Perturbative Results)

Home , Vacuum expectation value

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 supersymmetric 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 √ φ(x, θ) = φ(x) + 2θψ + 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 ψ(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 superpotential 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 superpotential 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) symmetries. 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 perturbation 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 renormalization 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 + |m|2 Ap2 + |m|2 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 problem 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 supersymmetric 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 |0i. We thus have that

P |0i = Q|0i = Q¯|0i (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. Thus, vacua are parametrized by vacuum expectation values of gauge invariant superfields h0|O(x, θ, θ¯)|0i (13) Translation invariance of the vacuum implies that ¯ ¯ h0|[Pµ, O(x, θ, θ)]|0i = 0 = i∂µh0|O(x, θ, θ)|0i (14) so that the vacuum expectation values are invariant under translations. Lorentz invariance implies that O is a scalar field, so it is a bosonic superfield. Similarly, we can use the supersymmetry of the vacuum to show that h0|D O(x, θ, θ¯)|0i = h0|[Q +2σµ θP¯ , O(x, θ, θ¯)]|0i = 0 = D h0|O(x, θ, θ¯)|0i α α α˙ β˙ µ α (15) In this equation Q, P are operators, while the θ, θ¯ are parameters and there- fore Q, P do not act on them. Similarly we can show that D¯h0|O(x, θ, θ¯)|0i = 0. This shows that on the space of vacua the superfields that can get vacuum expectation values are both chiral and antichiral. If the operators are already chiral, the antichirality of the vacuum expectation value is a property of the vacuum solution, but not of the general correlator of operators. Moreover, if a superfield can be written as ¯ ¯ O(x, θ, θ) = {Dα,G(x, θ, θ)} (16) then the same type of manipulations as done above show that the expectation value of O vanishes. This is why the order parameters of a vacuum manifold can be parametrized by equivalence classes of chiral operators: the cohomology of D¯. This is in the sense above, that D¯ derivatives vanish, so theory do not contribute to expectation values of order parameters. Also, he product of the vacuum expectation value of two chiral operators a different positions is chiral and independent of the position. This is, ¯ ¯ ¯ ¯ h0|O(x1, θ, θ)i∂x2 O(x2, θ, θ)|0i = h0|O(x1, θ, θ)[P, O(x2, θ, θ)]|0i (17) ¯ ¯ ¯ = h0|O(x1, θ, θ)[{Dα, Dα˙ }, O(x2, θ, θ)]|(18)0i ¯ ¯ ¯ = Dα˙ h0|O(x1, θ, θ)[Dα, O(x2, θ, θ)]|0i (19) = 0 (20)

6 ¯ where the D acts in the usual way in θ, but ∂x gets replaced by P = i∂x1 +i∂x2 Since the θ and θ2 component of a chiral superfield vanish in a supersymmetric vacuum (they can be show to be given by fields that are of the form DG¯ ), we find that supersymmetric vacua are characterized just by the vacuum expectation values of the lowest components of the chiral superfields. Moreover, the vacuum expectation value of the product of two (or many) such chiral operators is independent of position. Because of Cluster decom- position, (we can separate the x1, x2 away from each other as far as we want to), the correlation function factorizes. This is

hO1(x1)O2(x2)i = hO1(x1)ihO2(x2)i = hO1(0)ihO2(0)i = hO1(0)O2(0)i (21) the last equation is because the operator is independent of (x2), so we can take the limit x2 → 0. This means that the chiral operators that form the cohomology of D¯ have a ring structure ( we can multiply them and obtain another operator in the coholomogy of D¯). This endows the set of chiral operators with a commutative ring structure. This structure is called the chiral ring. Notice that since vacuum expectation values of chiral superfields are complex variables, the ring is a ring over the complex numbers. Also, by definition, a vacuum is an assignment of numbers to the chiral ring vacuum expectation values so that the relations of the chiral ring are satisfied. The claim of holomorphy is that the set of vacuum expectation values of the chiral ring suffices to classify all supersymmetric vacua and that the com- plete set of representations of the chiral ring is in one to one correspondence with the different vacua of the theory. The next problem is to understand how many vacua there are and how to calculate the chiral ring relations. The simplest example we can think of is of a theory with a single scalar field φ and a superpotential W (φ). The equations of motion make it so that

¯ 2 ¯ 0 D φ = ∂φW = W (22)

The order parameter we have is a = hφi. By taking vacuum expectation values and using the chiral ring properties we ﬁnd that

W 0(a) = 0 (23)

7 So these must be the relations in the chiral ring. The conditions W 0 = 0 are those that calculate the F ∗ expectation values. These must vanish in order for supersymmetry to be unbroken, because ¯ h[ξQ, ψ]i ∼ hδξψi ∼ hξF + ξ∂φi ∼ ξhF i = 0 (24)

If W is a polynomial of degree d, then W 0 has degree d − 1, and there are d − 1 solutions of a, which are the roots of W 0(a) = 0. This is the result in perturbation theory, when the coupling constants of W are small. Thus we would ﬁnd d − 1 supersymmetric vacua. The concept of the Witten index is a useful topological invariant. This counts vacua with a sign +1 for bosonic vacua, and −1 for fermionic vacua. This is preserved under small deformation of W , and are independent of the Kahler potential ( so long as it is regular). Because of spin statistics, all Lorentz invariant vacua must be bosonic states. However, we can sometimes have families of vacua characterized by continuous parameters. These continuous parameters are called moduli. Small ﬂuctuations in those parameters cost no energy, so the moduli indicate massless bosonic particles. By Supersymmetry, they have fermionic massless su- perpartners. If we take a limit with zero momentum, we can produce a fermionic state with zero energy. Thus in this situation the Witten index contribution is zero. A proof that the Witten index is a topological invariant is beyond the scope of the present discussion. The sketch of the proof is done with a quantum mechanical system as follows. We consider the function

I(t) = tr((−1)F exp(−tH)) (25) where the Hamiltonian is H = Q2, Q is hermitian, and we have a fermion number modulo 2 which anticommutes with Q.(−1)F Q = −Q(−1)F Since H is hermitian, we can diagonalize it, and we can also check that (−1)F commutes with H, so we can diagonalize them simultaneously. Notice that I(t) is independent of t, because for non-zero eigenvalues, given a state |ii, the state Q|ii has the same energy, but opposite statistics. These cancel in the evaluation of I(t) = I(0). Now, for the proof, we keep (−1)F , but we are allowed to deform Q and H, such that the rest of the algebraic relations are maintained. This will allow the non-zero energy states to migrate, but so long as the deformation is small

8 (this is where one needs to be very careful on the technical refinements), the ones that reach zero will get there in pairs of opposite statistics and we will find that I is also independent of the small parameters (hence it is called topological). For the case above, so long as we don’t change the degree of the polynomial in W , the Witten index is governed by the perturbative result, even if we make the coupling constants large. The true chiral ring will thus be given by a single generator a = hφi, and an exact operator relation ad−1 + ··· = 0 which is valid on all the d − 1 vacua. This can be integrated out to produce a W non-perturbatively (which happens to coincide with the perturbative results). One can also show that on taking the term with highest degree to zero, one of the vacua migrates towards infinity, thus changing the Witten index. Now, let us assume that we have just SU(N) gauge theory. How do we proceed? We first need to turn the problem of vacua to quantum mechanics. We do this by using dimensional reduction on a tours. We can do that and maintain supersymmetry, but we break Lorentz invariance. If the torus is very large, the theory is confined and is described by non-perturbative results. However, shrinking the size of the torus should not affect the Witten index. For very small torus, the theory is perturbative, and the analysis can be done classically. The classical vacua are characterized by vanishing F . This means that the connection can be reduced to the Cartan and to be constant. However, we still have Wilson lines along the circles of the torus (thus we have a non-tricial family problem). Because of large gauge transformations, the gauge field is unique up to periodic identifications (which put us in the fundamental cell of the root lattice of SU(N)). One also has Weyl reflection identifications, so one ends up with the Weyl chamber of SU(N). The index then reduces to a topological calculation on this space, which gives a Witten index N. Thus, SYM should have N vacua. How should we understand them?