cc.redberry.physics.oneloopdiv

Class OneLoopCounterterms

java.lang.Object

cc.redberry.physics.oneloopdiv.OneLoopCounterterms

public final class OneLoopCounterterms
extends Object

This class is a container of the calculated one-loop counterterms. It has no constructors and can be created using the static method calculateOneLoopCounterterms(OneLoopInput), which performs the whole calculation of the one-loop counterterms.

Here is the example of the calculation of one-loop counterterms of vector field:

      //setting symmetries to tensor P
      Tensors.addSymmetry("P_lm", IndexType.LatinLower, false, 1, 0);

      //input expressions
      Expression iK = Tensors.parseExpression("iK_a^b=d_a^b+ga*n_a*n^b");
      Expression K = Tensors.parseExpression("K^{lm}_a^{b}=g^{lm}*d_{a}^{b}-ga/(2*(1+ga))*(g^{lb}*d_a^m+g^{mb}*d_a^l)");
      Expression S = Tensors.parseExpression("S^p^l_m=0");
      Expression W = Tensors.parseExpression("W^{a}_{b}=P^{a}_{b}+ga/(2*(1+ga))*R^a_b");
      //F is equal to Riemann for vector field
      Expression F = Tensors.parseExpression("F_lmab=R_lmab");

      //tensors M and N are null, since operator order is 2
      OneLoopInput input = new OneLoopInput(2, iK, K, S, W, null, null, F);

      //performing the main calculation
      OneLoopCounterterms action = OneLoopCounterterms.calculateOneLoopCounterterms(input);
      Tensor counterterms = action.counterterms();
      //here some transformations can be performed to simplify counterterms
      ...
      System.out.println(counterterms);
 

The above code will produce the counterterms, which after some simplifications can be written in form

     (1/24*ga**2+1/4*ga+1/2)*P_\mu\nu*P^\mu\nu + 1/48*ga**2*P**2 + (1/12*ga**2+1/3*ga)*R_\mu\nu*P^\mu\nu +
     +(1/24*ga**2+1/12*ga+1/6)*R*P + (1/24*ga**2+1/12*ga-4/15)*R_\mu\nu*R^\mu\nu + (1/48*ga**2+1/12*ga+7/60)*R**2
 

The divergent part of the one-loop effective action can be obtained by multiplying the resulting counterterms on factor 1/(16*\pi**2*(d-4)) and integrating over the space volume.

Since:

1.0

Author:

Dmitry Bolotin, Stanislav Poslavsky

Method Summary

Methods

Modifier and Type	Method and Description
static OneLoopCounterterms	calculateOneLoopCounterterms(OneLoopInput input) This method performs the calculation of the one-loop counterterms.
cc.redberry.core.tensor.Expression	DELTA_1() Returns \Delta^{\mu ...} tensor, where dots mean 'matrix' indices.
cc.redberry.core.tensor.Expression	DELTA_2() Returns \Delta^{\mu\nu ...} tensor, where dots mean 'matrix' indices.
cc.redberry.core.tensor.Expression	DELTA_3() Returns \Delta^{\mu\nu\alpha ...} tensor, where dots mean 'matrix' indices.
cc.redberry.core.tensor.Expression	DELTA_4() Returns \Delta^{\mu\nu\alpha\beta ...} tensor, where dots mean 'matrix' indices.
cc.redberry.core.tensor.Expression	FF() Returns the FF counterterms part
cc.redberry.core.tensor.Expression	Flat() Returns the Flat counterterms part
cc.redberry.core.tensor.Expression	FR() Returns the FR counterterms part
cc.redberry.core.tensor.Expression	getCounterterms() Return resulting counterterms, i.e.
cc.redberry.core.tensor.Expression	RR() Returns the RR counterterms part
cc.redberry.core.tensor.Expression	SR() Returns the SR counterterms part
cc.redberry.core.tensor.Expression	SSR() Returns the SSR counterterms part
cc.redberry.core.tensor.Expression	WR() Returns the WR counterterms part

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Method Detail

Flat

public cc.redberry.core.tensor.Expression Flat()

Returns the Flat counterterms part

Returns:

Flat counterterms part

WR

public cc.redberry.core.tensor.Expression WR()

Returns the WR counterterms part

Returns:

WR counterterms part

SR

public cc.redberry.core.tensor.Expression SR()

Returns the SR counterterms part

Returns:

SR counterterms part

SSR

public cc.redberry.core.tensor.Expression SSR()

Returns the SSR counterterms part

Returns:

SSR counterterms part

FF

public cc.redberry.core.tensor.Expression FF()

Returns the FF counterterms part

Returns:

FF counterterms part

FR

public cc.redberry.core.tensor.Expression FR()

Returns the FR counterterms part

Returns:

FR counterterms part

RR

public cc.redberry.core.tensor.Expression RR()

Returns the RR counterterms part

Returns:

RR counterterms part

getCounterterms

public cc.redberry.core.tensor.Expression getCounterterms()

Return resulting counterterms, i.e. the Flat + WR + SR + SSR + FF + FR + RR. In order to obtain the divergent part of the one loop effective action, one should integrate counterterms over space volume and multiply on 1/(16*\pi^2*(d-4)) factor.

Returns:

resulting counterterms

DELTA_1

public cc.redberry.core.tensor.Expression DELTA_1()

Returns \Delta^{\mu ...} tensor, where dots mean 'matrix' indices.

Returns:

\Delta^{\mu ...} tensor, where dots mean 'matrix' indices.

DELTA_2

public cc.redberry.core.tensor.Expression DELTA_2()

Returns \Delta^{\mu\nu ...} tensor, where dots mean 'matrix' indices.

Returns:

\Delta^{\mu\nu ...} tensor, where dots mean 'matrix' indices.

DELTA_3

public cc.redberry.core.tensor.Expression DELTA_3()

Returns \Delta^{\mu\nu\alpha ...} tensor, where dots mean 'matrix' indices.

Returns:

\Delta^{\mu\nu\alpha ...} tensor, where dots mean 'matrix' indices.

DELTA_4

public cc.redberry.core.tensor.Expression DELTA_4()

Returns \Delta^{\mu\nu\alpha\beta ...} tensor, where dots mean 'matrix' indices.

Returns:

\Delta^{\mu\nu\alpha\beta ...} tensor, where dots mean 'matrix' indices.

calculateOneLoopCounterterms

public static OneLoopCounterterms calculateOneLoopCounterterms(OneLoopInput input)

This method performs the calculation of the one-loop counterterms. It also prints the interim results to standard output, during the calculation.

Parameters:

input - input parameters container.

Returns:

resulting counterterms container.