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Perturbative Results) Week 17: Holomorphy 1 The following books and articles are useful for this discussion: • J. Wess and J Bagger \Supersymmetry and Supergravity", chapters 8{ 13. • N. Seiberg,\The power of holomorphy: exact results in 4D SUSY field theories" http://arXiv.org/abs/hep-th/9408013 • F. Cachazo, M. R. Douglas, N. Seiberg and E. Witten, \Chiral Rings and Anomalies in Supersymmetric Gauge Theory," JHEP 0212, 071 (2002) [arXiv:hep-th/0211170]. • G. Veneziano and S. Yankielowicz, \An Effective Lagrangian For The Pure N=1 Supersymmetric Yang-Mills Theory," Phys. Lett. B 113, 231 (1982). 1 Holomorphy (perturbative results) So far we have seen how matter field in supersymmetric field theories are well described by chiral superfields and antichiral superfields. The chiral fields are special in that they can be thought of as holomorphic coordinates, while the antichiral superfields can be thought of as antiholomorphic coordinates. The superpotential describing part of the interactions of a supersym- metric field theory is only holomorphic. This means it depends only on holomorphic fields, but not their complex conjugates. Let us call the general class of holomorphic fields φ. There can be many of them. This is a shorthand notation where the details of the fields are not needed for the general argument. If we consider the effective action for matter, if will consist of various pieces. Z Z Z d4θK(φ, φ¯) + d2W (φ) + d2θ¯W¯ (φ¯) (1) We will assume that W (φ) is regular about zero, so that it admits a power series representation with coupling constants giving the Taylor expansion coefficients of W . X ai W (φ) = φi (2) tree i! i≥1 1 c David Berenstein 2009 1 This can work in the presence of more than one field so long as we recognize i that both ai and φ can be interpreted as multi-indices. For simplicity, we can shift the field so that a1 = 0, and we call a2 = m (this is the mass of the field). Notice that the superfield φ(x; θ) has two symmetries. Since φ(x; θ) is complex there is a U(1) charge that φ carries that counts the number of φ. Under this charge all components of the superfield carry charge one. There is a second charge (the R-charge), under which the components of φ transform with different weights p φ(x; θ) = φ(x) + 2θ + θ2F (3) so that Under this U(1) R-charge, the θ transform to make the superfield Field U(1)φ U(1)R φ(x) 1 0 (x) 1 1 F (x) 1 2 φ(x; θ) 1 0 θ 0 −1 R d2θ 0 2 have fixed charge. However, the superpotential in general does not preserve either of these charges. There is a very useful trick to use these symmetries anyhow to do the analysis. The idea is that the coupling constants ai should not be treated as constants, but as background fields that happen to be constant., One should imagine that there is a bigger unknown theory where the ai are dynamical fields. We treat them in the background method, so as far as calculations go, they have the same behavior as coupling constants. The point of treating them as dynamical fields is that we can promote them to superfields. Thus, in order to be in the superpotential, the ai must be chiral superfields, while thea ¯i must be antichiral. We can choose the U(1)φ and U(1)R charges of the ai so that the su- perpotential preserves both the U(1)R and the U(1)φ symmetries. Vacuum expectation values of the ai break this symmetry spontaneously. With these symmetries, the ai have charges 2 Field U(1)φ U(1)R ai −i -2 The U(1)R and U(1) symmetries are now symmetries of the superpo- tential and the full theory with the ai superfields. This is the tree level superpotential. Let us consider the full superpotential with radiative corrections to the superpotential included. Because the superpotential only depends on chiral fields, it is holomorphic. This means that @a¯W (φ) = 0 (4) Here this result is on the full superpotential and not just the tree level results. This is a really powerful statement, as we shall see. Now, since the original theory has two U(1) symmetries that act on field by rotations, we can find a regulator that preserves both symmetries. Di- mensional regularization will do that or us. Hence, the possible counterterms to the lagrangian (and the superpotetnial) must be polynomial in the φ fields and the coupling consatts ai>2. They must also preserve both U(1) symme- tries. Since the fields ai are background fields, a2 can appear in denominators. The most general correction must be of the form X Z d2θam1 : : : amk φman (5) i1 ik 2 The symmetries restrict the power series. The R-charge must be equal to −2, so we have that X −2(n + ikmk) = −2 (6) Remember that the mk are positive integers. Thus we find that n is neces- sarily negative, unless only one mk is non-zero, and identical to one (these are tree level superpotential terms). Also, the U(1) φ charge must vanish, so that X m − kikmk − 2n = 0 (7) P The first equation ?? tells us that for each vertex added ( ikmk), we add one denominator in a2 = m, except for the standard tree level superpotential term, which has no denominators in a2. The shift is exactly the one that 3 would be required to form a tree diagram with ai as vertices with i legs that are connected with propagators where 1=a2 is the propagator. The second equation tells us that the number of external φ legs corre- sponds to such tree level expressions. This is, the allowed counterterms in W only arise from tree diagrams. It is impossible to form a loop and to respect the symmetries of the lagrangian and at the same time contribute to W in any other way. However, this means that Wtree is not renormalized at any order in per- turbation theory. This is because in standard renormalization theory the counterterms are 1PI, and tree levels with propagators are not 1PI. Notice that the superpotential allows for integrating out massive fields in a low energy effective action: they get integrated out by tree diagrams only. 1.1 Beta functions Since the superpotential is not renormalized, one can imagine a renormaliza- tion group scheme where the β functions for the ai couplings vanish. In a general scheme, the renormalized fields depend on wave-function renromalzi- ation φren = Zφ. There are no loop counterterms to W , but the wavefunction renormalization would enter into the definition of φ with anomalous dimen- sions, etc. There is a scheme where Z = 1. In such a scheme the values of the fields are scale independent. This renormalization group scheme is called the Wilsonian action, and the holomorphy theorem is stated by saying that the Wilsonian couplings in the superpotential are not renormalized. In this setup the role of Z is played by the kinetic term of the theory, which does not have canonical normalization any longer. Thus, the value m is not a physical mass, but a holomorphic mass. The pole mass of the particle can get renormalized from the tree level expression, because the propagator gets modified to 1 1 ! (8) p2 + jmj2 Ap2 + jmj2 etc. However, if m = 0, then the corresponding particle is massless (the shift in the pole mass is multiplicative). If a particle is massless somewhere, then we can not integrate it out everywhere, and a superpotential where such a massless particle is integrated out should be singular at the position where the particle is massless. 4 In the Wilsonian RG scheme, we find that β(ai) = 0 Also, if we use other renormalization schemes, then we get that @a @Z(φ) k ' −k a (9) @ log(µ) @ log(µ) i i so that the beta function for Z ai is zero. Notice that this non-renormalization theorem solves the hierarchy prob- lem in a technically natural way. The idea is that if the mass terms mi are small at a high scale, they are naturally small at a low scale as well. They differ from the ones at a high scale by a wave function renormalzation factor (these are of order 1 because of the logarithmic running). Thus, the masses of the scalar bosons are protected from quadratic divergences that give rise to the usual fine- tuning and hierarchy problems. It is not a completely natural solution to the hierarchy problem because it still relies on assumptions about values at high energies. It just does not require fine tuning to huge precision relative to the size of couplings at the high scale. 2 Holomorphy: nonperturbative resutls Let us study in detail the structure of supersymmetric vacua in supersym- metric field theories. The supercharges are given by standard expressions @ Q = − iσµ θ¯β_ @ (10) α @θα αβ_ µ And the covariant derivatives are given by @ D = + iσµ θ¯β_ @ = Q + 2σµ θP¯ (11) α @θα αβ_ µ α αβ_ µ where we are identifying Pµ = i@µ as the generator of translations. We want to think of Q and P as operators in the Hilbert space of states. Assume that we have a supersymmetric vacuum j0i. We thus have that P j0i = Qj0i = Q¯j0i (12) 5 The different vacua can be parametrized by order parameters that distinguish them. The order parameters are local gauge invariant operators in the field theory.
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