Two particle elastic scattering at 1-loop

Transcription

1 Two particle elastic scattering at 1-loop based on S-20 Let s use our rules to calculate two-particle elastic scattering amplitude in, theory in 6 dimensions including all one-loop corrections: For the amplitude at tree level we have found: positive negative

2 The exact scattering amplitude is given by the diagrams: exact 4-point vertex exact 3-point vertex exact propagator external lines contribute only the residue of the pole at ; which is one

3 At one loop level we have: two external momenta are on shell

4 We can get some intuition by looking at high-energy limit (neglecting mass terms where possible): Propagator: neglecting m s is positive correct branch obtained by

5 Three-point vertex: we get: Four-point vertex:

6 Note: when evaluating integrals over Feynman variables we use:

7 some useful integrals:

8 Putting it all together: we get: typical result: tree level amplitude is corrected by powers of logs of kinematic variables!

9 Infrared divergences based on S-26 We calculated two-particle elastic scattering amplitude in, theory in 6 dimensions including all one-loop corrections: which in the high-energy limit can be written as: includes everything without a large logarithm that blows up when.

10 blows up in theories with massless particles. Where did we make a mistake? Up to this point we assumed that we can isolate individual particles. However there is no energy gap between one-particle states and multiparticle continuum in theories with massless particles: In this case we cannot distinguish between a one particle state and a state with an extra very low energy (soft) particle that we cannot detect. And so the possibility of having extra soft particles in the process should be included in the calculation! In addition there could be colinearly moving particles that would look like just one particle in the detector.

11 Let s account for the possibility that in a given process, described by the amplitude T, one of the final state particles splits into two: the amplitude for this process can be written as: in the massless limit the propagator can diverge! the amplitude for this process can be written as:

12 If the detector cannot tell whether or not the one particle split into two, then we should add probabilities for the two events (which are in principle distinguishable); we can define an effectively observable squared amplitude as: multiplying the second term on the RHS by identical particles other similar processes we can remove and write it as:

13 Note, that splitting of one particle into two gives a contribution comparable to one-loop corrections! In the massless limit, : thus, for small : diverges for soft collinear diverges for these integrals would be finite for non-zero m!

14 In 6 dimensions we have to worry only about collinear divergences; let s assume the detector cannot distinguish particles for smaller than some small angle (characteristic for the detector) : it is useful to define: everywhere except the propagator we can take :

15 expanding the propagator to the leading order in m and theta: changing integration variables, and to and we get:

16 integrating over x there is an identical correction from splitting the second particle in the final state; and also the corrections from the two incoming particles are identical; since we have 4 particles the total correction due to the failure of our detector to separate two particles with nearly parallel momenta is:

17 combining our result with 1-loop corrections we get: this log is still present, it blows up in the massless limit, we are still doing something wrong; we will discuss it next time. if we have a very good detector for which this log is not small we need to calculate higher order corrections.

18 Other renormalization schemes based on S-27 The origin of divergencies in the massless limit is our choice of the renormalization scheme! For the propagator we have found (at one-loop): and we imposed: but, in the massless limit satisfied for any A and B! ill defined! to ensure the exact propagator has a pole at. to ensure the residue of the pole is one

19 The problem is that there is no energy gap between one-particle states and multi-particle continuum in theories with massless particles: Lehmann-Källén form of the exact propagator: In the massless limit: the pole at merges with the branch point at! There is no isolated pole at with residue one, and so the conditions are not meaningful!

20 The renormalization scheme we have discussed so far: Let s try a different one: is called the on-shell or OS scheme. A and B have no finite part is called the modified minimal-subtraction or scheme. The correction to the propagator: has a well-defined limit, but it explicitly depends on the fake parameter!

21 Choosing the scheme leads to important changes in our calculations: the exact propagator will no longer have a pole at ; by definition the physical mass of the particle is determined by the location of the pole: ; the lagrangian parameter is no longer the same as the physical mass. The residue of the pole,, is no longer 1! The LSZ formula must be corrected by multiplying the right hand side by a factor of for each external particle because the field is properly normalized to create 1-particle state. In the LSZ formula each Klein-Gordon wave operator should be when the K-G operator acts on an external propagator it cancels the pole and leaves behind the residue These changes result in the following Feynman rules: assign factors: for each external line for each internal line with momentum k for each vertex

22 Let s calculate and : We have: by definition and we find: the difference is

23 Putting it together we find: and evaluating the integral we get: should not depend on the fake parameter RHS depends explicitly on Solution: and must depend on!

24 We can easily find this dependance: does not depend on the fake parameter we get: assumes we will verify this soon = is called anomalous dimension of the mass parameter

25 Let s now calculate the residue : The residue of a function that has a simple pole is given by: The residue of a function f(z)=g(z)/h(z) at a simple pole is given by:

26 The vertex function in the scheme: depends explicitly on

27 Now we can repeat the calculation of in, theory in 6 dimensions including all one-loop corrections (in the low-mass limit): Before we have found: In the scheme we get: includes everything without a large logarithm that blows up when.

28 To get an observable amplitude-squared we have to correct for an imperfect detector; we found: In the scheme we get: has a well-defined limit, but it explicitly depends on the fake parameter! should not depend on the fake parameter Solution: must depend on!

29 should not depend on the fake parameter we find: = is called the beta function

30 Note: we can choose any should choose! we might like, but to avoid large logs we To compare results at different energies we can solve the differential equation: decreases as increases theories with this property are said to be asymptotically free (the tree level approximation becomes better and better at higher and higher energies) (at lower and lower energies the theory becomes more and more strongly coupled)

31 The renormalization group based on S-28 We are going to derive, in a systematic way, how lagrangian parameters and other object that are not directly measurable vary with : equations of the renormalization group Let s consider the theory in dimensions: The same theory can be also written as: renormalized fields and parameters (in scheme) The dictionary: bare fields and parameters must be independent of!

32 In the scheme we choose the Zs to cancel the divergent parts of loop integrals and so in general: functions of at one loop we found: for all and are at least!

33 Let s derive consequences of bare gauge coupling being Let s define: independent: in particular:

34 does not depend on the fake parameter regrouping the terms we get: we can find the solution by matching terms at with given power of : matching terms should be finite as matching terms: matching terms with higher powers of leads to conditions among, e.g. matching the we get,... can be checked order by order in perturbation theory!

35 Putting it together we find: the same result as we obtained before from requiring that a particular cross section is independent of!

36 Let s repeat the same thing for the bare mass parameter: Let s define: in particular, we find: taking the log:

37 does not depend on the fake parameter rearranging, we find should be finite as terms with powers of these terms must all be zero!

38 For the anomalous dimension of the mass we get: the same result as we obtained before from requiring that a particular cross section is independent of!

39 We can repeat the same procedure for the propagator in the scheme: first, we obtain the bare propagator: should be independent of

40 should be finite as terms with powers of these terms must all be zero!

41 Defining the anomalous dimension of the field: We obtain: Callan-Symanzik equation for the propagator