunku Mrzeżyna. Ok. 1 km od ujścia, wpada od lewego brzegu duży rów melioracyjny. Zgniła Rega uchodzi do Starej Regi przy jazie i przepompowni przed Mrzeżynem.

Tereny nad Zgniłą Regą są podmokłe i zabagnione, porosłe wysokimi krzewami. Wody gruntowe na tym obszarze mają znaczny odsetek soli pochodzenia morskiego. W rejonie strugi występują rośliny solniskowe, m.in. aster solny, babka nadmorska, świbka morska.

Według Mapy Podziału Hydrograficznego Polski struga należy do zlewni rzeki Regi.

Zgniła Rega w całości należy do dwóch nachodzących na siebie obszarów programu Natura 2000 tj. obszaru specjalnej ochrony ptaków "Wybrzeże Trzebiatowskie" oraz obszaru ochrony siedlisk Trzebiatowsko-Kołobrzeskiego Pasa Nadmorskiego.

Nazwę Zgniła Rega wprowadzono urzędowo w 1949 roku, zastępując poprzednią niemiecką nazwę strugi – Faule Rega.

Zobacz też
 Rega
 Rzeki Polski

Przypisy

Dorzecze Regi

Strugi

redpajama_set_name: RedPajamaWikipedia


Previous post: …but don't you ever get bored?
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text:
\section{Introduction}
Despite the enormous success of the Standard Model (SM), we are certain that the SM
is an effective theory with a cutoff  that is much smaller than the Planck scale.
 A lot of effort is being devoted to constructing and studying extensions
 of the SM that predict new particles having TeV scale masses.
 An integral part of this effort is constraining the parameters of new models
 using experimental data. In many cases, the constraints are obtained by
 directly computing the deviations from the SM that are induced by new particles
 in a specific model.

 An effective field theory approach is a two step process. First, one integrates
 out all new heavy states and obtains effective interactions involving only fields of the SM.
 These new effective higher-dimensional operators are then used to compute the deviations
 from the SM  and compare with the experimental data. Following this
 approach one needs to make contact with experimental data only
 once---by computing the effects of higher-dimensional operators on different experiments and
obtaining bounds on the coefficients of such operators using the data. Once this
step is completed, one can constrain any model just by calculating the coefficients
of new effective operators.

The effective theory approach is by no means new, and has been applied to the
electroweak data many times.
Lucid explanation on applying  effective Lagrangians to precision electroweak
measurements  can be found, for example, in  Refs.~\cite{Buchmuller,GW}. Perhaps the best
known example of the effective approach are the so-called oblique corrections~\cite{oblique,STU}.
The oblique corrections only modify the SM gauge boson propagators. The formalism
of oblique corrections has been extended in Refs.~\cite{beyondSTU,LEP12}.
While in many models new physics contributions are limited to two point functions of gauge bosons,
it is not always the case.

In this article we study bounds on the coefficients of effective operators that could be
relevant for the physics of electroweak symmetry breaking. We assume that just above the
electroweak scale the $SU(2)_L\times U(1)_Y$ gauge symmetry is linearly realized and
therefore the field content  involves a scalar electroweak doublet.
This determines the power counting and our  effective Lagrangian
 \begin{equation}
 \label{eq:Lagrangian}
   {\mathcal L}=  {\mathcal L}_{SM} + a_i \, O_i
 \end{equation}
contains operators of dimension 6 in addition to the SM Lagrangian. Therefore
each coefficient $a_i$ has the dimension of inverse mass squared and can be conveniently
represented as  $a_i=\frac{1}{\Lambda_i^2}$. In our analysis,
we include 21 operators---two of which correspond to the $S$ and $T$ parameters~\cite{STU}.
Our choice of operators $O_i$ is explained in the next section.
In summary, we choose operators that conserve $U(3)^5$ flavor symmetry of the SM,
as well as conserve CP. We also restrict ourselves to operators that  are stringently constrained
by the usual set of electroweak precision measurements. There is a number of dimension
6 operators that are flavor and CP conserving but the available data is not accurate enough
to place stringent bounds on the corresponding $\Lambda_i$'s. The bounds on the scales
$\Lambda_i$ of the 21 individual operators in our basis  are all about 1 TeV or higher.
In contrast, the bounds on flavor conserving four-quark contact interactions or operators
involving quarks and gluons are usually much lower. While it is exciting that operators suppressed
by relatively low scales $\Lambda$ are allowed, such operators are not very useful for
constraining new models.

Because of the spectacular agreement between the SM and precision experiments
the allowed deviations from the SM are small. Therefore, when computing the effects of operators $O_i$
in Eq.~(\ref{eq:Lagrangian}) we can restrict ourselves to computing only the interference
terms between ${\mathcal L}_{SM} $ and the operators $O_i$. In other words, we work to the
linear order in the coefficients $a_i$. Expansion in powers of $a_i$ corresponds to expansion
in $\frac{E^2}{\Lambda_i^2}$ or $\frac{v^2}{\Lambda_i^2}$, where $E$ is the characteristic
energy scale of a given process and $v$ is the Higgs vacuum expectation value (vev).
Given that $E,v \ll \Lambda_i$, neglecting the terms quadratic
in $a_i$ is a good approximation. The SM predictions can be computed
to arbitrary accuracy since they do not depend on $a_i$. We then compare the results with
the experimental data and compute the $\chi^2$ distribution as a function of $a_i$.
Since our results are linear in the coefficients $a_i$, $\chi^2$ is quadratic in $a_i$
and can be written as
\begin{equation}
\label{eq:main}
  \chi^2(a_i)=\chi^2_{min} + (a_i - \hat{a}_i)  {\mathcal M}_{ij} (a_j-\hat{a}_j),
\end{equation}
 where $\hat{a}_i$ are the values of $a_i$ that minimize $\chi^2$.

The main result of our paper is the matrix ${\mathcal M}_{ij}$ and the vector $\hat{a}_i$
in Eq.~(\ref{eq:main}). We would like to stress an obvious point here---the coefficients
${\mathcal M}_{ij}$ and $\hat{a}_i$ are constants that we obtain by fitting to experimental data.
Given this, admittedly, large set of numbers one can compute
the bounds on any linear combination of operators $O_i$ in our basis.
Integrating out heavy particles from an  extension of the SM will, in general,
lead to a set of operators $O_i$ whose coefficients are correlated by the underlying theory.
Since we are working in the linear approximation in terms of the coefficients $a_i$, the effects
of correlated operators are also linear and the analysis of bounds on a new model
remains very simple. Readers interested in applying our work to their favorite model
can use  Eq.~(\ref{eq:main}) and skip all the details on how we obtain the $\chi^2$ distribution.

An effective Lagrangian analysis has been performed in many cases in the past.
For examples see Refs.~\cite{Buchmuller,GW,CGB,
Appelquist:1993ka,BarbieriStrumia,BGKLM,Arzt:1994gp}.
Ref.~\cite{BarbieriStrumia} is perhaps the closest to our
approach, but it only includes the operators constrained by the Z-pole measurements
and considers bounds on individual operators.
We focus on a broader set of effective operators and
use the results from LEP1, SLD, LEP2, as well as lower energy experiments to bound
the coefficients of the effective operators. The most important set of precision
electroweak measurements come from LEP and SLD. Since these experiments
no longer collect data, the precision of the data relevant for our analysis is not going
to significantly improve in the near future.

In the next section we enumerate the operators in our basis. We further explain what motivates
our choice of the 21 operators.  In Sec.~\ref{sec:Experiments},  we list the experiments that
we use to obtain the bounds on the effective operators. We outline our calculations of the
interference terms between the SM and the additional operators in Sec.~\ref{sec:Calculations},
and describe the numerical analysis in Sec.~\ref{sec:chi2}.
We summarize the results in Sec.~\ref{sec:Summary}.
Our numerical results are presented in Appendix~\ref{app:Matrix}. We show how to
use these results in Appendix~\ref{app:ST} by reproducing the bounds on the $S$ and
$T$ parameters. Another example is provided in Appendix~\ref{app:Zprime}, where
we illustrate our procedure on $Z'$ gauge bosons and compare our results
with Ref.~\cite{Zprime}. We performed the numerical analysis using Mathematica, and we make
the notebooks available online. A few comments about our code are contained
in Appendix~\ref{app:program}.

 \section{Operators}
 \label{sec:Operators}
We assume that just above the electroweak symmetry breaking scale, the effective
theory is that of the SM with one Higgs doublet.
In our effective theory the $SU(2)_L \times U(1)_Y$ symmetry is linearly realized.
The assumption of one Higgs doublet is not at all relevant since
it is  only the vev of the Higgs that enters the analysis.
Since the electroweak symmetry is linearly realized the physical Higgs and the
electroweak breaking  vev are assigned mass dimension one.
The vev appearing in the Lagrangian is always raised to positive powers.
In case of nonlinearly realized electroweak symmetry, the expansion
parameter is the momentum divided by the vev times $4 \pi$.
A number of authors studied electroweak precision data in nonlinear
realizations of electroweak symmetry,
for examples see Refs.~\cite{Longhitano,Appelquist:1993ka,lectures}.

A complete set of dimension-6 operators consistent with
the $SU(3)\times  SU(2)_L \times U(1)_Y$, baryon and lepton number conservation
and the linearly realized electroweak symmetry has been presented in Ref.~\cite{Buchmuller}.
There are 80 operators in the basis of Ref.~\cite{Buchmuller}
after the leading order equations of motion are used
to obtain an independent set of operators~\cite{Politzer,Arzt:1993gz}.

We are interested in constraining models of new physics pertinent to the electroweak
symmetry breaking. Processes that contribute to flavor or CP violation have to  be
suppressed by  scales much higher than the electroweak scale. A typical suppression for
 four fermion operators that contribute to the $K-\bar{K}$ mass difference is about
$10^3~{\rm TeV}$~\cite{Buchmuller}. The bounds are even more stringent in the lepton
sector: about $10^4~{\rm TeV}$ suppression is required for the electric dipole moment
of the electron~\cite{Regan:2002ta,Bernabeu:2004ww}; the limits on the $\mu\rightarrow e\gamma$
decay also imply $10^4~{\rm TeV}$ suppression for the contributing operators~\cite{Buchmuller}.
Thus it is natural to assume that the electroweak scale and the scales associated with flavor
and CP violation are well separated.  Processes involving the third generation are an
obvious exception. Experiments have limited statistics and new flavor-dependent physics
at TeV scale is possible. It is plausible that the third generation actively participates
in the electroweak symmetry breaking. This interesting case merits a separate study
but is beyond the scope of this article. We therefore impose $U(3)^5$ flavor symmetry
on our operators. A different $U(3)$ acts on the  left-handed quarks and leptons as well as on
the right handed quarks and leptons. Consequently, our operators are
unchanged when written in terms of either mass or gauge eigenstates.

Among the 80 operators of dimension 6 listed in Ref.~\cite{Buchmuller}, there are 28 operators
that do not conserve CP or flavor $U(3)^5$, or violate both. Among the remaining 52 operators
there are 18 operators that only involve quark and/or gluon fields. The bounds on such operators
are poor since the precision of hadron experiments is not comparable to that of the $e^+ e^-$ machines.
Such operators are therefore not helpful in constraining models with new physics at the TeV scale.

Thus, there are 34 operators that conserve flavor and CP and contain either electroweak gauge
bosons or leptons and perhaps some quark fields. Six of these 34 operators are not observable
in current experiments: either they contribute to dimension 4 couplings in the SM
Lagrangian or they involve the Higgs doublet only.

We are finally left with 28 operators. Seven operators in this set are of the form
\begin{equation}
\label{eq:GammaoverMZ}
 O_{fF}= i (\bar{f} \gamma^\mu D^\nu f) \, F_{\mu \nu},
\end{equation}
where $f$ represents a fermion and $F_{\mu \nu}$ is the  field strength for
the hypercharge or weak gauge bosons. These operators can only contribute to the
Z-pole measurements. At other energies the interference term between the SM contribution and
the contribution of operators  $O_{fF}$ vanishes since one of the contributions is real
and the other is imaginary. For the same reason the interference term at the Z pole
is suppressed by the ratio of the Z width to the Z mass.
We therefore neglect the operators $O_{fF}$ in our analysis.

We choose the following basis for the remaining 21 operators that are the focus of our work.
Our notation is standard: $q$ and $l$ represent the three families of the
left-handed quark and lepton fields, respectively. The right handed fields are labeled $u$, $d$, and $e$.
We omit the family index which is always summed over due to the flavor $U(3)^5$ symmetry.
We adopt the notation of Ref.~\cite{Buchmuller} with minor modifications
and  in a few cases our operators do differ from Ref.~\cite{Buchmuller} by a numerical factor.

The operators that contain only the gauge bosons and Higgs doublets are
\begin{equation}
\label{eq:owbh}
 O_{W\!B}=(h^\dagger \sigma^a h) W^a_{\mu \nu} B^{\mu \nu}, \  \  \   O_h = | h^\dagger D_\mu h|^2,
\end{equation}
where $W^a_{\mu \nu}$ is the $SU(2)$ field strength, $B_{\mu \nu}$ the hypercharge field strength,
and $h$ represents the Higgs doublet.
There are 11 four-fermion operators. These are
\begin{eqnarray} &&
  O_{ll}^s=\frac{1}{2} (\overline{l} \gamma^\mu l) (\overline{l} \gamma_\mu l), \ \ \
  O_{ll}^t=\frac{1}{2} (\overline{l} \gamma^\mu \sigma^a l) (\overline{l} \gamma_\mu \sigma^a l),
      \label{eq:oll} \\ &&
  O_{lq}^s= (\overline{l} \gamma^\mu l) (\overline{q} \gamma_\mu q), \ \ \
  O_{lq}^t= (\overline{l} \gamma^\mu \sigma^a l) (\overline{q} \gamma_\mu \sigma^a q),
     \label{eq:olq} \\ &&
  O_{le}= (\overline{l} \gamma^\mu l) (\overline{e} \gamma_\mu e),  \ \ \
  O_{qe}=(\overline{q} \gamma^\mu q) (\overline{e} \gamma_\mu e),
     \label{eq:olqe}  \\ &&
  O_{lu}= (\overline{l} \gamma^\mu l) (\overline{u} \gamma_\mu u),  \ \ \
  O_{ld}= (\overline{l} \gamma^\mu l) (\overline{d} \gamma_\mu d),
      \label{eq:olud} \\ &&
  O_{ee}=\frac{1}{2} (\overline{e} \gamma^\mu e) (\overline{e} \gamma_\mu e), \ \ \
  O_{eu}=(\overline{e} \gamma^\mu e) (\overline{u} \gamma_\mu u),  \ \ \
  O_{ed}=(\overline{e} \gamma^\mu e) (\overline{d} \gamma_\mu d).
      \label{eq:oeeud}
  \end{eqnarray}
  The operators in Eqs.~(\ref{eq:oll}) and (\ref{eq:olq}) involve only left-handed fields,
  in Eqs.~(\ref{eq:olqe}) and (\ref{eq:olud}) 2 left-handed and 2 right-handed,
  while  in Eq.~(\ref{eq:oeeud})  all right-handed fields. There are 7 operators containing
  2 fermions that alter the couplings of fermions to the gauge bosons
 \begin{eqnarray} &&
   O_{hl}^s = i (h^\dagger D^\mu h)(\overline{l} \gamma_\mu l) + {\rm h.c.}, \ \ \
   O_{hl}^t = i (h^\dagger \sigma^a D^\mu h)(\overline{l} \gamma_\mu \sigma^a l)+ {\rm h.c.},
     \label{eq:ohl} \\ &&
   O_{hq}^s = i (h^\dagger D^\mu h)(\overline{q} \gamma_\mu q)+ {\rm h.c.}, \ \ \
   O_{hq}^t = i (h^\dagger \sigma^a D^\mu h)(\overline{q} \gamma_\mu \sigma^a q)+ {\rm h.c.},
    \label{eq:ohq} \\ &&
      O_{hu} = i (h^\dagger D^\mu h)(\overline{u} \gamma_\mu u)+ {\rm h.c.}, \ \ \
   O_{hd} = i (h^\dagger D^\mu h)(\overline{d} \gamma_\mu d)+ {\rm h.c.},
           \label{eq:Ohud}  \\ &&
   O_{he} = i (h^\dagger D^\mu h)(\overline{e} \gamma_\mu e)+ {\rm h.c.}\,.
     \label{eq:ohe}
\end{eqnarray}
Finally, there is an operator that  modifies the triple gauge  boson interactions
 \begin{equation}
 \label{eq:ow}
  O_W = \epsilon^{abc} \, W^{a \nu}_{\mu} W^{b\lambda}_{\nu} W^{c \mu}_{\lambda}.
\end{equation}
Eqs.~(\ref{eq:owbh}) through (\ref{eq:ow}) define our basis of the 21 operators.

We denote the coefficients $a_i$ in the Lagrangian in Eq.~(\ref{eq:Lagrangian}) using the same
indices as the corresponding operators, so that the effective Lagrangian is
 \begin{equation}
 \label{eq:effLagrangian}
   {\mathcal L}=  {\mathcal L}_{SM} + a_{W\!B} \, O_{W\!B} + a_h \, O_h + \ldots + a_W\,   O_W.
 \end{equation}
 Note that the first two operators in our basis, $O_{W\!B} $ and $O_h$, are in a one-to-one
relation with the $S$ and $T$ parameters~\cite{STU}. The correspondence is
 \begin{equation}
 \label{eq:ST}
  a_{W\!B}= \frac{1}{4 s c} \frac{\alpha}{v^2} S, \\\ a_h = - 2 \frac{ \alpha}{v^2} T,
 \end{equation}
 where $\langle h \rangle =\left( \begin{array}{c}  0 \\ v/\sqrt{2} \end{array} \right)$,
 $\alpha$ is the fine-structure constant, $s$ and $c$ are the sine and cosine
 of the weak mixing angle, respectively.
 The $U$ parameter is related to a dimension-8 operator in our power counting scheme
 and therefore does not appear in our analysis.

 \section{Experiments}
  \label{sec:Experiments}

The three most precisely measured electroweak sector observables: $\alpha$, $G_F$,
and $M_Z$ are taken to be the input parameters, from which the SM gauge couplings
and the Higgs vev are inferred. Predictions for experiments are
computed in terms of the inputs and the coefficients of the new operators.
 The experimental quantities we use to constrain the coefficients of operators are listed in
Table~\ref{table:experiments}. Detailed descriptions and references for individual
experiments can be found in many reviews, for example in Refs.~\cite{erler+langacker}
and \cite{LEPII}.

\begin{table}[htb]
\begin{tabular}{|c|c|c|c|}
\hline
 & Standard Notation &  Measurement &Reference\\
 \hline
 Atomic parity &$Q_W(Cs)$&Weak charge in Cs&\cite{QWCs}\\
 violation& $Q_W(Tl)$& Weak charge in Tl&\cite{QWTl}\\
 \hline
   DIS&$g_L^2,g_R^2$&$\nu_\mu$-nucleon scattering from NuTeV&\cite{NuTeV}\\
      & $R^\nu$&$\nu_\mu$-nucleon scattering from CDHS and CHARM&\cite{CDHS,CHARM}\\
      &$\kappa$&$\nu_\mu$-nucleon scattering from CCFR&\cite{CCFR}\\
      &$g_V^{\nu e},g_A^{\nu e}$&$\nu$-$e$ scattering from CHARM II&\cite{CHARMII}\\

 \hline
   Z-pole&$\Gamma_Z$&Total $Z$ width&\cite{LEPII}\\
         &$\sigma_h^0$ &$e^+e^-$ hadronic cross section at $Z$ pole&\cite{LEPII}\\
         &$R_f^0(f=e,\mu,\tau,b,c)$& Ratios of decay rates &\cite{LEPII}\\
         &$A_{FB}^{0,f}(f=e,\mu,\tau,b,c)$ &Forward-backward asymmetries&\cite{LEPII}\\
         &$\sin^2\theta_{eff}^{lept}(Q_{FB})$&Hadronic charge asymmetry&\cite{LEPII}\\
         &$A_f(f=e,\mu,\tau,b,c)$&Polarized asymmetries&\cite{LEPII}\\
 \hline
 Fermion pair    &$\sigma_f(f=q,\mu,\tau)$& Total cross sections for
$e^+e^-\rightarrow f\overline f$&\cite{LEPII}\\
 production at   &$A_{FB}^f(f=\mu,\tau)$& Forward-backward asymmetries for
$e^+e^-\rightarrow f\overline f$&\cite{LEPII}\\
 LEP2            &$d\sigma_e/d\cos\theta$&Differential cross section for
$e^+e^-\rightarrow e^+e^-$&\cite{OPALfpair}\\
 \hline
 $W$ pair&$d\sigma_W/d\cos\theta$& Differential cross section for $e^+e^-\rightarrow W^+W^-$&\cite{L3Wpair}\\
 \hline
 &$M_W$ & W mass &\cite{LEPII,WmassTevatron}\\

 \hline
\end{tabular}
\caption{\label{table:experiments} Relevant measurements}
\end{table}

The list of experiments in Table~\ref{table:experiments} does not include
the anomalous magnetic moment of the muon~\cite{gminus2}, one of the
most precisely measured electroweak quantities. The operators that contribute
directly to $(g-2)$ involve left and right-handed fields and are not $U(3)^5$ invariant.
There are also loop contributions from operators like $O_{W\!B}$, $O_W$,
and many four-fermion operators. Such loop contributions are divergent and
require introducing  counterterms in the form of  operators excluded from our analysis
due to their lack of $U(3)^5$ invariance. An operator analysis of contributions to the
muon $(g-2)$ can be found in Ref.~\cite{AEW}.

For a given observable $X$, our prediction can be written as:
\begin{equation}
\label{prediction}
   X_{th}=X_{SM}+\sum_i a_i X_i,
\end{equation}
where $X_{th}$ is the prediction in the presence of additional operators,
$X_{SM}$ is the standard model prediction and $\sum_ia_iX_i$
are corrections from our new operators. In practice, the SM predictions
are computed to the required accuracy in perturbation theory
and are well known for all the measurements we use.
Note that the corrections $X_i$  arise in two different ways.
First, an operator can generate a new Feynman diagram contributing
to a given physical process.  For example, a four-fermion operator
$O_{le}$ enters the $e^+e^-\rightarrow \mu^+\mu^-$ process as a new diagram,
in addition to the $Z$ and $\gamma$ exchange
diagrams. We call this ``direct'' correction. Second, some operators
can shift the input parameters, because they add new
diagrams to the physical processes based on which $\alpha$,
$G_F$, and $M_Z$ are measured. Thus, the input parameters
determined from these observables are different from their SM values.
Since all the other observables are calculated from these
input parameters, they will inevitably receive indirect corrections
from the shifts. We summarize the direct and indirect effects of our operators
in Table~\ref{table:operators}.

Because of their high statistics, the Z-pole data and several best
measured low-energy observables dominate the bounds on the coefficients $a_i$
whenever such measurements constrain an operator. This is the case for the
operators that shift the input parameters and the operators of the
form $O_{hf}$, which change the couplings between the $Z$ boson and
the fermions. The four-fermion operators do not contribute to
the $Z$-pole measurements at the linear order,
that is the interference term between the SM contribution
and  the four-fermion operators vanishes at the Z pole.
Therefore, to constrain four-fermion operators we have to include the
cross sections for fermion-pair production at LEP2. We also include the differential
cross sections for the $W$ pair production to constrain the operator
$O_W$. There are several operators, in addition to $O_W$, that alter the cross section
for $W$ pair production. However, these operators are well bounded by other
measurements and the $W$ pair production does not contribute significantly to
the bounds on their coefficients.

\begin{table}[hbt]
\begin{tabular}{|c|c|c|c|c|c|c|c|}
\hline
 Operator(s)&shift&$M_W$&Z-pole&DIS&$ Q_W$&$e^+e^-\rightarrow f\overline f$
 (LEP2)&$e^+e^-\rightarrow W^+W^-$\\
 \hline
 $O_{W\!B}$&$\alpha, \, M_Z$& &$\surd$&$\surd$&$\surd$&$\surd$&$\surd$\\
 \hline
 $O_h$&$M_Z$&&&&&&\\
 \hline
 $O_{ll}^t$&$G_F$&&&$\surd$&&$\surd$&\\
 \hline
 $O_{ll}^s$, $O_{le}$ &&&&$\surd$&&$\surd$&\\ \hline
 $O_{ee}$ & & & & & & $\surd$ & \\ \hline
 $O_{lq}^s,O_{lq}^t,O_{lu},O_{ld}$&&&&$\surd$&$\surd$&$\surd$&\\ \hline
 $O_{eq},O_{eu},O_{ed}$&&&&&$\surd$&$\surd$&\\
 \hline
 $O_{hl}^t$&$G_F$&&$\surd$&$\surd$&$\surd$&$\surd$&$\surd$\\
 \hline
 $O_{hl}^s,O_{he}$&&&$\surd$&$\surd$&$\surd$&$\surd$&$\surd$\\
 \hline
 $O_{hu},O_{hd},O_{hq}^s,O_{hq}^t$&&&$\surd$&$\surd$&$\surd$&$\surd$&\\
 \hline
 $O_W$&&&&&&&$\surd$\\
\hline
\end{tabular}
\caption{ \label{table:operators}
Measurements influenced by different operators.
The check marks, $\surd$,  indicate  ``direct" corrections only. When an operator
contributes to one of the input parameters, the corresponding shift of the input parameter
does affect all measurements. }
\end{table}

\section{Calculations}
\label{sec:Calculations}
In this section we describe the computation of the effects of
dimension-6 operators. Since a lot of work on this topic is already
available in the literature, we quote the results whenever available.
We have independently verified all the quoted results.

We work in the linear approximation
in terms of the coefficients $a_i$. As we indicated in the previous
section, there are two ways that terms linear in $a_i$ arise.
First, as a result of additional Feynman diagrams due to the
dimension-6 operators. In this case we simply compute the
interference terms between the new operators and the tree-level
contribution in the SM. Second, a few of our operators
redefine the input parameters inferred from the measurements
of $\alpha$, $M_Z$, and $G_F$. We use the tree-level SM results
and expand to the linear order in the deviations induced by the
coefficients $a_i$. One of the most transparent ways of dealing with
the shifts of the input parameters is described in Ref.~\cite{BGKLM}.

To track down all the shifts, we
find it convenient to use the following parameters in the SM
Lagrangian: $e$, $s$, and $M_Z$; or equivalently $g$, $g'$, and $M_Z$.
They are related to the input
parameters and the new operators as
\begin{eqnarray}
e^2&=&e^2_0 (1-2v^2s c \, a_{W\!B}),\label{Sshift}\\
M_Z^2&=&M_{Z0}^2(1+2v^2sc\, a_{W\!B}+ \frac{v^2 }{2 }a_h),\label{STshift}\\
\frac{1}{v^2}&=&\frac{1}{v_0^2}+ 2 a_{hl}^t -a_{ll}^t,
\end{eqnarray}
where
\begin{equation}
\alpha=\frac{e_0^2}{4\pi},\quad
G_{F}=\frac{1}{\sqrt{2}v_0^2}=\frac{e^2_0}{4 \sqrt{2} s_0^2 c_0^2 M_{Z0}^2}.
\end{equation}
Parameters with subscripts $0$ are the values derived in the absence
of any additional operators.
Eqs.~(\ref{Sshift}) and (\ref{STshift}) are the shifts
due to the $S$ and $T$ parameters.

The corrections induced by $S$ and $T$ are given in Ref.~\cite{STU},
and  can be easily translated to our notation using Eq.~(\ref{eq:ST}).
Ref.~\cite{STU} does not provide formulas for LEP2, but the extension to
LEP2 is simple. The operator $O_{W\!B}$ also contains a triple gauge boson coupling,
which contributes to the $e^+ e^- \rightarrow W^+ W^-$ process. We discuss this
at the end of this section.

 The operators with 2 Higgs doublets and 2 fermions, $O_{hf}$, alter
the couplings between gauge bosons and fermions. The changes to the
$Z$-fermion couplings,  $g_V^f$ and $g_A^f$, are given in
Ref.~\cite{BarbieriStrumia}. (The $v^2$ in Ref.~\cite{BarbieriStrumia}
is defined as one half of our value.) These changes affect all
measurements involving a $Z$-exchange diagram. We have
calculated the SM tree-level predictions in terms of arbitrary
$g_V^f$ and $g_A^f$, so it is easy to obtain the corrections induced by $O_{hf}$.
In addition, the operators $O_{hl}^t$ and $O_{hq}^t$ change the couplings of
the $W$ boson to the leptons and quarks:
\begin{eqnarray}
 g&\rightarrow&g(1+v^2
       a_{hl}^t)\qquad(W-\mbox{leptons coupling}),\\
 g&\rightarrow&g(1+v^2
 a_{hq}^t)\qquad(W-\mbox{quarks coupling}).
\end{eqnarray}
These changes enter, besides the shift to $G_F$, the calculations of
the $e^+e^-\rightarrow W^+W^-$ cross section in the $\nu$ exchange
channel, and the $\nu$-nucleon scattering charged current
cross sections.

The operator $O_{ll}^t$ shifts the value of the input parameter inferred from
$G_F$. All other four-fermion operators do not contribute to the $Z$-pole measurements.
However, they contribute to the low-energy measurements and LEP2 measurements.
 We now enumerate the effects of four-fermion operators on the relevant  observables:
\begin{enumerate}
 \item The weak charges measured in atomic parity violation
 experiments
 \begin{equation}
 Q_W(Z,N)=-2[(2Z+N)C_{1u}+(Z+2N)C_{1d}].
 \end{equation}
 $C_{1u}$ and $C_{1d}$ receive corrections
 \begin{eqnarray}
 \Delta
C_{1u}&=&\frac{\sqrt{2}}{4G_F}(-a_{lq}^s+a_{lq}^t+a_{eu}+a_{qe}-a_{lu}),\\
 \Delta
C_{1d}&=&\frac{\sqrt{2}}{4G_F}(-a_{lq}^s-a_{lq}^t+a_{ed}+a_{qe}-a_{ld}).
 \end{eqnarray}
\item $\nu$-nucleon scattering. The 4-fermion operators affect both the
neutral
 current and the charged current cross sections. The corrections to the
couplings $g_L^u, g_L^d,g_R^u,g_R^d$ are
 \begin{eqnarray}
 \Delta g_{L,eff}^u&=&\frac{1}{2\sqrt{2}G_F} [- a_{lq}^s + (2 g_L^u-1) a_{lq}^t],\\
 \Delta g_{L,eff}^d&=&\frac{1}{2\sqrt{2}G_F} [- a_{lq}^s+ (2 g_L^d+1) a_{lq}^t],\\
 \Delta g_{R,eff}^u&=&\frac{1}{2\sqrt{2}G_F} (- a_{lu}^s+2 g_R^u a_{lq}^t),\\
 \Delta g_{R,eff}^d&=&\frac{1}{2\sqrt{2}G_F} (- a_{ld}^s+2 g_R^d a_{lq}^t).
\end{eqnarray}
 These corrections are``effective" in the sense that they are not
corrections to the $Z$-fermion couplings and thus the formulas above only
apply to the $\nu$-nucleon scattering. The corrections to the measured
quantities  can be easily calculated from the above equations, for example,
 $g_L^2=(g_L^u)^2+(g_L^d)^2$ measured at NuTeV receive the correction
 \begin{equation}
 \Delta(g_L^2)=2g_L^u \Delta g_{L,eff}^u+2g_L^d\Delta g_{L,eff}^d.
 \end{equation}
\item $\nu$-$e$ scattering. The corrections to the coupling
 $g_V^{\nu e}$ and $g_A^{\nu e}$ are
 \begin{eqnarray}
 \Delta g_{V,eff}^{\nu
e}&=&\frac{1}{2\sqrt{2}G_F} (-a_{ll}^s+a_{ll}^t -a_{le}),\\
 \Delta g_{A^eff}^{\nu e}&=&\frac{1}{2\sqrt{2}G_F} (-a_{ll}^s+a_{ll}^t+a_{le}).
 \end{eqnarray}
 Again, these corrections are ``effective'' that is only apply to the
 $\nu$-$e$ scattering process.
 \item Fermion pair production at LEP2 energies. The differential
 cross sections in the presence of the contact operators are given,
 for example, in Ref.~\cite{Kroha:1991mn}.
\end{enumerate}

 Finally, the operators $O_W$ and $O_{W\!B}$ alter the
 triple gauge boson couplings.  After substituting the vev for the Higgs doublet,
 the two operators yield the following couplings
 \begin{equation}
 \label{eq:3gauge}
\Delta \mathcal{L}= ia_{W\!B} v^2 gW^+_\mu W^-_\nu(c A^{\mu\nu}-s
  Z^{\mu\nu})+6ia_WW^{-\mu}_\nu
  W^{+\lambda}_\mu(sA^\nu_\lambda+cZ_\lambda^\nu).
 \end{equation}
The tree-level $e^+e^-\rightarrow W^+W^-$
differential cross section is calculated in
Ref.~\cite{triplegauge} for arbitrary triple gauge boson couplings.
Our effective couplings, Eq.~(\ref{eq:3gauge}), correspond to the terms multiplying
 $\kappa_V$ and $\lambda_V$ in Eq.~(2.1) in Ref.~\cite{triplegauge}.
 To obtain the  cross section we substitute
\begin{eqnarray}
 \Delta\kappa_\gamma&=&\frac{ v^2 c}{s} \,a_{W\!B} ,\\
 \Delta\kappa_Z&=&-\frac{v^2s}{c} \, a_{W\!B},\\
 \Delta\lambda_\gamma&=&\Delta\lambda_Z=\frac{3v^2 g}{2} \,a_W .
\end{eqnarray}
where $\Delta$'s denote the deviations from the SM values.

\section{Total $\chi^2$ distribution}
\label{sec:chi2}
In the previous section, we described how to compute
the changes of observable quantities induced by the operators.
Given theses results we calculate the total $\chi^2$ distribution. For non-correlated measurements,
\begin{equation}
\label{eq:chi2uncorr}
  \chi^2(a_i)=\sum_X\frac{(X_{th}(a_i)-X_{exp})^2}{\sigma_X^2},
\end{equation}
where $X_{exp}$ is the experimental value for observable $X$ and
$\sigma_X$ is the total error both experimental and theoretical. The
experimental values for the observables are obtained from the
references cited in Table~\ref{table:experiments}.
The input parameters and the SM
 predictions, except those for LEP2, are obtained from
Ref.~\cite{erler+langacker}, where the following input parameters
are used:
\begin{equation}
\label{eq:inputs}  m_{Higgs}=113~\mbox{GeV},\
m_{top}=176.9~\mbox{GeV},\ \alpha_s(M_Z)=0.1213
\end{equation}
in addition to $M_Z$, $\alpha$, and $G_F$. The uncertainty in the values of the input parameters
are incorporated as theory errors on the SM predictions and combined later
with experimental errors.
 For the DIS measurements of CDHS, CHARM and CCFR, we use the SM
predictions in Ref.~\cite{Erler+Langacker2002}, but have corrected
them for the small differences of input parameters~\cite{Langacker:1991zr}.
The sensitivities of the SM predictions for the $e^+e^-\rightarrow f^+f^- (f\neq e)$
cross sections at LEP2 and the $e^+e^-\rightarrow W^+W^-$ cross section
to small changes of the input parameters are negligible compared to experimental
errors, as we have verified using ZFITTER~\cite{zfitter} and RacoonWW~\cite{RacoonWW}.
Therefore, we use the SM predictions provided in the corresponding experimental references.
The SM prediction for the $e^+e^-\rightarrow e^+e^-$ (LEP2) differential
cross section is calculated using the program BHWIDE~\cite{BHWIDE},
assuming the same input parameters as in Eq.~(\ref{eq:inputs}).

The SM predictions agree with the experimental values
well, except for a  significant discrepancy for
$g_L^2$ obtained by the NuTeV collaboration~\cite{NuTeV}.
We include the NuTeV result in our calculation, but
one could easily omit this result from the  $\chi^2$ calculation.
We also note that the LEP2 results for the
$e^+e^-\rightarrow q\overline q$ total cross sections are
larger than the SM predictions fairly consistently across different
energies probed by LEP2. If we tried to constrain the coefficients
of four-fermion operators with two leptons and two quarks using
only the  $e^+e^-\rightarrow q\overline q$ total cross sections,
we would get relatively weak bounds on the
coefficients $a_i$ of such operators. Weak bounds mean
that the the contributions quadratic in the corrections in
$a_i$'s should not be neglected, contrary to what we do.
However, this discrepancy is not supported by other data that also constrains the
same operators. A combined fit to all observables does not yield any coefficients
$a_i$ large enough to invalidate the linear approximation.
Therefore we suspect this pattern is caused by a  systematic error.

Higher order terms in perturbation theory alter our tree-level
calculation of the interference terms between the SM and the
additional operators. The most important effect in electron-positron
scattering is the initial state QED radiation. In order to assess
the effect of the radiative corrections, we compare the bounds on
the 4-fermion contact operators  for the $e^+e^-\rightarrow
l^+l^-(l\neq e)$ and $e^+e^-\rightarrow q\overline q$ channels with
the results given in Ref.~\cite {LEPII}, Table 8.13. In Ref.~\cite{LEPII}, the $e^+e^-
\rightarrow q\overline q$ channel is constrained using the inclusive
hadronic total cross sections. The $e^+e^- \rightarrow
l^+l^-(l\neq e)$ channel is constrained using the total cross
sections and asymmetries for $l=\mu,\tau$ and assuming equal
coefficients for the contact terms with $\mu$ and $\tau$. For the
purpose of comparison we use the same data sets and carry out the
same fits. In our final results,
 of course, all the data is taken into account to obtain
the bounds on contact operators.

The comparison was carried out for different contact interactions.
(The coefficients of 4-fermion operators in Ref.~\cite {LEPII}
differ from our coefficients $a_i$ by a factor
of $4 \pi$.) Several radiative corrections and second order terms in
$a_i$ are considered when obtaining bounds on these
coefficients\footnote{The radiative corrections are not mentioned in
Ref.~\cite{LEPII}, but are described in results of individual
experiments at LEP2. See, for example,
Refs.~\cite{OPALfpair}~\cite{DELPHIfpair}.}. We have obtained bounds
for the coefficients of operators neglecting radiative corrections
but including second order terms in $a_i$. Except for one case, the
differences between our bounds and the bounds in Ref.~\cite{LEPII}
for errors on $a_i$ are less than 20\%, and for centra values
of $a_i$ are less than $0.2\sigma$. When translated to the
scale $\Lambda$, the difference is less than 10\%, which is
satisfactory for our purposes. The exception is the bound for the
``LR" interaction for $e^+e^- \rightarrow q\overline q$ channel,
which corresponds to $a_{lu}$ and $a_{ld}$ in our notation, in which
case the errors quoted in Ref.~\cite{LEPII} are much more asymmetric
than our estimate of errors. However, the central value of the
coefficients differs less than $0.2\sigma$. Since the contact
terms with two leptons and two quarks can be constrained much more
stringently by the low energy measurements, this discrepancy
cannot significantly affect our global fit.

For $e^+e^-\rightarrow f^+f^- \quad(f\neq e)$ channels, we have also
implemented the initial-state photonic correction to the order
$O(\alpha)$, which includes initial state soft photon exponentiation
and hard photon emission~\cite{zfitter}. This correction is the
largest radiative correction and we have obtained better agreement
with Ref.~\cite{LEPII} by including it. Except for the ``LR"
interaction in the  $e^+e^-\rightarrow q\overline q$ channel
mentioned previously, the discrepancies between our errors and
central values of $a_i$ and the results of \cite{LEPII} are within
10\% and $0.1\sigma$ respectively. The remaining discrepancy likely
arises from the final state radiative correction, pair production
correction and higher order corrections, that we have not
implemented.

We can safely neglect effects that contribute at a 10\% level
to our estimates of the coefficients $a_i$, we have nevertheless
included the first order initial state QED corrections to
$e^+e^-\rightarrow f^+f^- \quad(f\neq e)$ channels in our
calculation of $\chi^2$. A factor of $(1+\alpha_s/\pi)$ is
also used to account for the QCD corrections for the hadronic final
states~\cite{OPALfpair130}.

Eq.~(\ref{eq:chi2uncorr}) must be modified to account for
the correlations between different measurements. There are three
categories of data for which the correlations between measurements cannot be
neglected. These are the correlations between $Z$-pole
observables~\cite{LEPII}, the experimental error correlations for the hadronic
total cross sections at LEP2 energies~\cite{LEPII}; and the theoretical
and experimental error correlations for $e^+e^-\rightarrow e^+e^-$
differential cross sections~\cite{Ward}. Including correlations,
\begin{equation}
 \label{eq:chi2corr}
  \chi^2(a_i)=\sum_{p,q}(X_{th}^p(a_i)-X_{exp}^p)(\sigma^2)^{-1}_{pq}(X_{th}^q(a_i)-X_{exp}^q),
\end{equation}
where the error matrix $\sigma^2$ is related to the standard deviations
$\sigma_p$ and the correlation matrix $\rho_{pq}$ as folows
\begin{equation}
 \sigma^2_{pq}=\sigma_p\rho_{pq}\sigma_q.
\end{equation}
Note that often the correlations for the theoretical, statistical and
systematic errors are different, and one should take this into account
when computing the final error matrix.

Numerical results for the $\chi^2$ distribution are presented in Appendix~\ref{app:Matrix}.

 \section{Summary}
  \label{sec:Summary}
We have obtained bounds on the coefficients of 21 dimension 6 operators in the SM\@.
Our analysis is linear in the coefficients of these operators. Therefore, the deviations from
the SM predictions arise as interference terms between the SM and the dimension 6 operators.
As is often the case, integrating out heavy particles leads not to just one but to several operators
whose coefficients are related in terms of the masses and coupling constants of the heavy states.
A global analysis of precision electroweak measurements must take into account all new operators
induced by integrating out heavy states and account for relations between the coefficients
of such operators. Our analysis allows obtaining bounds not just on each individual operator, but
on their linear combinations as well. Doing so, in the linear approximation, does not require
complicated numerical analysis, and can be done efficiently using our results.
Of course, if the new physics contributions are ``oblique" or ``universal" only, \cite{STU,LEP12}, one
does not need the whole set of 21 operators. A subset of our operators that only modify SM
gauge boson propagators has already been considered in Refs.~\cite{STU,LEP12}

Our analysis is accomplished by computing the $\chi^2$ distribution as a function of the coefficients
$a_i$. In the linear approximation, $\chi^2$ takes the form shown in Eqs.~(\ref{eq:main})
and (\ref{eq:main2}). We concentrated on flavor and CP conserving operators.
Such operators are allowed in the SM when suppressed by scales of the order of a few TeV\@.
Generic flavor and CP violating operators must be suppressed by much higher scales. This wide
separation of characteristic scales suggests that the electroweak symmetry breaking
and the flavor and CP violating sectors can be analyzed independently of one another.
We excluded from our analysis operators that are not tightly constrained by the data,
for example operators involving only quarks and gluons. Such operators are not helpful
in constraining extensions of the SM.

The bounds on the coefficients of individual operators, by which we mean that the
SM Lagrangian is amended by only one operator at a time, can be easily obtained
from Eq.~(\ref{eq:main}). The $1\, \sigma$ bound on a coefficient $a_k$ is $\hat{a}_k \pm
\sqrt{{\mathcal M}_{kk}^{-1}}$,
where ${\mathcal M}_{kk}$ indicates a diagonal element of ${\mathcal M}$
and is not summed over $k$. The fourth roots of the diagonal elements, $\sqrt[4]{{\mathcal M}_{kk}}$,
vary from $1.3~{\rm TeV}$ to  $17~{\rm TeV}$, which is a measure of how rapidly $\chi^2$ changes
as a function of $a_k=\frac{1}{\Lambda_k^2}$.

What is interesting is that the eigenvalues of ${\mathcal M}$ vary over a much
wider range, their fourth roots span from $180~{\rm GeV}$ to $21~{\rm TeV}$.
In particular, the fourth roots of the four smallest eigenvalues are $180$, $250$, $390$
and $420~{\rm GeV}$. This means that four linear
combinations of the operators are much more weakly constrained than the individual
operators. The emergence of these ``weakly bounded directions" in the operator  space is an
interesting byproduct of our analysis. Of course, one can not trust the exact bounds on the
``weakly bounded" operators. The linear analysis is not applicable when the suppression
scales are so low. One needs to work to the quadratic order in the coefficients to reliably
determine the bounds. However, it is clear that the bounds on such linear combinations
of operators are quite weak and below 1~TeV\@.
It is interesting to find out if there are heavy particles that yield one of
the weakly constrained combination of operators when integrated out. This
possibility is currently being investigated.

\section*{Acknowledgments}
We thank Jens Erler, Alex Pomarol, and Pat Ward for helpful correspondences. We are
grateful to Tom Appelquist and Martin Schmaltz for discussions and comments.
This research was supported in part by the US Department of Energy
under grant  DE-FG02-92ER-40704. WS is also supported in part by the DOE OJI program.

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by The Frontier Post
John O. McGinnis
Hardly a day goes by without hearing s-ome impassioned paean to "our democrac-y" and concerns about the dangers it faces. But do we have a common understanding of democracy?
The left's concept of democracy contains an apparent tension: On the one hand, it wants to make sure that everyone has an equal voice in elections to assure government accountability to the people. Thus, it seeks to end gerrymandering, reduce, if not eliminate, the political influence of the rich, and even reform the Senate. On the other hand, once representatives are elected, the left is perfectly happy, indeed eager, for them to delegate large swaths of power to the administrative state, where bureaucrats, not elected representatives, formulate the governing rules. The concern for making sure that everyone has a precisely equal voice sits uneasily with empowering a government bureaucracy that is unlikely to be closely accountable to that electorate for its decisions. The tension is particularly acute because modern government has become administrative government.
Conservatives, by contrast, seem much less concerned about equalizing citizens' voices in the electoral process to make representatives accountable to the people, but instead have focused on increasing the administrative state's accountability to representatives. Thus, conservatives may be thought to have a tension of their own when it comes to democracy. Are these different combinations of positions explicable only in realist terms or are there more principled reasons lying behind two distinct versions of democratic ideals?
Realist Explanations
A realist perspective on politics can explain the left's lack of concern for making legislatures accou-ntable for the administrat-ive state. Bureaucrats stand to the left of the median D-emocrat, to say nothing of the median voter. More-over, public choice theoris-ts have shown that bureaucrats are interested in growing the bureaucracy. Thus, giving more power to the bureaucracy is a force multiplier for moving society to the left.
It might be objected that the President controls the administrative state and that voters elect the President, assuring democratic accountability. But that is an imperfect control of the substantial discretion provided by large delegations. Congress has insulated from removal the heads of many of the most important agencies, like the Federal Trade Commission and the Federal Communi-cations Commission, making them only liable to dismissal for cause. Not surprisingly, from a realist perspective, it is the left that defends this insulation from presidential control. And even when the President can dismiss agency heads, it is politically costly to do so: he is implicitly admitting that he made a mistaken appointment and then must spend political capital to confirm a successor.
As a result, any agency head has substantial freedom of action. Moreover, given that it is important for his career that he be considered a success, he has substantial incentives to compromise with the career bureaucrats that staff his agency. Government employees have effective life tenure, and getting along with them makes for an easier life and more productive agenda. Thus, the political valence of the bureaucracy matters.
The left's enthusiasm for purifying the electoral process can also be explained through a realist lens. Reducing the influence of the wealthy in politics will increase the power of the left-leaning chattering classes—the media and academia—in setting the political agenda. Thus, although the left now receives a roughly equal amount of money from the rich, particularly as the rich have become more socially progressive, banning it again has a left valence.
Making it easier to vote by mail facilities the power of organized groups to influence ballots. Because Democrats have more dependent voters (those whose vote is influenced by entities such as unions and community organizers), legal ballot harvesting may advantage them. To be sure, political scientists have suggested that most voter access laws have relatively little partisan effect. The vast majority of people can vote if they want, and there is little voter fraud. But the left may be able to use election laws to energize its base with the slogan that the Republicans are suppressing votes.
Of course, the realist explanation can be turned on the right as well. They want to highlight voter fraud to energize their base. They want to limit bureaucratic power because it leans left. Every realist explanation of the left's position also explains the right's opposition to it.
Principled Differences about Democracy
Though self-interest partly explains the combination of positions held by both the left and right, there are also deeper and more principled rationales that suggest a fundamental conflict in their concepts of democracy. First, the left sees democracy as creating a general will that the bureaucracy with its greater reasoning ability applies to concrete situations. These ideas have deep philosophical roots. The notion that majority rule can capture the enduring general will of society rather than just a snapshot of the constellation of interests that happens to exist at a particular time goes back to Rousseau. The view that the bureaucracy can convert that general will into concrete solutions through reason goes back to Hegel. American progressivism helped such notions cross the Atlantic.
The left's combination of views leads to a focus on purifying the general will by eliminating extraneous influences like money. The claim that bureaucracy is a social apparatus distinguished by its reasoning capacity depends on the notion that it is a meritocratic institution whose employees depend on salary from the state, guaranteeing that they will follow the public interest, not special interests. This conception then becomes a justification for delegating to it the power to carry out the general will. Democracy ultimately brings about a historicized rationality to social affairs. Democracy must be rational if the arc of history is to bend toward justice, to use a favorite phrase of the left. Under this view, the press and academia are also reasoning institutions, being in the left's view relatively meritocratic and not compensated by particular interests. If restricting the influence of the wealthy (who are partial to the sources and maintenance of their wealth) increases the influence of academia and the press, so much the better.
Moreover, the left—at least the very substantial portion that is invested in democracy—has a principled view of the world that requires the general will be the dominant social force. The left believes that human flourishing is threatened less by government than by private power, such as the market, which needs to be regulated, and conventional norms, from which governmental power may need to liberate us. Government can constrain the market and promote liberation in today's world only through substantial delegation because of the self-interest of legislators who must be concerned always with status vis-a-vis one another in power politics. And if the democratic will is so foundational, it must be purified: hence the concern about gerrymandering and the unequal popular representation in the Senate.
The right—at least the very substantial portion that still adheres to democracy—has a much more modest view of both the nature and function of democracy. Democracy does not capture any general will. An election provides only a snapshot of people's vacillating interests. Moreover, citizens do not even systemically review the policies that are at issue in an election. Because their vote is so unlikely to change the result, they do not gather the pertinent information. For the most part, they are rationally ignorant of politics and easily influenced by events irrelevant to governance.
Conservatives also have a higher opinion of social ordering from non-democratic sources, from the market to the family to civic associations like religious institutions. They worry that government will disturb this more natural ordering. To be sure, since government is required, the right recognizes the need for constraints on rulers and for feedback information about the preferences of citizens, but that is all that can be expected from democracy. If the left idolizes the perfected social power of democracy, the right celebrates only its protective one.
As result, the right is very concerned about delegation to administrative agencies. The deliberation and consensus required by forcing a bicameral legislature to agree on detailed rules help assure that democracy will not disturb or destroy other forms of social ordering without sufficient consideration. Far from being repositories of reason, bureaucrats are another group with interests that need to be kept on a short leash because of their powerful institutional position.
The right's limited view of the function of democracy makes it much less concerned with perfecting representation. It performs its protective function quite well even without perfect equality of influence. And some of the unequal features themselves protect important elements of an organic society. It is good that the rich have more influence: in a commercial republic, they know how commerce works. Because of its unequal representation, the Senate protects small states from being ignored and thus preserves federalism. Some of the suggested electoral reforms are worse than any disease. Drawing election districts by commission gives power to commissioners or their bureaucratic staff—groups that pursue their interests all the more effectively because they do so under the veneer of the public interest.
To be sure, one can imagine concepts of democracy between these two antitheses. For instance, John Hart Ely in Democracy and Distrust was enthusiastic about making sure that democracy has more egalitarian inputs. But in a portion of the book rarely mentioned, Ely also called for a revival of the delegation doctrine. Ely was a pluralist who nevertheless was more optimistic about democracy than most conservatives and more pessimistic about traditional forms of social ordering.
Because most of the right and left today disagree about the nature of democracy, however, it is too facile to say, as the President recently suggested, that the Americans can be united by adherence to the democratic form of government. To be sure, there can be hope that they will accept the results of elections. But that is democracy at its most mechanical. The spirit of democracy cannot be separated from the substance of one's view of how society works. The left and right have entirely different understandings of society and cannot be expected to agree in any deep sense on its mode of governance. Belief in "democracy" won't bridge our differences.
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redpajama_set_name: RedPajamaCommonCrawl


text:
// Copyright (c) Charlie Poole, Rob Prouse and Contributors. MIT License - see LICENSE.txt

#if NETFRAMEWORK
using System;
using System.Collections.Generic;
using System.Reflection;
using NUnit.Engine.Extensibility;
using NUnit.Engine.Internal;

namespace NUnit.Engine.Drivers
{
    public class NUnit2DriverFactory : IDriverFactory
    {
        private const string NUNIT_FRAMEWORK = "nunit.framework";
        private const string NUNITLITE_FRAMEWORK = "nunitlite";
        private ExtensionNode _driverNode;

        // TODO: This should be a central service but for now it's local
        private ProvidedPathsAssemblyResolver _resolver;
        bool _resolverInstalled;

        public NUnit2DriverFactory(ExtensionNode driverNode)
        {
            _driverNode = driverNode;
            _resolver = new ProvidedPathsAssemblyResolver();
        }

        /// <summary>
        /// Gets a flag indicating whether a given assembly name and version
        /// represent a test framework supported by this factory.
        /// </summary>
        /// <param name="reference">An AssemblyName referring to the possible test framework.</param>
        public bool IsSupportedTestFramework(AssemblyName reference)
        {
            return NUNIT_FRAMEWORK.Equals(reference.Name, StringComparison.OrdinalIgnoreCase) && reference.Version.Major == 2
                || NUNITLITE_FRAMEWORK.Equals(reference.Name, StringComparison.OrdinalIgnoreCase) && reference.Version.Major == 1;
        }

        /// <summary>
        /// Gets a driver for a given test assembly and a framework
        /// which the assembly is already known to reference.
        /// </summary>
        /// <param name="domain">The domain in which the assembly will be loaded</param>
        /// <param name="reference">The name of the test framework reference</param>
        /// <returns></returns>
        public IFrameworkDriver GetDriver(AppDomain domain, AssemblyName reference)
        {
            if (!IsSupportedTestFramework(reference))
                throw new ArgumentException("Invalid framework", "reference");

            if (!_resolverInstalled)
            {
                _resolver.Install();
                _resolverInstalled = true;
                _resolver.AddPathFromFile(_driverNode.AssemblyPath);
            }

            return _driverNode.CreateExtensionObject(domain) as IFrameworkDriver;
        }
    }
}
#endif

redpajama_set_name: RedPajamaGithub