DOI: 10.1063/1.1724268. Mass Singularities of Feynman Amplitudes

Mass Singularities of Feynman Amplitudes

10.1063/1.1724268

Crossref journal-article

AIP Publishing

Journal of Mathematical Physics (317)

Abstract

Feynman amplitudes, regarded as functions of masses, exhibit various singularities when masses of internal and external lines are allowed to go to zero. In this paper, properties of these mass singularities, which may be defined as pathological solutions of the Landau condition, are studied in detail. A general method is developed that enables us to determine the degree of divergence of unrenormalized Feynman amplitudes at such singularities. It is also applied to the determination of mass dependence of a total transition probability. It is found that, although partial transition probabilities may have divergences associated with the vanishing of masses of particles in the final state, they always cancel each other in the calculation of total probability. However, this cancellation is partially destroyed if the charge renormalization is performed in a conventional manner. This is related to the fact that interacting particles lose their identity when their masses vanish. A new description of state and a new approach to the problem of renormalization seem to be required for a consistent treatment of this limit.

Bibliography

Kinoshita, T. (1962). Mass Singularities of Feynman Amplitudes. Journal of Mathematical Physics, 3(4), 650â677.

Authors 1

Toichiro Kinoshita (first)

References 69 Referenced 1,197

10.1063/1.1703676 / J. Math. Phys. (1960)
10.1016/0003-4916(61)90151-8 / Ann. Phys. (1961)
10.1007/BF02731243 / Nuovo Cimento (1961)
10.1103/PhysRev.113.1652 / Phys. Rev. (1959)
10.1103/PhysRev.112.267 / Phys. Rev. (1958)
10.1103/PhysRevLett.2.477 / Phys. Rev. Letters (1959)
10.1103/PhysRevLett.1.468 / Phys. Rev. Letters (1958)
10.1103/PhysRev.95.1300 / Phys. Rev. (1954)
10.1103/PhysRev.77.536 / Phys. Rev. (1950)
10.1017/S0305004100027651 / Proc. Cambridge Phil. Soc. (1952)
10.1007/BF02747390 / Nuovo Cimento (1957)
10.1143/PTP.17.401 / Progr. Theoret. Phys. (Kyoto) (1957)
{'key': '2024020601530481000_r7'}
10.1103/RevModPhys.21.434 / Revs. Modern Phys. (1949)
10.1143/PTP.20.690 / Progr. Theoret. Phys. (Kyoto) (1958)
10.1016/0029-5582(59)90154-3 / Nuclear Phys. (1959)
10.1143/PTP.22.128 / Progr. Theoret. Phys. (Kyoto) (1959)
{'key': '2024020601530481000_r10b'}
10.1103/PhysRev.111.1187 / Phys. Rev. (1958)
{'key': '2024020601530481000_r12'}
{'key': '2024020601530481000_r12a'}
{'issue': '6', 'key': '2024020601530481000_r13', 'volume': '1', 'year': '1958', 'journal-title': 'Kgl. Danske Videnskab. Selskab, Mat.-Fys. Skrifter'} / Kgl. Danske Videnskab. Selskab, Mat.-Fys. Skrifter (1958)
{'key': '2024020601530481000_r13a'}
{'key': '2024020601530481000_r14'}
{'key': '2024020601530481000_r15'}
{'key': '2024020601530481000_r16'}
{'key': '2024020601530481000_r17'}
{'key': '2024020601530481000_r18'}
10.1103/PhysRev.117.886 / Phys. Rev. (1960)
{'key': '2024020601530481000_r20'}
{'key': '2024020601530481000_r21'}
{'key': '2024020601530481000_r22'}
{'key': '2024020601530481000_r22a'}
{'key': '2024020601530481000_r23'}
10.1143/ptp/5.6.1045 / Progr. Theoret. Phys. (Kyoto) (1950)
{'key': '2024020601530481000_r24a'}
{'key': '2024020601530481000_r25'}
{'key': '2024020601530481000_r26'}
10.1143/PTP.19.159 / Progr. Theoret. Phys. (Kyoto) (1958)
{'key': '2024020601530481000_r28'}
{'key': '2024020601530481000_r29'}
{'key': '2024020601530481000_r30'}
10.1103/PhysRev.109.193 / Phys. Rev. (1958)
10.1103/PhysRevLett.5.323 / Phys. Rev. Letters (1960)
10.1103/PhysRevLett.5.22 / Phys. Rev. Letters (1960)
{'key': '2024020601530481000_r32b'}
10.1103/PhysRev.123.627 / Phys. Rev. (1961)
{'key': '2024020601530481000_r32d'}
{'key': '2024020601530481000_r33'}
10.1103/PhysRevLett.4.620 / Phys. Rev. Letters (1960)
{'key': '2024020601530481000_r35', 'first-page': '383', 'volume': '1', 'year': '1956', 'journal-title': 'Bull. Am. Phys. Soc.'} / Bull. Am. Phys. Soc. (1956)
{'key': '2024020601530481000_r35a', 'first-page': '775', 'volume': '8', 'year': '1958', 'journal-title': 'Nuovo Cimento'} / Nuovo Cimento (1958)
{'key': '2024020601530481000_r36', 'first-page': '2050', 'volume': '119', 'year': '1961', 'journal-title': 'Phys. Rev.'} / Phys. Rev. (1961)
{'key': '2024020601530481000_r37'}
10.1515/zna-1959-5-601 / Z. Naturforsch. (1959)
10.1103/PhysRev.122.345 / Phys. Rev. (1961)
10.1007/BF02725099 / Nuovo Cimento (1958)
10.1007/BF02859738 / Nuovo Cimento (1960)
10.1007/BF02727566 / Nuovo Cimento (1960)
10.1103/PhysRevLett.4.380 / Phys. Rev. Letters (1960)
10.1016/0003-4916(57)90015-5 / Ann. Phys. (1957)
10.1016/0003-4916(60)90126-3 / Ann. Phys. (1960)
10.1103/PhysRev.123.1065 / Phys. Rev. (1961)
10.1103/PhysRev.125.1067 / Phys. Rev. (1962)
{'key': '2024020601530481000_r41'}
{'key': '2024020601530481000_r42'}
{'key': '2024020601530481000_r43'}
{'key': '2024020601530481000_r44'}
{'key': '2024020601530481000_r45'}

Dates

Type	When
Created	20 years, 7 months ago (Jan. 6, 2005, 12:54 p.m.)
Deposited	1 year, 6 months ago (Feb. 5, 2024, 8:53 p.m.)
Indexed	2 days, 12 hours ago (Aug. 23, 2025, 1:01 a.m.)
Issued	63 years, 1 month ago (July 1, 1962)
Published	63 years, 1 month ago (July 1, 1962)
Published Print	63 years, 1 month ago (July 1, 1962)

Funders 0

None

BibTeX

@article{Kinoshita_1962, title={Mass Singularities of Feynman Amplitudes}, volume={3}, ISSN={1089-7658}, url={http://dx.doi.org/10.1063/1.1724268}, DOI={10.1063/1.1724268}, number={4}, journal={Journal of Mathematical Physics}, publisher={AIP Publishing}, author={Kinoshita, Toichiro}, year={1962}, month=jul, pages={650–677} }

JSON

{
  "indexed": {
    "date-parts": [
      [
        2025,
        8,
        23
      ]
    ],
    "date-time": "2025-08-23T05:01:00Z",
    "timestamp": 1755925260313
  },
  "reference-count": 69,
  "publisher": "AIP Publishing",
  "issue": "4",
  "content-domain": {
    "domain": [
      "pubs.aip.org"
    ],
    "crossmark-restriction": true
  },
  "published-print": {
    "date-parts": [
      [
        1962,
        7,
        1
      ]
    ]
  },
  "abstract": "<jats:p>Feynman amplitudes, regarded as functions of masses, exhibit various singularities when masses of internal and external lines are allowed to go to zero. In this paper, properties of these mass singularities, which may be defined as pathological solutions of the Landau condition, are studied in detail. A general method is developed that enables us to determine the degree of divergence of unrenormalized Feynman amplitudes at such singularities. It is also applied to the determination of mass dependence of a total transition probability. It is found that, although partial transition probabilities may have divergences associated with the vanishing of masses of particles in the final state, they always cancel each other in the calculation of total probability. However, this cancellation is partially destroyed if the charge renormalization is performed in a conventional manner. This is related to the fact that interacting particles lose their identity when their masses vanish. A new description of state and a new approach to the problem of renormalization seem to be required for a consistent treatment of this limit.</jats:p>",
  "DOI": "10.1063/1.1724268",
  "type": "journal-article",
  "created": {
    "date-parts": [
      [
        2005,
        1,
        6
      ]
    ],
    "date-time": "2005-01-06T17:54:10Z",
    "timestamp": 1105034050000
  },
  "page": "650-677",
  "update-policy": "http://dx.doi.org/10.1063/aip-crossmark-policy-page",
  "source": "Crossref",
  "is-referenced-by-count": 1197,
  "title": "Mass Singularities of Feynman Amplitudes",
  "prefix": "10.1063",
  "volume": "3",
  "author": [
    {
      "given": "Toichiro",
      "family": "Kinoshita",
      "sequence": "first",
      "affiliation": [
        {
          "name": "Laboratory of Nuclear Studies, Cornell University, Ithaca, New York"
        }
      ]
    }
  ],
  "member": "317",
  "reference": [
    {
      "key": "2024020601530481000_r1",
      "doi-asserted-by": "crossref",
      "first-page": "429",
      "DOI": "10.1063/1.1703676",
      "volume": "1",
      "year": "1960",
      "journal-title": "J. Math. Phys."
    },
    {
      "key": "2024020601530481000_r2",
      "doi-asserted-by": "crossref",
      "first-page": "379",
      "DOI": "10.1016/0003-4916(61)90151-8",
      "volume": "13",
      "year": "1961",
      "journal-title": "Ann. Phys."
    },
    {
      "key": "2024020601530481000_r2a",
      "doi-asserted-by": "crossref",
      "first-page": "1010",
      "DOI": "10.1007/BF02731243",
      "volume": "19",
      "year": "1961",
      "journal-title": "Nuovo Cimento"
    },
    {
      "key": "2024020601530481000_r3",
      "doi-asserted-by": "crossref",
      "first-page": "1652",
      "DOI": "10.1103/PhysRev.113.1652",
      "volume": "113",
      "year": "1959",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r3a",
      "doi-asserted-by": "crossref",
      "first-page": "267",
      "DOI": "10.1103/PhysRev.112.267",
      "volume": "112",
      "year": "1958",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r3b",
      "doi-asserted-by": "crossref",
      "first-page": "477",
      "DOI": "10.1103/PhysRevLett.2.477",
      "volume": "2",
      "year": "1959",
      "journal-title": "Phys. Rev. Letters"
    },
    {
      "key": "2024020601530481000_r3c",
      "doi-asserted-by": "crossref",
      "first-page": "468",
      "DOI": "10.1103/PhysRevLett.1.468",
      "volume": "1",
      "year": "1958",
      "journal-title": "Phys. Rev. Letters"
    },
    {
      "key": "2024020601530481000_r4",
      "doi-asserted-by": "crossref",
      "first-page": "1300",
      "DOI": "10.1103/PhysRev.95.1300",
      "volume": "95",
      "year": "1954",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r5",
      "doi-asserted-by": "crossref",
      "first-page": "536",
      "DOI": "10.1103/PhysRev.77.536",
      "volume": "77",
      "year": "1950",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r6",
      "doi-asserted-by": "crossref",
      "first-page": "300",
      "DOI": "10.1017/S0305004100027651",
      "volume": "48",
      "year": "1952",
      "journal-title": "Proc. Cambridge Phil. Soc."
    },
    {
      "key": "2024020601530481000_r6a",
      "doi-asserted-by": "crossref",
      "first-page": "1064",
      "DOI": "10.1007/BF02747390",
      "volume": "6",
      "year": "1957",
      "journal-title": "Nuovo Cimento"
    },
    {
      "key": "2024020601530481000_r6b",
      "doi-asserted-by": "crossref",
      "first-page": "401",
      "DOI": "10.1143/PTP.17.401",
      "volume": "17",
      "year": "1957",
      "journal-title": "Progr. Theoret. Phys. (Kyoto)"
    },
    {
      "key": "2024020601530481000_r7"
    },
    {
      "key": "2024020601530481000_r8",
      "doi-asserted-by": "crossref",
      "first-page": "434",
      "DOI": "10.1103/RevModPhys.21.434",
      "volume": "21",
      "year": "1949",
      "journal-title": "Revs. Modern Phys."
    },
    {
      "key": "2024020601530481000_r9",
      "doi-asserted-by": "crossref",
      "first-page": "690",
      "DOI": "10.1143/PTP.20.690",
      "volume": "20",
      "year": "1958",
      "journal-title": "Progr. Theoret. Phys. (Kyoto)"
    },
    {
      "key": "2024020601530481000_r10",
      "doi-asserted-by": "crossref",
      "first-page": "181",
      "DOI": "10.1016/0029-5582(59)90154-3",
      "volume": "13",
      "year": "1959",
      "journal-title": "Nuclear Phys."
    },
    {
      "key": "2024020601530481000_r10a",
      "doi-asserted-by": "crossref",
      "first-page": "128",
      "DOI": "10.1143/PTP.22.128",
      "volume": "22",
      "year": "1959",
      "journal-title": "Progr. Theoret. Phys. (Kyoto)"
    },
    {
      "key": "2024020601530481000_r10b"
    },
    {
      "key": "2024020601530481000_r11",
      "doi-asserted-by": "crossref",
      "first-page": "1187",
      "DOI": "10.1103/PhysRev.111.1187",
      "volume": "111",
      "year": "1958",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r12"
    },
    {
      "key": "2024020601530481000_r12a"
    },
    {
      "issue": "6",
      "key": "2024020601530481000_r13",
      "volume": "1",
      "year": "1958",
      "journal-title": "Kgl. Danske Videnskab. Selskab, Mat.-Fys. Skrifter"
    },
    {
      "key": "2024020601530481000_r13a"
    },
    {
      "key": "2024020601530481000_r14"
    },
    {
      "key": "2024020601530481000_r15"
    },
    {
      "key": "2024020601530481000_r16"
    },
    {
      "key": "2024020601530481000_r17"
    },
    {
      "key": "2024020601530481000_r18"
    },
    {
      "key": "2024020601530481000_r19",
      "doi-asserted-by": "crossref",
      "first-page": "886",
      "DOI": "10.1103/PhysRev.117.886",
      "volume": "117",
      "year": "1960",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r20"
    },
    {
      "key": "2024020601530481000_r21"
    },
    {
      "key": "2024020601530481000_r22"
    },
    {
      "key": "2024020601530481000_r22a"
    },
    {
      "key": "2024020601530481000_r23"
    },
    {
      "key": "2024020601530481000_r24",
      "doi-asserted-by": "crossref",
      "first-page": "1045",
      "DOI": "10.1143/ptp/5.6.1045",
      "volume": "5",
      "year": "1950",
      "journal-title": "Progr. Theoret. Phys. (Kyoto)"
    },
    {
      "key": "2024020601530481000_r24a"
    },
    {
      "key": "2024020601530481000_r25"
    },
    {
      "key": "2024020601530481000_r26"
    },
    {
      "key": "2024020601530481000_r27",
      "doi-asserted-by": "crossref",
      "first-page": "159",
      "DOI": "10.1143/PTP.19.159",
      "volume": "19",
      "year": "1958",
      "journal-title": "Progr. Theoret. Phys. (Kyoto)"
    },
    {
      "key": "2024020601530481000_r28"
    },
    {
      "key": "2024020601530481000_r29"
    },
    {
      "key": "2024020601530481000_r30"
    },
    {
      "key": "2024020601530481000_r31",
      "doi-asserted-by": "crossref",
      "first-page": "193",
      "DOI": "10.1103/PhysRev.109.193",
      "volume": "109",
      "year": "1958",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r32",
      "doi-asserted-by": "crossref",
      "first-page": "323",
      "DOI": "10.1103/PhysRevLett.5.323",
      "volume": "5",
      "year": "1960",
      "journal-title": "Phys. Rev. Letters"
    },
    {
      "key": "2024020601530481000_r32a",
      "doi-asserted-by": "crossref",
      "first-page": "22",
      "DOI": "10.1103/PhysRevLett.5.22",
      "volume": "5",
      "year": "1960",
      "journal-title": "Phys. Rev. Letters"
    },
    {
      "key": "2024020601530481000_r32b"
    },
    {
      "key": "2024020601530481000_r32c",
      "doi-asserted-by": "crossref",
      "first-page": "627",
      "DOI": "10.1103/PhysRev.123.627",
      "volume": "123",
      "year": "1961",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r32d"
    },
    {
      "key": "2024020601530481000_r33"
    },
    {
      "key": "2024020601530481000_r34",
      "doi-asserted-by": "crossref",
      "first-page": "620",
      "DOI": "10.1103/PhysRevLett.4.620",
      "volume": "4",
      "year": "1960",
      "journal-title": "Phys. Rev. Letters"
    },
    {
      "key": "2024020601530481000_r35",
      "first-page": "383",
      "volume": "1",
      "year": "1956",
      "journal-title": "Bull. Am. Phys. Soc."
    },
    {
      "key": "2024020601530481000_r35a",
      "first-page": "775",
      "volume": "8",
      "year": "1958",
      "journal-title": "Nuovo Cimento"
    },
    {
      "key": "2024020601530481000_r36",
      "first-page": "2050",
      "volume": "119",
      "year": "1961",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r37"
    },
    {
      "key": "2024020601530481000_r38",
      "doi-asserted-by": "crossref",
      "first-page": "441",
      "DOI": "10.1515/zna-1959-5-601",
      "volume": "14",
      "year": "1959",
      "journal-title": "Z. Naturforsch."
    },
    {
      "key": "2024020601530481000_r38a",
      "doi-asserted-by": "crossref",
      "first-page": "345",
      "DOI": "10.1103/PhysRev.122.345",
      "volume": "122",
      "year": "1961",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r39",
      "doi-asserted-by": "crossref",
      "first-page": "433",
      "DOI": "10.1007/BF02725099",
      "volume": "9",
      "year": "1958",
      "journal-title": "Nuovo Cimento"
    },
    {
      "key": "2024020601530481000_r39a",
      "doi-asserted-by": "crossref",
      "first-page": "705",
      "DOI": "10.1007/BF02859738",
      "volume": "16",
      "year": "1960",
      "journal-title": "Nuovo Cimento"
    },
    {
      "key": "2024020601530481000_r39b",
      "doi-asserted-by": "crossref",
      "first-page": "757",
      "DOI": "10.1007/BF02727566",
      "volume": "17",
      "year": "1960",
      "journal-title": "Nuovo Cimento"
    },
    {
      "key": "2024020601530481000_r39c",
      "doi-asserted-by": "crossref",
      "first-page": "380",
      "DOI": "10.1103/PhysRevLett.4.380",
      "volume": "4",
      "year": "1960",
      "journal-title": "Phys. Rev. Letters"
    },
    {
      "key": "2024020601530481000_r40",
      "doi-asserted-by": "crossref",
      "first-page": "407",
      "DOI": "10.1016/0003-4916(57)90015-5",
      "volume": "2",
      "year": "1957",
      "journal-title": "Ann. Phys."
    },
    {
      "key": "2024020601530481000_r40a",
      "doi-asserted-by": "crossref",
      "first-page": "1",
      "DOI": "10.1016/0003-4916(60)90126-3",
      "volume": "11",
      "year": "1960",
      "journal-title": "Ann. Phys."
    },
    {
      "key": "2024020601530481000_r40b",
      "doi-asserted-by": "crossref",
      "first-page": "1065",
      "DOI": "10.1103/PhysRev.123.1065",
      "volume": "123",
      "year": "1961",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r40c",
      "doi-asserted-by": "crossref",
      "first-page": "1067",
      "DOI": "10.1103/PhysRev.125.1067",
      "volume": "125",
      "year": "1962",
      "journal-title": "Phys. Rev."
    },
    {
      "key": "2024020601530481000_r41"
    },
    {
      "key": "2024020601530481000_r42"
    },
    {
      "key": "2024020601530481000_r43"
    },
    {
      "key": "2024020601530481000_r44"
    },
    {
      "key": "2024020601530481000_r45"
    }
  ],
  "container-title": "Journal of Mathematical Physics",
  "original-title": [],
  "language": "en",
  "link": [
    {
      "URL": "https://pubs.aip.org/aip/jmp/article-pdf/3/4/650/19167464/650_1_online.pdf",
      "content-type": "application/pdf",
      "content-version": "vor",
      "intended-application": "syndication"
    },
    {
      "URL": "https://pubs.aip.org/aip/jmp/article-pdf/3/4/650/19167464/650_1_online.pdf",
      "content-type": "unspecified",
      "content-version": "vor",
      "intended-application": "similarity-checking"
    }
  ],
  "deposited": {
    "date-parts": [
      [
        2024,
        2,
        6
      ]
    ],
    "date-time": "2024-02-06T01:53:42Z",
    "timestamp": 1707184422000
  },
  "score": 1,
  "resource": {
    "primary": {
      "URL": "https://pubs.aip.org/jmp/article/3/4/650/227944/Mass-Singularities-of-Feynman-Amplitudes"
    }
  },
  "subtitle": [],
  "short-title": [],
  "issued": {
    "date-parts": [
      [
        1962,
        7,
        1
      ]
    ]
  },
  "references-count": 69,
  "journal-issue": {
    "issue": "4",
    "published-print": {
      "date-parts": [
        [
          1962,
          7,
          1
        ]
      ]
    }
  },
  "URL": "http://dx.doi.org/10.1063/1.1724268",
  "relation": {},
  "ISSN": [
    "0022-2488",
    "1089-7658"
  ],
  "subject": [],
  "published-other": {
    "date-parts": [
      [
        1962,
        7
      ]
    ]
  },
  "published": {
    "date-parts": [
      [
        1962,
        7,
        1
      ]
    ]
  }
}