Algebro-geometric Feynman rules. (English) Zbl 1225.81101
In theoretical particle physics, Feynman rules are used to translate (Feynman) graphs into a mathematical formula (the “value” or the “result” of this graph). Within dimensional regularisation, the result can be viewed as a Laurent series in the dimensional regularisation parameter. In this sense, Feynman rules define a map from a set of graphs into the commutative ring of Laurent series. One can consider different sets of Feynman rules, which will map the same graph to different results. An example would be a set of Feynman rules for a scalar theory with massless particles and a second set of Feynman rules for a scalar theory with particles of mass \(m\).
From the basics of quantum field theory it follows that, whenever a graph is disconnected, the result of this graph is given as the product of the results of the connected components.
In this paper the authors study this situation mathematically. They consider maps from graphs into a commutative ring which satisfy the multiplicative property above. They call these maps “algebro-geometric Feynman rules”, although the suggestion “multiplicative graph maps in algebraic geometry” captures the set-up more precisely.
The main question of the paper is the following: Is it possible to assign a quantity related to the Euler characteristic of the graph hypersurface complement to a graph, and at the same time satisfy the multiplicative property? This is a non-trivial question, raised by a referee of a previous publication of the authors. A straightforward construction, which satisfies the multiplicative property, will assign a zero value to every graph which is not a forest. The authors carefully trace back the reason for this zero to a torus prefactor \((\mathbb L-1)\), where \(\mathbb L\) is the Lefschetz motive. In order to give a positive answer to the question above, the authors show that it is possible to define a multiplicative map from graphs into a polynomial ring such that the derivative of the polynomial at zero is the Euler characteristic. This is the main result of the paper.
The paper also includes a section on renormalisation and a section on motivic zeta functions obtained from the partition function.
From the basics of quantum field theory it follows that, whenever a graph is disconnected, the result of this graph is given as the product of the results of the connected components.
In this paper the authors study this situation mathematically. They consider maps from graphs into a commutative ring which satisfy the multiplicative property above. They call these maps “algebro-geometric Feynman rules”, although the suggestion “multiplicative graph maps in algebraic geometry” captures the set-up more precisely.
The main question of the paper is the following: Is it possible to assign a quantity related to the Euler characteristic of the graph hypersurface complement to a graph, and at the same time satisfy the multiplicative property? This is a non-trivial question, raised by a referee of a previous publication of the authors. A straightforward construction, which satisfies the multiplicative property, will assign a zero value to every graph which is not a forest. The authors carefully trace back the reason for this zero to a torus prefactor \((\mathbb L-1)\), where \(\mathbb L\) is the Lefschetz motive. In order to give a positive answer to the question above, the authors show that it is possible to define a multiplicative map from graphs into a polynomial ring such that the derivative of the polynomial at zero is the Euler characteristic. This is the main result of the paper.
The paper also includes a section on renormalisation and a section on motivic zeta functions obtained from the partition function.
Reviewer: Stefan Weinzierl (Mainz)
MSC:
| 81T18 | Feynman diagrams |
| 81Q30 | Feynman integrals and graphs; applications of algebraic topology and algebraic geometry |
| 58D30 | Applications of manifolds of mappings to the sciences |
| 18F30 | Grothendieck groups (category-theoretic aspects) |
| 13D15 | Grothendieck groups, \(K\)-theory and commutative rings |
| 16S34 | Group rings |
Keywords:
Feynman rules; parametric Feynman integrals; graph hypersurfaces; Grothendieck ring of varieties; Chern-Schwartz-MacPherson classesReferences:
| [1] | DOI: 10.1016/j.crma.2006.01.002 · Zbl 1085.14006 · doi:10.1016/j.crma.2006.01.002 |
| [2] | DOI: 10.1017/S0305004109990247 · Zbl 1187.14008 · doi:10.1017/S0305004109990247 |
| [3] | DOI: 10.4310/CNTP.2009.v3.n1.a1 · Zbl 1171.81384 · doi:10.4310/CNTP.2009.v3.n1.a1 |
| [4] | Aluffi P., Adv. Theor. Math. Phys. |
| [5] | Aluffi P., Lett. Math. Phys. |
| [6] | Bjorken J., Relativistic Quantum Mechanics (1965) |
| [7] | DOI: 10.1007/s11537-007-0648-9 · Zbl 1161.11021 · doi:10.1007/s11537-007-0648-9 |
| [8] | S. Bloch, The Geometry of Algebraic Cycles (AMS, 2010) pp. 137–144. |
| [9] | DOI: 10.1007/s00220-006-0040-2 · Zbl 1109.81059 · doi:10.1007/s00220-006-0040-2 |
| [10] | DOI: 10.4310/CNTP.2008.v2.n4.a1 · Zbl 1214.81095 · doi:10.4310/CNTP.2008.v2.n4.a1 |
| [11] | DOI: 10.1007/s00220-009-0740-5 · Zbl 1196.81130 · doi:10.1007/s00220-009-0740-5 |
| [12] | DOI: 10.1007/s002200050779 · Zbl 1032.81026 · doi:10.1007/s002200050779 |
| [13] | DOI: 10.1007/PL00005547 · Zbl 1042.81059 · doi:10.1007/PL00005547 |
| [14] | Connes A., Amer. Math. Soc. 55 |
| [15] | DOI: 10.1016/j.geomphys.2005.04.004 · Zbl 1092.81050 · doi:10.1016/j.geomphys.2005.04.004 |
| [16] | DOI: 10.4310/CNTP.2010.v4.n2.a3 · Zbl 1219.81201 · doi:10.4310/CNTP.2010.v4.n2.a3 |
| [17] | DOI: 10.1088/0305-4470/37/45/020 · Zbl 1062.81113 · doi:10.1088/0305-4470/37/45/020 |
| [18] | DOI: 10.1007/978-3-662-02421-8 · doi:10.1007/978-3-662-02421-8 |
| [19] | Gillet H., J. Reine Angew. Math. 478 pp 127– |
| [20] | Itzykson C., Quantum Field Theory (2006) |
| [21] | DOI: 10.1002/sapm197961293 · Zbl 0471.05020 · doi:10.1002/sapm197961293 |
| [22] | DOI: 10.1007/978-3-642-56478-9_10 · doi:10.1007/978-3-642-56478-9_10 |
| [23] | Kwieciński M., C. R. Acad. Sci. Paris Sér. I Math. 314 pp 625– |
| [24] | Larsen M., Mosc. Math. J. 3 pp 85– |
| [25] | DOI: 10.2307/1971080 · Zbl 0311.14001 · doi:10.2307/1971080 |
| [26] | Marcolli M., Feynman Motives (2010) |
| [27] | Marcolli M., J. Phys. A 41 |
| [28] | G. C. Rota, Problèmes combinatoires et théorie des graphes, Colloq. Internat. CNRS (CNRS, Paris, 1978) pp. 363–365. |
| [29] | Schwartz M. H., C. R. Acad. Sci. Paris 260 pp 3535– |
| [30] | DOI: 10.1016/S0166-8641(98)00037-6 · Zbl 0928.14010 · doi:10.1016/S0166-8641(98)00037-6 |
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