Contents

FTPI-MINN-11/10, UMN-TH-2943/11

July 26, 2019

Supersymmetry and a Nonrenormalization Theorem

Xiaoyi Cui and M. Shifman

Department of Physics, University of Minnesota, Minneapolis, MN 55455, USA

[1mm] William I. Fine Theoretical Physics Institute, University of Minnesota, Minneapolis, MN 55455, USA

Abstract

In this paper we continue the study of perturbative renormalizations in an supersymmetric model. Previously we analyzed one-loop graphs in the heterotically deformed CP models. Now we extend the analysis of the function and appropriate factors to two, and, in some instances, all loops in the limiting case . The field contents of the model, as well as the heterotic coupling, remain the same, but the target space becomes flat. In this toy model we construct supergraph formalism. We show, by explicit calculations up to two-loop order, that the function is one-loop-exact. We derive a nonrenormalization theorem valid to all orders. This nonrenormalization theorem is rather unusual since it refers to (formally) terms. It is based on the fact that supersymmetry combined with target space symmetries and “flavor” symmetries is sufficient to guarantee the absence of loop corrections. We analyze the supercurrent supermultiplet (i.e., the hypercurrent) providing further evidence in favor of the absence of higher loops in the function.

## 1 Introduction

In this paper we discuss multiloop calculations in a specific linear sigma model. The motivation is two-folded. On the one hand, this is a continuation of our previous study [1] of a class of two-dimensional CP nonlinear sigma models (heterotic CP models for short). On the other hand, the linear model we suggest has its own field-theoretical significances, among which the most interesting are a peculiar supergraph technique and a version of nonrenormalization theorem. Surprisingly, it is a renoramlization theorem for terms!

Two-dimensional CP models emerged as effective low-energy theories on the world sheet of non-Abelian strings in a class of four-dimensional   gauge theories [2, 3, 4, 5] (for reviews see [6]). Deforming these models in various ways (i.e. breaking supersymmetry down to ) one arrives at heterotically deformed CPmodels [7, 8, 9, 10], a very interesting and largely unexplored class of models characterized by two coupling constants: the original asymptotically free coupling and an extra one describing the strength of the heterotic deformation. These two-dimensional models exhibit highly nontrivial dynamics, with a number of phase transitions. This fact was recently revealed [11] in the large- solution of the model.

Our task is the study of perturbation theory in two-dimensional heterotic models. Many general aspects of models in perturbation theory were discussed in [12, 13]. The problem we address is more concrete. In [1] we studied particular renormalization properties and calculated the one-loop functions in the CP models heterotically deformed in a special way. Written in components111The superfield expression for the heterotically deformed CP models can be found e.g. in [1]., the Lagrangian of the heterotic CP model takes the following form [8]:

 Lhet=L(2,2)+L(0,2), (1)

where

 L(2,2)=G{∂μϕ∂μϕ†+i¯ψ⧸∂ψ−2i1χ¯ψγμψϕ†∂μϕ−2χ2ψ†LψLψ†RψR}, (2)

and 222The sign in front of the term in (3) is opposite to that in [8] due to a typo in [8]. Also notice that the definition of in this paper corresponds to in [8]. The reason for rescaling of the deformation parameter compared to [8] is that both and here are genuine loop expansion parameters, as the reader will see later.

 L(0,2)=ζ†Ri∂LζR+[γg2ζRR(i∂Lϕ†)ψR+H.c.] +|γ|2g2(ζ†RζR)(Rψ†LψL)+G{2|γ|2g2χ2ψ†LψLψ†RψR}. (3)

We denote by the Kähler metric on the target space,

 G=2g2χ2, (4)

is the Ricci tensor,

 R=2χ2, (5)

and we use the notation

 χ≡1+ϕϕ†. (6)

The coupling enters through the metric, while the deformation coupling appears in Eq. (3). In the previous paper [1] we determined the one-loop functions

 β(g2) ≡ ∂∂lnμg2(μ)=−g42π+..., (7) β(γ) ≡ ∂∂lnμγ(μ)=γ2π(γ2−g2)+..., (8)

where the dots stand for two-loop and higher-order terms. The heterotic deformation does not affect which stays the same as in the CP(1) model. Among other results, we calculated the law of running of the ratio . If in the ultraviolet (UV) limit is chosen to be smaller than , in the infrared (IR) it runs to , which is the fixed point for this parameter. With in the UV, the theory is asymptotically free.

Now we undertake the next step: multiloop graphs. However, this is not easy. At two and higher loops interplay between and is contrived. The impact of the deformation term was not studied before. It seems reasonable to start from untangling from the nonlinear target space. In the heterotic CP model there is a fermion flavor symmetry. From a practical point of view, we want to understand this symmetry by probing it in a simpler setup. So we will focus on a simpler, linear version of the sigma model, (setting ) which serves our purposes at this stage.

We start from developing an appropriate supergraph technique to carry our an explicit two-loop calculation. The result is as follows: the interaction term proportional to is not renormalized, and so are the factors of the superfield (see Eq. (16)). The factors of the superfields and are renormalized, but this is just an iteration of the one-loop contribution. Then we prove the nonrenormalization theorem, which extends the first result to all orders. What is remarkable is the fact that the nonrenormalization theorem emerges for a term provided there are certain target space conditions. Thus, up to two-loop order, the function in the heterotic model at hand is

 β(γ)=γ32π. (9)

This is compatible with (8), of course. Due to the fact that the nonrenormalization theorem generally fails to detect the geometric progression in the factors of and , at the moment we can not directly extend this result to three loops and higher in . But it is reasonable to conjecture that this is the case. An argument substantiating this statement is presented in Sec. 7.

The paper is organized as follow. In Sec. 2 we introduce the simplified heterotic linear model, which captures in full the quantum behavior of the deformation strength . In Sec. 3 we give the Feynman rules for supergraph calculations in theories. In Sec. 4, we calculate the two-loop contribution to . Vanishing of certain diagrams provides us with an indication of the nonrenormalization theorem. In Sec. 5, we give the term nonrenormalization theorem, which is valid perturbatively. In Sec. 6 we extend this statement beyond perturbation theory. In Sec. 7 we analyze the supercurrent supermultiplet of this model (the so-called hypercurrent), following the line of reasoning of [14].

## 2 An N=(0,2) linear model

In our previous work [1] we showed that in the CP model, there is a fermionic SU flavor symmetry, which mixes the chiral fields and (see Eq. (19)). To mimic this phenomenon, we introduce a simplified linear model, which emphasizes the mechanism of the deformation and retains the fermion flavor symmetry.

We begin by briefly reviewing supersmmetry and some notations. We define the left moving and right moving derivatives as

 ∂L=∂t+∂z,∂R=∂t−∂z, (10)

and use the following definition for the superderivatives:

 DR=∂∂θR−iθ†R∂L,¯DR=−∂∂θ†R+iθR∂L. (11)

Their commutator gives , as it should. All integrations and differentiations are understood as acting from the left, if not stated to the contrary. The shifted space-time coordinates that satisfy the chiral condition are

 y0=t+iθ†RθR,y1=z+iθ†RθR. (12)

The antichiral counterparts are

 ~y0=t−iθ†RθR,~y1=z−iθ†RθR. (13)

Under supersymmetric transformation

 θR→θR+ϵ, θ†R→θ†R+¯ϵ, yμ→yμ+2i¯ϵθR, ~yμ→~yμ−2iθ†Rϵ, (14)

where .

We can now define the chiral superfields in our model,

 A(yμ,θR)=ϕ(yμ)+√2θRψL(yμ), B(yμ,θR)=ψR(yμ)+√2θRF(yμ), B(yμ,θR)=ζR(yμ)+√2θRF(yμ). (15)

Here , , and describe physical degrees of freedom, while and will enter without derivatives and, thus, can be eliminated by virtue of equations of motion.

In the superfield formalism the Lagrangian of the simplified model is as follow:

 L=12∫d2θR[12(iA†∂RA−iA∂RA†)+B†B+B†B−(γBBA†+H.c.)]. (16)

In the component language, after eliminating and , we have

 L = ∂μϕ†∂μϕ+i¯ψ⧸∂ψ+iζ†R∂LζR+[γζRψR∂Lϕ†+H.c.] (17) +γ2(ζ†RζR)(ψ†LψL)+γ2(ψ†RψR)(ψ†LψL).

Note that supersymmetry completely fixes the second line in terms of the first line.

The Lagrangian is invariant under SU rotations of and . Actually, if we define an SU superfield doublet

 Ψ=(BB), (18)

the part of the Lagrangian that involves all right-handed fermions can be rewritten as

 12∫d2θRΨ†aΨa+[γ2A†εabΨaΨb+H.c.], (19)

which is obviously SU invariant.

Comparing with Eq. (3), we indeed see that Eq. (17) is the limiting case of the former with . The opposite limiting case, , is well-understood; it is just the undeformed model in Eq. (2). The model in Eq. (17) can be viewed as a preparatory step to developing perturbation theory in the heterotic CP models. We will show that this model exhibits a nonrenormalization theorem. The proof of the latter strengthens our understanding of heterotic supersymmetry.

## 3 Supergraph method

In this section we explicitly formulate superfield/supergraph calculus for the given model. Calculations in the language were previously discussed in the literature, see e.g. [15, 16]. We feel that it is worth developing a similar formalism for theories, for the following reasons. First, most models can be obtained as deformations from , where holomorphic structures are crucial. It would be best if we preserve them explicitly. Second, this language is useful in deriving the nonrenormalization theorem of Sec. 5, a phenomenon not so easy to see when manipulating with superalgebras. Third, so far no calculations were performed at two-loop level. The tools we develop here are expected to be helpful in the heterotic CP models too.

To derive the superpropagator, we define the functional variation for a bosonic chiral and antichiral superfields,

 δδA(y,θR)A′(y′,θ′R) = δ(y−y′)δ(θR−θ′R), δδA†(~y,θ†R)A′†(~y′,θ′†R) = δ(~y−~y′)δ(θ†R−θ′†R), (20)

where and are defined in Eq. (12) and (13). For a generic function , we have

 δδA(y,θR)∫d2x′dθ′Rdθ′†RA(y′,θ′R)F(x′,θ′R,θ′†R) (21) = ∫d2y′dθ′Rdθ′†Rδ(y−y′)δ(θR−θ′R)F(y′−iθ′†Rθ′R,θ′R,θ′†R) = −∫dθ†RF(y−iθ†RθR,θR,θ†R)=¯DRF(x,θR,θ†R).

Similarly,

 δδA†(~y,θ†R)∫d2x′dθ′Rdθ′†RF(x′,θ′R,θ′†R)A†(~y′,θ′†R) (22) = ∫d2y′dθ′Rdθ′†RF(~y′+iθ′†Rθ′R,θ′R,θ′†R)δ(~y−~y′)δ(θ†R−θ′†R) = ∫F(~y+iθ†RθR,θR,θ†R)dθR=F(x,θR,θ†R)←DR.

Note that we intentionally write acting from the right, because we want our expression to be explicitly Hermitean-conjugate to the previous result.

On the other hand, we compare the result with

 ∫d2x′dθ′Rdθ′†RF(x′,θ′R,θ′†R)¯DRδ(x−x′)δ(θ†R−θ′†R)δ(θR−θ′R) (23) = −∫d2x′dθ′Rdθ′†Rδ(x−x′)δ(θ†R−θ′†R)δ(θR−θ′R)¯DRF(x′,θ′R,θ′†R) = −¯DRF(x,θR,θ†R),

which implies that, upon integration,

 δδA(x,θR,θ†R)A′(x′,θ′R,θ′†R) = −¯DRδ(x−x′)δ(θ†R−θ′†R)δ(θR−θ′R), δδA†(x,θR,θ†R)A′†(x′,θ′R,θ′†R) = −δ(x−x′)δ(θ†R−θ′†R)δ(θR−θ′R)←DR. (24)

For a chiral field , we have the projection

 ¯DRDR2i∂LJA={¯DR,DR}2i∂LJA=JA. (25)

Using this we can conveniently pass from the term to the integration over the full superspace, namely

Here and in what follows in this section we use to denote the triplet of (super)coordinates . Note that the currents and are Grassmannian. We can write the partition function as

and, by virtue of the functional integration, we get

 (28) = exp∫d3z−i2(J†A1□JA+JA−1□J†A).

As a result we get the Feynman propagator for the chiral field in the form

 ⟨0|T{A(x,θR,θ†R),A†(y,ηR,η†R)}|0⟩=i□δ(x−y)δ(θ†R−η†R)δ(θR−ηR). (29)

Using the same line of reasoning now we will determine the propagators for the superfields and . Note that due to the fermionic symmetry (see Eq. (19)), they are exactly the same. Take for example; the partition function is

 Z[JB,J†B]=∫DBDB†exp(i∫d3z12B†B+BDR2i∂LJB+B†−¯DR2i∂LJ†B). (30)

By virtue of the functional integration, we arrive at

 (31) = exp∫d3z−i2(J†B1i∂LJB+JB−1i∂LJ†B).

As a result,

 ⟨0|T{B(x,θR,θ†R),B†(y,ηR,η†R)}|0⟩=−1∂Lδ(x−y)δ(θ†R−η†R)δ(θR−ηR). (32)

The same applies to .

Now, let us pass to the interaction vertices. They can be obtained by considering

 Sint[δδJ]JBJBJ†A=−γ2∫d3z⎛⎝δδJBδδJBδδJ†A⎞⎠JB(z1)JB(z2)J†A(z3) (33) = γ2∫d3z¯DRδ(z−z1)¯DR(z−z2)δ(z−z3)←DR.

We can summarize the Feynman rules for the model at hand in the momentum space:

• For each propagator , write

 −ip2δ(θ12)

where ; for each propagator or , write

 −ipLδ(θ12),

with the momentum flowing from to .

• For each vertex, write .

• For each propagator that connects a chiral field to the vertex, put acting on it; for that connecting an antichiral field, put acting on it, where is the momentum that flows into the vertex through the propagator.

• Integrate over and impose momentum conservation at each vertex, integrate over the momentum for each loop.

• For each external chiral or antichiral line, we have a factor for the field, but no or factors.

This set of the Feynman rules is displayed in Fig. 1.

To facilitate our calculation, let us present here some useful identities. Verification of these identities is straightforward and is left as an exercise for the reader. In what follows, we will omit the subscript in and ,

 δ(θ1−θ2)←D2(θ2,p) = −D1(θ1,−p)δ(θ1−θ2), ¯D1D1(θ1,p)δ(θ12)|θ1=θ2 = −D2¯D2(θ2,−p)δ(θ12)|θ1=θ2=1. (34)

## 4 One and two-loop results

Now we are ready to undertake the loop calculations using the superfield technique. We start from the Lagrangian with the bare coupling in UV, and evolve it down, where we have

 L=12∫d2θR12ZA(iA†∂RA−iA∂RA†)+ZBB†B+ZBB†B−Zγ(γ0BBA†+H.c.). (35)

First, we would like to calculate the one-loop correction to the factors. The diagrams to be considered are collected in Fig 2.

For diagram (a), we get

 ∫dθ1dθ2d2q(2π)2A†1A2−iqL−ipL−qL¯D1δ(θ12)←D2¯D1δ(θ12)←D2 (36) = ∫dθ1d2q(2π)2{D1,¯D1}A†1A11pL(pL−qL).

In the above calculation we used integration by parts to move all ’s on one delta-function, and then do the integration over . One can show that the integration over the momentum is finite, and, hence, this graph does not contribute to .

As for diagram (b), we obtain

 (iγ2)2∫dθ1dθ2d2q(2π)2B1B†2−ipL−qL−iq2¯D1δ(θ12)←D2¯D2δ(θ12)←D1 (37) = −(iγ2)2∫dθ1dθ2d2q(2π)2B†2B11q2(qL−pL)(¯D1D1¯D1D1δ(θ12))δ(θ12) = (iγ2)2∫dθ1d2q(2π)2B†1B12(qL−pL)(qL−pL)q2.

Finally, it is not difficult to see that

 ZB=1+iγ2I, (38)

where the integral is defined as

 I≡∫d2q(2π)21q2, (39)

which gives a single pole in the UV.

Due to the fermion flavor symmetry , we do not need a separate calculation here. Also, at one-loop level there is no diagram contributing to , hence is totally determined by the factors. In this way, we recover the result of our previous paper [1],

 βone-loop(γ)=γ32π. (40)

Now we are ready to move on to the two-loop calculation. We would like to prove a version of nonrenormalization theorem, stating that the interaction term is not renormalized. First, we will verify it at the two-loop level, by considering the diagram depicted in Fig 3.

To this end it is sufficient to manipulate a little bit with the -algebras,

 (¯D1δ(θ12)←D2)(¯D3δ(θ23)←D2)(¯D3δ(θ34)←D4)(¯D5δ(θ45)←D4)(¯D4δ(θ14)←D1)(¯D2δ(θ25)←D5) (41) = ¯D1D1δ(θ12)¯D3D3δ(θ23)¯D3D3δ(θ34)¯D5D5δ(θ45)¯D4D4δ(θ14)¯D2D2δ(θ25) = −¯D1D1δ(θ12)¯D3¯D3D3δ(θ23)D3δ(θ34)¯D5D5δ(θ45)¯D4D4δ(θ14)¯D2D2δ(θ25) = 0,

Here we need to emphasize that the canceling is independent of the ways of regularization one takes, as we have not come to the stage of doing actual momentum integration. One can also see this explicitly from component field calculation. Q.E.D.

With some extra work, one can show that due to the very same reason, the two-loop correction to , as shown in Fig 4, vanishes.

There are, however, corrections to and (see Fig. 4). After a straight-forward calculation we get (the subscript 0 labels the bare coupling)

 ZB=ZB=1+iγ20I+12γ40I2. (42)

The two-loop term is an iteration of the one-loop term and has no impact on the at the two-loop level. Indeed,

 γ2=γ20/Z2B, (43)

and

 1γ2=1γ20+2iI+(possibly)O(γ4), (44)

with no terms . The right-hand side leads us back to as in Eq. (9), with no two-loop contribution.

## 5 Nonrenormalization theorem in full

Since we have both and not corrected up to two-loop order, one can expect that they do not receive higher loop corrections at all. We will show that this is guaranteed by a nonrenormalization theorem, based on supersymmetry in conjunction with the target space symmetry of this model. Moreover, the nonrenormalization theorem is about a term rather than an term!

Generally speaking, each term in the Lagrangian can be treated as an term, by replacing the integration over s by s acting on the integrand. Then, following the argument of the term nonrenormalization, one could ask: is it possible to find some background that preserves a half of supersymmetry on which the given term does not vanish? Can one deduce, on these grounds, that a nonrenormalization appears? The answer is negative.

Let us first understand why nonrenormalization theorems lose their validity for terms. Assume we want to choose a background, preserved by the supertransformation . Then, for a chiral superfield and its antichiral counterpart, we have

 ¯DRϕ=0,DRϕ†=0, (45)

and

 δ¯ϵϕ=0,δ¯ϵϕ†=0, (46)

where we define the supertransformation to be

 δ¯ϵ=¯ϵ∂∂θ†R+i¯ϵθR∂L. (47)

This implies strong constraints on the background field that we could choose. Indeed, has to satisfy

 ∂∂θ†Rϕ=0,θR∂Lϕ=0. (48)

Equation (48) implies, in turn a general solution of the following form:

 ϕ=f(t−z,θR)+g(t,z)θR, (49)

where and could be arbitrary functions. Similarly, for , we have

 ϕ†=h(t−iθ†RθR,z−iθ†RθR). (50)

Now, both and satisfy the chiral condition. Therefore, if we have a combination of and and take the integral over , it vanishes!

Needless to say, if one first integrates over, say, , and is left with the “fake” term, the proof of the nonrenormalization theorem also fails. The above term,

 ¯DRA†BB,

will be a total derivative, of necessity, and, hence, the integral over will vanish (assuming the background to decay at infinity).

This is merely a recap of what we knew before, in a little bit fancy language. We can generalize the logic of the proof, however. In our problem the target space symmetry reveals itself in the invariance of the action under the shift of ,

 A→A+a(t−z),A†→A†+a†(t−z), (51)

where and are generic functions of . (Note that they do not need to be Hermitean-conjugate to each other.) The reason is that the function can be understood as being both chiral and antichiral, since both and vanish when acting on it. This makes it possible to combine the target space symmetry with the requirement of the supertransformation symmetry. Namely, we will require the background field to be invariant under the shift by supplemented by the target space symmetry.

We can say that what enters in the kinetic term for and the interaction term, is in fact not the field itself, but, rather, its equivalence classes under the aforementioned transformation (51). Let us denote by the equivalence class to which belongs. The key idea is that by claiming so, our constraints for the background field get weaker, and we have a “thickening” of our domain of possible solutions to Eqs. (45) Eq. (46). At the end of the day, a nontrivial background is possible.

In fact, since both the supertransformation symmetry and that of Eq. (51) are valid symmetries of , if we pick one element in , say, , and apply , it may end up being another element in , without changing the whole equivalence class it belongs to. Thus we can relaxe our condition (46),

 δ¯ϵA=¯ϵa(t−z),δ¯ϵA†=¯ϵa†(t−z), (52)

where and are functions of , and is a small supertransformation parameter.

Now, this will lead us to a more general solution for the background field ,

 A†=θ†Ra†(t−z)+h(t−iθ†RθR,z−iθ†RθR). (53)

Furthermore, one can also show that the allowed background for the field is not “thickened.” It is straightforward to verify that by taking, for example, the following background fields:

 A=g(t,z)θR, A†=θ†Ra†(t−z), B=1, B†=0, B=f(t−z,θR), B†=0, (54)

we indeed have a desirable nontrivial background for both the kinetic term and the interaction term.

We can then apply the argumentation which leads us to the nonrenormalization theorem. To calculate effective action we decompose the superfields into the background and the quantum parts. Due to the linearity of the target space symmetry, the symmetry transformation can be assigned only to the background part of the field (and, of course that of , too), leaving the quantum part intact. The chosen background fields are invariant under the transformation of supplemented by the target space shift. The symmetry is exact, it translates to the quantum level in form of a supersymmetry shift of . Therefore, the integrand in the loop calculations is homogeneous in , and, hence, is independent of .

On the other hand, we learn from the Feynman rules listed in Sec. 3 that all loop calculations 333Strictly speaking, this does not include the one-loop correction, since the ultraviolet contribution does not involve integrations over . However, the one-loop calculation is easy to carry out explicitly in the way we did it. should involve the integration over . Thus, finally we have to obtain zero in two and higher-loop perturbative calculation.

At the moment we are aware of no way to predict quantum corrections for and without explicit calculations, since the constraints (45) and (46) hold “as is”, leaving us with no nontrivial background for their kinetic terms.

Indeed, from the loop calculation in Sec. 4 we can see that they get renormalized at two-loop order. Strictly speaking, the two-loop effects contain only double poles, and are merely manifestations of the one-loop terms. However, the background field method can not distinguish between a geometric progression and genuine two-loop effects.

## 6 Generalization to nonperturbative regime a lá Seiberg

In this section we will extend the nonrenormalization theorem of Sec. 5 beyond perturbatiion theory. We show that and do not receive nonperturbative corrections either.

Following arguments similar to that in [17], we promote to a chiral superfield. It is important to note that the chirality of is protected by the target space symmetry.

Indeed, let us inspect the term . It must be invariant under the shift . Then must vanish. This is impossible unless is a chiral superfield.

Now, we can assign appropriate -charges to all fields. They are collected in Table 1.

Using these charge assignments one can show that independent -neutral combinations of , , and are

 γBBA†,|B|2,|B|2,|A|2,and|γ|2. (55)

Therefore, we could the renormalized interaction term in the effective Lagrangian in the most general case takes the form

 ∫d2θRf(γBBA†,|A|2,|B|2,|B|2,|γ|2)+H.c.. (56)

Let us suppress the dependence of on , and for a short while. For a generic function of and , it does no harm to express its dependence on these variables as

 f(γBBA†,γBBA). (57)

Now let us check the symmetry: under the shift symmetry for a constant , we have

 δc∫d2θRf(